Quantum Cosmology - The Supersymmetric Perspective - Vol. 2: Advanced Topic (Lecture Notes in Physics)

Lecture Notes in Physics Founding Editors: W. Beiglböck, J. Ehlers, K. Hepp, H. Weidenmüller Editorial Board R. Beig,...

Author: Paulo Vargas Moniz

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Lecture Notes in Physics Founding Editors: W. Beiglböck, J. Ehlers, K. Hepp, H. Weidenmüller Editorial Board R. Beig, Vienna, Austria W. Beiglböck, Heidelberg, Germany W. Domcke, Garching, Germany B.-G. Englert, Singapore U. Frisch, Nice, France F. Guinea, Madrid, Spain P. Hänggi, Augsburg, Germany W. Hillebrandt, Garching, Germany R. L. Jaffe, Cambridge, MA, USA W. Janke, Leipzig, Germany H. v. Löhneysen, Karlsruhe, Germany M. Mangano, Geneva, Switzerland J.-M. Raimond, Paris, France M. Salmhofer, Heidelberg, Germany D. Sornette, Zurich, Switzerland S. Theisen, Potsdam, Germany D. Vollhardt, Augsburg, Germany W. Weise, Garching, Germany J. Zittartz, Köln, Germany

The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new developments in physics research and teaching – quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research and to serve three purposes: • to be a compact and modern up-to-date source of reference on a well-defined topic • to serve as an accessible introduction to the field to postgraduate students and nonspecialist researchers from related areas • to be a source of advanced teaching material for specialized seminars, courses and schools Both monographs and multi-author volumes will be considered for publication. Edited volumes should, however, consist of a very limited number of contributions only. Proceedings will not be considered for LNP. Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive being available at springerlink.com. The series content is indexed, abstracted and referenced by many abstracting and information services, bibliographic networks, subscription agencies, library networks, and consortia. Proposals should be sent to a member of the Editorial Board, or directly to the managing editor at Springer: Christian Caron Springer Heidelberg Physics Editorial Department I Tiergartenstrasse 17 69121 Heidelberg / Germany [email protected]

Paulo Vargas Moniz

Quantum Cosmology The Supersymmetric Perspective - Vol. 2 Advanced Topics

ABC

Paulo Vargas Moniz Universidade da Beira Interior Depto. Fisica Rua Marquês d’Avila e Bolama 6200-307 Covilhã Portugal [email protected] http://www.dfis.ubi.pt/∼pmoniz (Also at CENTRA-IST, Lisboa, Portugal)

Moniz, P.V.: Quantum Cosmology - The Supersymmetric Perspective - Vol. 2: Advanced Topics, Lect. Notes Phys. 804 (Springer, Berlin Heidelberg 2010), DOI 10.1007/978-3-642-11570-7

Lecture Notes in Physics ISSN 0075-8450 e-ISSN 1616-6361 ISBN 978-3-642-11569-1 e-ISBN 978-3-642-11570-7 DOI 10.1007/978-3-642-11570-7 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010922992 c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Integra Software Services Pvt. Ltd., Pondicherry Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To all my cats!

Preface

We read in order to know we are not alone, I once heard, and perhaps it could also be suggested that we write in order not to be alone, to endorse, to promote continuity. The idea for this book took about 10 years to materialize, and it is the author’s hope that its content will constitute the beginning of further explorations beyond current horizons. More specifically, this book appeals to the reader to engage upon and persevere with a journey, moving through the less well explored territories in the evolution of the very early universe, and pushing towards new landscapes. Perhaps, during or after consulting this book, this attitude and this willingness will be embraced by someone, somewhere, and this person will go on to enrich our quantum cosmological description of the early universe, by means of a clearer supersymmetric perspective. It is to these creative and inquisitive ‘young minds’ that the book is addressed. The reader will not therefore find in this book all the answers to all the problems regarding a supersymmetric and quantum description of the early universe, and this remark is substantiated in the book by a list of unresolved and challenging problems, itself incomplete. Consequently, the idea is to provide a description of the many features present in a supersymmetric perspective of quantum cosmology. The book is split into two volumes: • In Vol. I, entitled Fundamentals, the reader will find an accessible primer. After a contextualized introduction and guidance through possible routes of exploration, the essential content starts with a chapter presenting quantum cosmology in general terms. It is then followed by a chapter summarizing the ideas and methods of supersymmetry and supergravity. It is only afterwards that a thorough supersymmetric analysis of some relevant (quantum) cosmological models is undertaken. More precisely, the reader and fellow explorer is introduced to the main ideas, techniques, achievements, and problems, which characterize the research subject of supersymmetric quantum cosmology (SQC). Different approaches that have been employed in SQC will be discussed, bringing them together for the first time in a book publication. • In Vol. II, entitled Advanced Topics, the scope for analyzing quantum cosmological models within a supersymmetric framework is broadened. The aim is vii

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to provide those who have worked through Vol. I with an introduction to more topics, as well as some complementary developments. Volume II adds furthers essentials and additional (optional) frameworks to employ within SQC. The above objectives for this book will be helped along with a detailed set of exercises, accompanied by the corresponding solutions. A display of summary boxes (outlining relevant features, concepts, and results in the form of reviewing questions) will be added at the end of each chapter. It is hoped that this book will stimulate the interest of (a) final year undergraduates, (b) graduate students, (c) lecturers designing a course that will include aspects of superstrings and supergravity from a quantum mechanical and cosmological point of view, and (d) researchers who would like to either initiate or apply SQC methods and ideas in their work. The reader is assumed to have a good working knowledge of mathematical analysis, quantum mechanics, modern cosmology, field theory, and general relativity theory. A prior knowledge of quantum cosmology is not a condition to start using this graduate textbook. In particular, concepts related to supersymmetry/supergravity or quantum cosmology will generally be presented within the book as indicated. Finally, before embarking upon the many technical details of this fascinating exploration, I would like to share with the reader the following words by Alberto Caeiro (Fernando Pessoa): Aceita o Universo Como to deram os deuses. Se os deuses te quisessem dar outro ter-to-iam dado. Se há outras matérias e outros mundos – Haja. London, Köln, and Covilhã

Paulo Vargas Moniz

Acknowledgments

This book would not have been possible if it were not for the warm support and considerable learning made available by so many people. The author is most grateful for the unfailing friendship of Reza Tavakol, Bernard Carr, Malcolm MacCallum, and Jim Lidsey, and also to D. Mulryne, J. Ward, C. Hidalgo, J. Taylor, W. Clavering, D. Seery, T. Harada, as well as the astronomy and mathematics staff of various institutes for their efficient and sincere hospitality. The superfriendly and stimulating work atmosphere at Queen Mary College of the University of London1 provided the necessary driving force just when this book project required crucial stamina. Similar thanks and equivalently unbounded gratitude are extended to Claus Kiefer and Tobias Lück, especially for the opportunity to work together on a special and long-sought project. In particular, Claus Kiefer’s guidance and assistance were paramount. In addition, the kind hospitality provided by the ITP at the Universität zu Köln was most appreciated. A particular word of thanks is due to Alfredo Henriques for his significant support in more strenuous and less confortable times. Particular words of gratitude are extended to J. Marto, P. Parada, E. Segre, and J. Velhinho at UBI, with very special thanks to Carolina de Almeida and Manuela Raposo for efficient ‘administrative filtering’. Most of my learning on superstring, supergravity, quantum cosmology, and then supersymmetric quantum cosmology was obtained at DAMTP, University of Cambridge. I spent five wonderful years there (1993–1998), learning a lot and getting better at it all the time. I would particularly like to thank G. Gibbons, P. Townsend, and M. Perry for useful discussions and hints. Special words go to S.W. Hawking and P.D. D’Eath for their time, patience, and sharing. In particular, I must thank P.D. D’Eath for making available his vast learning with regard to supersymmetric quantum cosmology. A.D.Y. Cheng was both a friend and a stimulating collaborator,

1 The author would like to acknowledge the support provided by the grant FCT (FEDER) SFRHBSAB-396/2003 during his sabbatical (September 2004–2005) at Queen Mary College of the University of London (QMUL). He is also grateful to a CRUP–DAAD bilateral cooperation grant that provided support during his stay in Köln.

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to whom the author is most grateful for the many hours of discussions and shared projects. An important thanks also goes to Gonville and Caius College, Cambridge, for the nice hospitality, brilliant atmosphere, ‘soul warming’ port, and a ravishingly delicious bread and butter pudding! Other methods and experiences in supersymmetric quantum cosmology were provided from conversations with M. Ryan, A. Macias, J. Socorro, H. Rosu, and U. Schreiber. The kind hospitality and availability of H. Luckock, O. Obregon, and R. Graham to discuss their work is acknowledged. Regarding his research work, the author is also most grateful to M. Cavaglia, M. Bouhmadi-López, A. Zhuk, U. Guenther, and D. Gal’tsov for the opportunity to learn from them. Visits to A. Vilenkin, Bei-Lok Hu, J. Louko, W. Kümmer, and P. Peter were likewise important. Particular thanks go to G. Gionti (S.J.) for his time and patience in discussing science and its ‘boundaries’. The comments, feedback, and words of encouragement from A. Barroso, M. Bojowald, E. Donets, D. Freedman, L. Garay, A.B. Henriques, Bei-Lok Hu, C. Kiefer, H. Kodama, J. Louko, A. Morisse, D. Mulryne, D. Page, Y. Tavakoli, J. Velhinho, A. Vilenkin, J. Ward, D. Wiltshire, and S. Winitzki are warmly acknowledged. Concerning the editorial component, the author is most grateful to Christian Caron of Springer-Verlag for his patience, positive feedback, guidance and sincere commitment to assist. A most special thanks indeed. In addition, the author’s gratitude is also extended to Stephen Lyle, for his most careful copy-editing of the manuscript. As a consequence, the text has been considerably improved. Finally, a few words are owed to the following persons, although no words could ever be sufficient. First to my dear mother and father, I hope this book will honour all the time, patience, dedication, effort, and teaching you unconditionally gave to me, providing me with the guidance to reach this far. And, of course, the author is deeply and immensely grateful to his dear wife, Teresa, for her love, patience, and support at all times and beyond what any words could express, but also to his dear daughter, Teresa, for keeping him awake regularly at the crack of dawn. So, now, on with the book. London, Köln, and Covilhã, November 2004–December 2008

Paulo Vargas Moniz

Contents

Part I Overview 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 7

Part II Further Essentials 2 ‘Observational’ Quantum Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Semi-Classical Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Hamilton–Jacobi Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Quantum Gravity Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 ‘Predictions’ from Quantum Cosmology . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Towards Structure Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Towards Spacetime Decoherence . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Towards Spacetime Correlations . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 15 15 17 19 19 21 25 28 31 32

3 Additional SUSY and SUGRA Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Fayet–Iliopoulos (FI) Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 SUSY Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Mechanisms for SUSY Breaking . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Spontaneous SUSY Breaking (in SUGRA) . . . . . . . . . . . . . . . . 3.3 N =2 Supersymmetric Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Main Features and Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 SQM, Topology, and Vacuum States . . . . . . . . . . . . . . . . . . . . . 3.4 Nicolai Maps and SQM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 36 37 39 43 43 52 55 56 57 xi

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4 Semiclassical N=1 Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Physical States in N =1 SUGRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Bosonic (n=0) States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 States with Finite Fermion Number . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Solutions with Infinite Fermion Number . . . . . . . . . . . . . . . . . . 4.2 Semiclassical Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 The Hamilton–Jacobi Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 The DeWitt Supermetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 The Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 The DeWitt Supermetric and the Schrödinger Equation . . . . . . 4.2.5 Quantum N =1 SUGRA Corrections . . . . . . . . . . . . . . . . . . . . . . 4.2.6 The DeWitt Supermetric and Quantum Gravity Corrections for the Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.7 Towards SQC with ‘Observational’ Insights? . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 62 63 63 66 68 71 72 74 75 78 81 81 82

Part III Complementary Frameworks in SQC 5 Further Explorations in SQC N=1 SUGRA . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.1 The Issue of Supersymmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.2 Supermatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.2.1 Generic Gauge Supermatter in FRW . . . . . . . . . . . . . . . . . . . . . 91 5.2.2 Scalar Supermultiplets in Bianchi IX . . . . . . . . . . . . . . . . . . . . . 96 5.2.3 Vector Fields in Bianchi I (N = 2 SUGRA) . . . . . . . . . . . . . . . 101 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6 Connections and Loops Within SQC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.1 A Brief Summary of the Connection Representation . . . . . . . . . . . . . . . 111 6.1.1 Two Spinor Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.1.2 Hamiltonian Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.1.3 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.2 N =1 SUGRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.2.1 Hamiltonian Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.2.2 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.2.3 FRW with Cosmological Constant . . . . . . . . . . . . . . . . . . . . . . . 118 6.2.4 Bianchi Class A Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7 SQC Matrix Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.1 Motivation and Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . 129

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7.2 Supersymmetric Bianchi Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.2.1 Bianchi I Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.2.2 Bianchi IX Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.2.3 Taub Minisuperspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.2.4 FRW Minisuperspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.3 Bianchi Models and Lorentz Invariance . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.3.1 Implications of the Lorentz Constraints . . . . . . . . . . . . . . . . . . . 148 7.3.2 Physical States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.3.3 Lorentz Constraint Components and Bianchi IX Diagonal Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 8 N =2 (Local) Conformal Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 163 8.1 Motivation and Superfield Description . . . . . . . . . . . . . . . . . . . . . . . . . . 163 8.1.1 Empty Matter Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 8.1.2 Complex Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 8.2 Quantum FRW Minisuperspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 8.2.1 Supersymmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 8.2.2 Towards an Inflationary Scenario . . . . . . . . . . . . . . . . . . . . . . . . 185 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Part IV More Points of Departure 9 More Obstacles and Results: From QC to SQC . . . . . . . . . . . . . . . . . . . . . 191 9.1 Quantum Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 9.2 Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 9.3 Supersymmetric Quantum Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . 193 9.3.1 FRW Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 9.3.2 Spatial Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 10 More Routes Beyond the Borders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 10.1 Beyond Minisuperspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 10.2 Other New Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 A List of Symbols, Notation, and Useful Expressions . . . . . . . . . . . . . . . . . . 207 A.1 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 A.2 Conventions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 A.3 About Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 A.3.1 Spinor Representations of the Lorentz Group . . . . . . . . . . . . . . 212

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A.3.2 Dirac and Majorana Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 A.4 Useful Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 A.4.1 Metric and Tetrad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 A.4.2 Connections and Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 A.4.3 Covariant Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 A.4.4 (Gravitational) Canonical Momenta . . . . . . . . . . . . . . . . . . . . . . 227 A.4.5 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 A.4.6 Decomposition with Four-Component Spinors . . . . . . . . . . . . . 229 A.4.7 Equations Used in Chap. 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 B Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

Chapter 1

Introduction

What is this book about? What is quantum cosmology with supersymmetry? How is supersymmetry implemented? Is it through the use of (recent developments in) a superstring theory? Why should the very early universe be explored in that manner? Are there enticing and interesting research problems left to solve? How relevant would it be to address and solve them? The above are just the kind of questions that a potential reader is likely to ask as she or he begins to browse through the first few pages of this book. Perhaps a clear picture will only emerge by the time the final chapters are reached. But still, we may say in simplified terms that investigating quantum cosmology with supersymmetry means using both bosonic and fermionic degrees of freedom, supersymmetrically intertwined, within a suitable quantum mechanical representation of the universe (e.g., where bosonic canonical momenta are represented by a differential operator and fermions by a matrix). Supersymmetry (SUSY) [1–8] will be implemented through different procedures, ranging from taking the bosonic sector of different string theories and then inserting fermionic partners in an appropriately consistent manner, up to using the full theory of N = 1 supergravity (SUGRA) [9–11] in 4D spacetime. Of course, this may seem to fall short of using the whole set of elements of 10-dimensional superstring or 11-dimensional SUGRA: N = 1 11-dimensional SUGRA is considered to be a low energy limit of M-theory, constituting each of the five 10-dimensional weak-coupling limits of superstring theories (all these limits are related by duality transformations) [12–14]. However, when initiating and proceeding to explore uncharted domains, it is often more efficient to get acquainted with the main features of simpler settings before attempting bolder routes,1 hoping the results found in the former will be in some way employed in the latter.

1 N = 1 4D SUGRA can follow from 11D or 10D SUGRA through suitable compactifications and restrictions. Other 4D limits are, e.g., N = 2 SUGRA (which constitutes the sought route to bring Einstein’s general relativity and Maxwell’s electromagnetism within a unified setting), and N = 8 SUGRA [9] (which was long considered the best hope for a unified theory of all interactions and quantum gravity).

Moniz, P.V.: Introduction. Lect. Notes Phys. 804, 3–10 (2010) c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-11570-7_1

4

1 Introduction

Superstring theory (and its extensions to SUGRA or M-theory) [15–20, 12–14], although still under development, is an excellent (perhaps the best!) candidate for a unification theory, constituting a fascinating quantum gravity framework [21]. The limits of investigation have been stretched impressively, in particular, allowing the proposal of novel scenarios that go deep into the very early universe. It is therefore hoped that superstring theory will indeed assist in the process of stepping into these and many other as yet unexplored landscapes (see, e.g., [22, 23]). Starting with superstring theory (and requiring the extra spatial dimensions to be compactified) to achieve a 4D spacetime description, it is found that the resulting quantum gravity scenario could retain at least some supersymmetry. Such a setting would surely be interesting for the cosmology of the very early universe. It could, on the one hand, assist in reducing our ignorance regarding the nature of the creation of the universe, and on the other hand, hopefully provide lateral information that would enrich our mastery of superstring theory, perhaps contributing indirectly to clarifying its central ideas and its complete formulation. To be more precise, exploring the very early universe with the help of features from a quantum gravity theory will hopefully bring more accurate explanations, elucidating observational issues and other perplexing problems in contemporary cosmology. For example, what is the inflaton? What is the nature of the dark energy effect? Why do we have this universe and not another? Moreover, the analysis of the very early universe (ranging from a quantum origin up to structure formation, and involving a crucial inflationary stage) currently also offers a noteworthy opportunity to test some of the features and predictions of such a quantum gravity theory. In fact, cosmology has quite recently reached a significant level of observational accuracy, prompting part of the scientific community to identify this stage as a ‘golden epoch’ [24–27]. Quantum gravity or its superstring version may eventually acquire an observational component in the context of future cosmological tests, quite apart from speculative hopes for the LHC in 2009 and beyond. Consequently, a quantum mechanical approach to cosmological theories retrieved via a quantum gravity theory may constitute a significant step in the study of the early evolution of the universe. This methodology is generally called quantum cosmology (QC). It aims to explain how and why our universe is the way it is. Basically, quantum cosmology employs quantum mechanics to investigate the universe as a whole [28–34]. Research on quantum cosmology has been through several periods. In the 1960s, the seminal work of C. Misner, J. Wheeler, and B. DeWitt built the foundations of canonical quantum gravity [35–43]. In particular, they established the conditions (constraint equations) that the quantum state of the universe ought to satisfy, together with a definition of its configuration space (superspace). In this context, the universe is studied by means of a wave function, rather than classical spacetime solutions. Almost all the models subsequently considered had all but a finite number of degrees of freedom frozen. This is achieved by restricting the fields to be spatially homogeneous, and such models correspond to a (finite-dimensional) ‘minisuperspace’ scenario [44, 30, 45, 31, 46, 33, 34]. It was only in the 1980s that

1 Introduction

5

quantum cosmology attracted renewed and active interest. The main reason was the rigorous debate and introduction of boundary conditions for the wave function of the universe. J. Hartle and S.W. Hawking, on the one hand, [29, 45], and A. Vilenkin [47–49], on the other, put forward the two main schemes for boundary conditions, with suitable variations added by A. Linde. The vast majority of research has been divided among what are known as the no-boundary proposal and the tunneling proposal, although recent work in superstring quantum cosmology and the landscape problem has made use of the so-called infinite wall proposal advanced earlier by B. DeWitt (see, e.g., [40]). The physical setting in which the overwhelmingly vast majority of publications in quantum cosmology can be found is that of Einstein’s general relativity within a Hamiltonian formulation or a Feynman path integral, employing a metric point of view [29, 30, 45, 21]. But, as mentioned above, if we wish to proceed within a more fundamental perspective, elements from superstring/supergravity theory would constitute an attractive and perhaps more fundamental angle from which to explore the very early universe and explain how the universe got its observational properties. For these reasons, the use of such supersymmetric frameworks for viewing quantum cosmology could not be ignored, and have indeed been explored over the past 25 years or so [50–54]. This setting, in which the methods of quantum cosmology are extended to embrace the techniques of SUSY, is called supersymmetric quantum cosmology (SQC) [50, 55]. SQC thus constitutes an interesting and rewarding research topic. On the one hand, it provides the opportunity to perform calculations that may be relevant for phenomenology, and on the other hand, it is closely linked to exciting new areas of fundamental research, such as superstring theory (and theoretical high energy physics in general). In brief, the fundamental purpose of SQC research (in the author’s opinion) is to determine whether a path can be consistently established from the description of a supersymmetric quantum universe (built upon some of the intrinsic elements of superstring and supergravity theories) toward the current view of a classical stage [56–58], possibly with observationally testable predictions. Let us add that the presence of SUSY has constituted an element of the utmost importance in quantum gravity investigations. For example, it plays a crucial role in SUGRA and superstring theory in removing divergences [9] or unstable states (e.g., tachyons) that would otherwise be present in bosonic quantum gravity theories (e.g., simply extending from general relativity or the bosonic string case) [18–20, 14]. It must also be emphasized that supergravity theories represent a kind of square root of Einsteinian gravity. To determine physical states it may be sufficient to employ the Lorentz and supersymmetry invariances [59, 60]. Moreover, it has been pointed out [61] that the presence in purely bosonic quantum cosmological models of string dualities can be related to the existence of quantum states that have invariance under SUSY. In more detail, the following properties and features further enhance a significant motivation for SQC:

6

1 Introduction

• Both pure quantum gravity and SUSY effects are treated as equally determinant, guaranteeing an improved description of the very early universe [50, 55, 57]. This contrasts with bosonic quantum cosmology, where quantum gravity is present but supersymmetry is not. In the SQC framework, we will therefore find a larger set of variables (bosonic and fermionic), as well as additional symmetries which increase the number of constraints, subsequently imposing a wider algebra. • N = 1 SUGRA [62, 9] is employed as a natural square root of gravity in the manner of Dirac [53, 54, 59, 60]. The analysis of a second order equation of the Klein–Gordon type (i.e., the Wheeler–DeWitt equation) could be replaced by the analysis of a supersymmetrically induced set of first order differential equations. This would then have profound consequences regarding what to take as a (SUSY) wave function of the universe and how to retrieve it [56, 63, 64, 58, 65] and any corresponding cosmological behaviour subsequently induced. • There remain many open issues in SQC [55]. A brief list is as follows: – Does SQC improve on the purely bosonic quantum cosmological formulations with matter fields but no SUSY? That is, does the presence of SUSY invariance contribute, and is it paramount to a more realistic (quantum mechanical) description of the very early universe? – How can spontaneous supersymmetry breaking be properly described within a SQC perspective [57]? – How can we analyse the retrieval of semiclassical features and the origin of structure formation in SQC? – Is it possible to consistently identify quantum-to-classical transitions in SQC? – Is there an imprint of an early SUSY quantum epoch on the (currently) observed universe [58]? The key purpose of this book will be to introduce and present the essentials, as well as pertinent details of SQC to the interested fellow explorer, in order to address some of the above challenges. The book is split into two volumes, each divided into four parts. Volume I conveys, in essence, the fundamentals of SQC. Part I constitutes a brief overview of the methods and results discussed throughout the two volumes and is provided identically in both books. It contains this introduction as Chap. 1. Part II of Vol. I presents some basic elements that will be required for the core part of the book. Chapter 2 provides a generic description of the main aspects of and methods used in quantum cosmology, namely, minisuperspace quantization. We introduce and explain the basic principles of SUSY and SUGRA in Chap. 3. The Hamiltonian formulation and canonical quantization of SUGRA are analysed in detail in Chap. 4. In Part III, Chap. 5 focuses on supersymmetric cosmological models extracted from N = 1 SUGRA, exploring a fermionic differential operator representation for SQC models, which has proved itself to be a suitable method. The emphasis is on FRW models, and in particular on obtaining physical states for quantum supersymmetric universes. Bianchi models are also dealt with, and the presence of matter fields is explored. Chapter 6 presents a thorough description of supersymmetric

References

7

minisuperspaces derived from the bosonic sectors of superstring theories. Some of these models are explored in a context where duality properties can be related to these hidden supersymmetries. Finally, in Part IV, within the context of Vol. I, we will present an appraisal of current accomplishments, and point to subsequent suggestions for investigation in SQC. In more detail, Chap. 7 will describe the main results that have been achieved, while Chap. 8 lists several lines of further enquiry. Two appendices conclude the volume, explaining the notation used along with some useful expressions (Appendix A), and describing canonical quantization procedures for theories with constraints (Appendix B). Volume II is entitled Advanced Topics. It begins with a carbon copy of Part I from Vol. I, the intention being to provide a consistent context and broad guidance throughout the book. Part II is entitled Further Essentials. These begin in Chap. 2, which corresponds to Chap. 2 of Vol. I, focusing again on quantum cosmology and possible semiclassical limits. Chapter 3 provides a brief treatment of SUSY breaking in SUSY and SUGRA theories, placing particular emphasis on supersymmetric quantum mechanics (SQM). Then in Chap. 4, counterpart to Chap. 4 of Vol. I, we detail a framework suitable for describing the semiclassical limit of quantum SUGRA. Part III of Vol. II discusses alternative frameworks for SQC. The presence of supermatter and SUSY breaking is presented in Chap. 5 with regard to supersymmetric cosmological models extracted from N = 1 SUGRA, with a fermionic differential operator representation for SQC. Then in Chap. 6 we describe connection and loop variables in SQC, with emphasis on the interesting developments they have stimulated. Chapter 7 deals with the matrix fermionic representation used in SQC for fermionic momenta. Chapter 8 reports on minisuperspace models extending from those discussed in Chap. 6 of Vol. I. We conclude with Part IV. Chapter 9 summarizes reported accomplishments concerning the frameworks introduced in Parts II and Part III of Vol. II, while Chap. 10 suggests more routes for exploration and investigation that follow on from the contents of Vol. II. An appendix concludes Vol. II, with a review of the notation and some further useful expressions. Having said all this, it is time to wish the reader a pleasant and stimulating journey of exploration within the current confines of supersymmetric quantum cosmology (SQC) and of course beyond, if she or he should so desire. Because there are indeed plenty of challenging and open problems still to address, whose resolution may change our knowledge and perspective of how a realistic very early universe might have come into being and subsequently evolved.

References 1. Bailin, D., Love, A.: Supersymmetric Gauge Field Theory and String Theory. Graduate Student Series in Physics, 322pp. IOP, Bristol (1994) 3 2. Binetruy, P.: Supersymmetry. Oxford University Press, Cambridge (2006) 3

8

1 Introduction

3. Freund, P.G.O.: Introduction to Supersymmetry. Cambridge Monographs on Mathematical Physics, 152pp. Cambridge University Press, Cambridge (1986) 3 4. Martin, S.P.: A Supersymmetry Primer. hep-ph/9709356 (1997) 3 5. Muller-Kirsten, H.J.W., Wiedemann, A.: Supersymmetry: An Introduction with Conceptual and Calculational Details, pp. 1–586. World Scientific (1987). Print-86–0955 (Kaiserslautern) (1986) 3 6. Polonsky, N.: Supersymmetry: Structure and phenomena. Extensions of the standard model. Lect. Notes Phys. M 68, 1–169 (2001) 3 7. Sohnius, M.F.: Introducing supersymmetry. Phys. Rep. 128, 39–204 (1985) 3 8. Weinberg, S.: The Quantum Theory of Fields. 3: Supersymmetry, p. 419. Cambridge University Press, Cambridge (2000) 3 9. Van Nieuwenhuizen, P.: Supergravity. Phys. Rep. 68, 189–398 (1981) 3, 5, 6 10. Wess, J., Bagger, J.: Supersymmetry and Supergravity, p. 259. Princeton University Press, Princeton, NJ (1992) 3 11. West, P.C.: Introduction to Supersymmetry and Supergravity, p. 425. World Scientific, Singapore (1990) 3 12. Polchinski, J.: String Theory. 1. An Introduction to the Bosonic String, p. 402. Cambridge University Press, Cambridge (1998) 3, 4 13. Polchinski, J.: String Theory. 2. Superstring Theory and Beyond, p. 531. Cambridge University Press, Cambridge (1998) 3, 4 14. Zwiebach, B.: A First Course in String Theory, p. 558. Cambridge University Press, Cambridge (2004) 3, 4, 5 15. Duff, M.J. (ed.): The World in Eleven Dimensions: Supergravity, Supermembranes, and M-Theory, p. 513. IOP, Bristol (1999) 4 16. Duff, M.J.: Ten big questions. AIP Conf. Proc. 758, 3–29 (2005) 4 17. Duff, M.J.: Top ten problems in fundamental physics. Int. J. Mod. Phys. A 16, 1012–1013 (2001) 4 18. Johnson, C.V.: D-Branes, p. 548. Cambridge University Press, New York (2003) 4, 5 19. Kaku, M.: Introduction to Superstrings and M-Theory, p. 587. Springer, Heidelberg (1999) 4, 5 20. Kaku, M.: Strings, Conformal Fields, and M-Theory, p. 531. Springer, Heidelberg (2000) 4, 5 21. Kiefer, C.: Quantum Gravity. International Series of Monographs on Physics, vol. 136, 2nd edn., pp. 1–308. Clarendon Press, Oxford (2007) 4, 5 22. Bouhmadi-López, M., Moniz, P.V.: Quantisation of parameters and the string landscape problem. hep-th/0612149 (2006) 4 23. Bousso, R.: Precision cosmology and the landscape. hep-th/0610211 (2006) 4 24. Easther, R., Greene, B.R., Kinney, W.H., Shiu, G.: Imprints of short distance physics on inflationary cosmology. Phys. Rev. D 67, 063508 (2003) 4 25. Kinney, W.H.: Cosmology, inflation, and the physics of nothing. astro-ph/0301448 (2003) 4 26. Tegmark, M.: What does inflation really predict? J. Cosmol. Astropart. Phys. 0504, 001 (2005) 4 27. Turner, M.S.: A sober assessment of cosmology at the new millennium. Publ. Astron. Soc. Pac. 113, 653 (2001) 4 28. Fang, L.-Z., Ruffini, R. (eds.): Quantum Cosmology. Advanced Series in Astrophysics and Cosmology 3, 329pp. World Scientific, Singapore (1987) 4 29. Halliwell, J.J.: Introductory lectures on quantum cosmology. In: Proceedings of Jerusalem Winter School on Quantum Cosmology and Baby Universes, Jerusalem, Israel, 27 December 1989–4 January 1990 4, 5 30. Hawking, S.W.: Lectures on quantum cosmology. In: De Vega, H.J., Sanchez, N. (eds.) Field Theory, Quantum Gravity and Strings, Lecture Notes in Physics, 246, pp. 1–45. Springer, Heidelberg (1986) 4, 5 31. Kiefer, C.: Quantum gravity: A general introduction. Lect. Notes Phys. 631, 3–13 (2003) 4 32. Kiefer, C.: Conceptual issues in quantum cosmology. Lect. Notes Phys. 541, 158–187 (2000) 4 33. Page, D.N.: Lectures on quantum cosmology. In: Mann, R.B., et al. (eds.) Proceedings of Banff Summer Institution on Gravitation, Banff, Canada, 12–15 August 1990. World Scientific, Singapore (1991) 4

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34. Wiltshire, D.L.: An introduction to quantum cosmology. In: Robson, B., Visvanathan, N., Woolcock, W.S. (eds.) Cosmology: The Physics of the Universe, Proceedings of the 8th Physics Summer School, A.N.U., January–February 1995, pp. 473–531. World Scientific, Singapore (1996). gr-qc/0101003 (1995) 4 35. Arnowitt, R., Deser, S., Misner, C.W.: Canonical variables for general relativity. Phys. Rev. 117, 1595–1602 (1960) 4 36. Arnowitt, R., Deser, S., Misner, C.W.: The dynamics of general relativity. gr-qc/0405109 (1962) 4 37. DeWitt, B.S.: Quantum theory of gravity. I. The canonical theory. Phys. Rev. 160, 1113–1148 (1967) 4 38. Dewitt, C.M., Wheeler, J.A. (eds.): Battelle rencontres. 1967 Lectures in Mathematics and Physics (Seattle), 557pp. W.A. Benjamin, New York (1968) 4 39. Misner, C.W.: Minisuperspace. In: Klauder, J.R. (ed.) Magic Without Magic, pp. 441–473. Freeman, San Francisco (1972) 4 40. Ryan, M.P., Shepley, L.C.: Homogeneous Relativistic Cosmologies. Princeton Series in Physics, 320pp. Princeton University Press, Princeton, NJ (1975) 4, 5 41. Wheeler, J.A.: Superspace. In: Gilbert, R., Newton, R. (eds.) Analytic Methods in Mathematical Physics, pp. 335–378. Gordon and Breach, New York (1970) 4 42. Wheeler, J.A.: Superspace and the nature of quantum geometrodynamics. In: Fang, L.-Z., Ruffini, R. (eds.) Quantum Cosmology, pp. 27–92. World Scientific, Singapore (1988) 4 43. Wheeler, J.A.: On the nature of quantum geometrodynamics. Ann. Phys. 2, 604–614 (1957) 4 44. Hartle, J.B.: Quantum cosmology: Problems for the 21st century. gr-qc/9701022 (1997) 4 45. Hawking, S.W.: Quantum cosmology. In: 300 Years of Gravity: A Conference to Mark the 300th Anniversary of the Publication of Newton’s Principia, Cambridge, England, 29 June–4 July 1987 4, 5 46. Kiefer, C.: Quantum cosmology: Expectations and results. Annalen Phys. 15, 316–325 (2006) 4 47. Vilenkin, A.: Quantum creation of universes. Phys. Rev. D 30, 509–511 (1984) 5 48. Vilenkin, A.: Creation of universes from nothing. Phys. Lett. B 117, 25 (1982) 5 49. Vilenkin, A.: The birth of inflationary universes. Phys. Rev. D 27, 2848 (1983) 5 50. D’Eath, P.D.: Supersymmetric Quantum Cosmology, p. 252. Cambridge University Press, Cambridge (1996) 5, 6 51. Esposito, G.: Quantum gravity, quantum cosmology, and Lorentzian geometries. Lect. Notes Phys. M 12, 1–326 (1992) 5 52. Macias, A.: The ideas behind the different approaches to quantum cosmology. Gen. Rel. Grav. 31, 653–671 (1999) 5 53. Macias, A., Obregon, O., Ryan, M.P.: Quantum cosmology: The supersymmetric square root. Class. Quant. Grav. 4, 1477–1486 (1987) 5, 6 54. Pilati, M.: The canonical formulation of supergravity. Nucl. Phys. B 132, 138 (1978) 5, 6 55. Moniz, P.V.: Supersymmetric quantum cosmology – shaken not stirred. Int. J. Mod. Phys. A 11, 4321–4382 (1996) 5, 6 56. Kiefer, C., Luck, T., Moniz, P.: The semiclassical approximation to supersymmetric quantum gravity. Phys. Rev. D 72, 045006 (2005) 5, 6 57. Moniz, P.V.: A tale of two symmetries or the quantum universe from supersymmetry and duality. Nucl. Phys. Proc. Suppl. 88, 57–66 (2000) 5, 6 58. Moniz, P.V.: Origin of structure in a supersymmetric quantum universe. Phys. Rev. D 57, 7071–7074 (1998) 5, 6 59. Tabensky, R., Teitelboim, C.: The square root of general relativity. Phys. Lett. B 69, 453 (1977) 5, 6 60. Teitelboim, C.: Supergravity and square roots of constraints. Phys. Rev. Lett. 38, 1106–1110 (1977) 5, 6 61. Lidsey, J.E.: Scale factor duality and hidden supersymmetry in scalar–tensor cosmology. Phys. Rev. D 52, 5407–5411 (1995) 5 62. Freedman, D.Z., van Nieuwenhuizen, P.: Supergravity and the unification of the laws of physics. Sci. Am. 238, 126–143 (1978) 6

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1 Introduction

63. Luckock, H., Oliwa, C.: Quantisation of Bianchi class A models in supergravity and the probability density function of the universe. In: 7th Marcel Grossmann Meeting on General Relativity (MG 7), Stanford, CA, 24–30 July 1994 6 64. Moniz, P.V.: Can the imprint of an early supersymmetric quantum cosmological epoch be present in our cosmological observations? In: COSMO 97: 1st International Workshop on Particle Physics and the Early Universe, Ambleside, England, 15–19 September 1997 6 65. Moniz, P.: FRW minisuperspace with local N = 4 supersymmetry and self-interacting scalar field. Annalen Phys. 12, 174–198 (2003) 6

Chapter 2

‘Observational’ Quantum Cosmology

Following on the appraisal presented in Chap. 2 of Vol. I, the reader may rightfully be asking: Given the framework of quantum cosmology (QC), where are the boundaries of our knowledge, i.e., what exactly constitutes these limits? What are the best directions to move in, and in particular, what predictions or (falsifiable) tests for the universe can be made using quantum cosmology? With regard to the latter question a vast subject has emerged [1–15] and in Sect. 2.1 we will introduce the main approach to a semiclassical picture. It will constitute an intermediate stage between quantum cosmology and a picture of how the classically observed universe could come into being from induced initial conditions and settings. It is a generic presentation and it will be useful to compare it with the content of Sect. 4.2. It is also an issue of considerable relevance, and has attracted the interest of a number of researchers involved in quantum cosmology. We shall move towards more specific features, with a view to establishing concrete predictions, and a route through which actual cosmological observations can be foreseen. The issue of probing quantum cosmological signatures is addressed in Sect. 2.2, where an ‘observational’ perspective for QC is explored, in particular regarding structure formation in a suitable inflationary stage.

2.1 Semi-Classical Gravity One way of facing the challenge of assigning an observational context to QC could be by establishing a suitable semi-classical approximation scheme for canonical quantum gravity [3, 6, 15]. In some respects, it will mean importing the structure of Sects. 2.2, 2.5, and 2.6 of Vol. I in order to prepare the ground for Sect. 2.2 of this volume, and more specifically Sect. 4.2 of this volume, where some elements of SUGRA will be brought in. Furthermore, it will be really important when reading Part IV (of either volume) and contemplating what has been achieved and what remains to be explored in SQC [16, 17]. Our starting point will be to rewrite (see Sects. 2.5 and 2.6 of Vol. I) the Wheeler– DeWitt equation and the momentum constraints for quantum geometrodynamics Moniz, P.V.: ‘Observational’ Quantum Cosmology. Lect. Notes Phys. 804, 13–33 (2010) c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-11570-7_2

14

2 ‘Observational’ Quantum Cosmology

(without torsion), having reinstated the Planck constant h¯ (see Appendix A for notation and conventions and see also [12, 18, 15] for more details): −16π G h¯ 2 Gi jkl

δ2 1 √ (3) m − h R + H⊥ Ψ [h i j , φ] = 0 , δh i j δh kl 16π G δ 2i + Him Ψ [h i j , φ] = 0 , − (3) ∇ j h ik δh jk h¯

(2.1) (2.2)

where (3) ∇ j denotes the covariant spatial derivative, the matter Hamiltonian density m is assumed for simplicity with a minimally coupled scalar field φ, and following H⊥ [16], we also adopt the parameter M≡

1 , 32π G

that is, M is proportional to the Planck mass squared. m can be taken as the Hamiltonian density of a miniNote 2.1 Generally, H⊥ mally coupled scalar field, i.e.,

√ ij √ 2 2 1 h¯ 2 δ2 + hh φ,i φ, j + h m φ + U (φ) , −√ Hm = 2 h δφ 2

(2.3)

with V (φ) arbitrary.

Note 2.2 The aim is to perform an expansion with respect to M. This will bring the Wheeler–DeWitt equation into a form similar to the Schrödinger equation in quantum mechanics, and thus allow the (formal) application of the Born–Oppenheimer scheme. (In quantum electrodynamics, one can perform an expansion with respect to the electric charge [9].) The gravitational variables are ‘slow’ and the remaining (‘matter’) variables (whose Hamiltonian is m ) are ‘fast’, following the analogy with atomic and molecular denoted by H⊥ physics (see Exercise 2.1). Equation (2.1) then becomes δ2 h¯ 2 g m + MU + H⊥ Ψ = 0 , Gi jkl − 2M δh i j δh jk

√ with U g ≡ −2 h (3)R.

(2.4)

2.1

Semi-Classical Gravity

15

2.1.1 Hamilton–Jacobi Equation The essential step is then to make the following ansatz for the wave functional:

Ψ [h i j , φ] = exp

i S[h i j , φ] h¯

,

(2.5)

with the expansion S[h i j , φ] ≡

∞

Sn [h i j , φ]M−n+1 .

n=0

Equation (2.4) then leads to the following: To Order M2 . We find the independence of S0 from the matter field φ, that is, S0 ≡ S0 [h i j ], i.e.,

δS0 δφ

2 =0.

(2.6)

To Order M1 . This yields the Hamilton–Jacobi ‘equation’ for the gravitational field1 1 δS0 δS0 + Ug = 0 , Gi jkl 2 δh i j δh kl

(2.8)

since it represents an infinite number of equations, one at every point of space (see Exercise 2.2).2

2.1.2 Schrödinger Equation To Order M0 . Here we obtain [12, 15] Gi jkl

1

δS0 δS1 δ2 S0 ih¯ m − Gi jkl + H⊥ =0. δh i j δh kl 2 δh i j δh kl

For the momentum constraints (2.2) at this order, we obtain

δS0 =0. h i j (3) ∇k δh ik

(2.9)

(2.7)

2 Every solution of (2.8) determines a family of solutions of the classical field equations. Equations (2.7) and (2.8) are equivalent to Einstein’s field equations.

16


Through the wave functional

F = K[h i j ] exp

iS1 [h i j , Φ] h¯

,

(2.10)

with the WKB prefactor K required to satisfy the ‘conservation law’ (which in quantum mechanics would just express the conservation of probability) δ Gi jkl δh i j

1 δS0 K2 δh kl

=0,

(2.11)

we proceed3 to recover the Tomonaga–Schwinger equation, also known as the local Schrödinger equation: ih¯ Gi jkl

δS0 δF δF m ≡ ih¯ F, = H⊥ δh i j δh kl δτ

(2.13)

where the ‘time’4 functional τ is implicitly defined by Gi jkl (x)

δS0 δτ (y; h i j ] = δ(x − y) . δh i j (x) δh kl (x)

(2.14)

Note 2.3 This time functional τ actually constitutes a many-fingered time: each spacetime is retrieved from a specific foliation, which is labeled by τ . It is defined on configuration space, but induces a time parameter in each of the spacetimes described by S0 . A potential problem is that τ is not a spacetime scalar: it depends on the chosen embedding. In the semi-classical approximation, however, we use a specific solution of the Hamilton–Jacobi equation, and therefore construct a spacetime from a given 3-geometry as above. The spacetime emerges within a specific foliation. Note that τ is also called WKB time.

3

Up to this order, the total wave functional thus reads Ψ ≈

1 K

exp (iMS0 [h ab ]/h¯ ) F [h ab , φ] ,

(2.12)

where F satisfies the Schrödinger equation. 4

‘Time’ is thus defined through the chosen solution S0 of the Hamilton–Jacobi equation. However, τ is not a spacetime scalar. Nevertheless, the semi-classical scheme can be implemented with the (functional) Schrödinger equation found by integrating (2.14) over three-dimensional space.

2.1

Semi-Classical Gravity

17

2.1.3 Quantum Gravity Corrections To the next order we can obtain quantum gravitational correction terms to (2.13): To Order M−1 . Here there are terms that act along the chosen classical spacetime [15] and others that act in a ‘transverse’ manner [3], eventually probing the structure of superspace (see Sect. 2.5 of Vol. I). These correction terms are of two types: • Breakdown of the classical background. • Quantum gravitational corrections for the master fields. In particular, we retrieve correction terms to the functional Schrödinger equation. The equation (involving S2 ) is:

δS0 δS2 1 δS1 δS1 ih¯ δ2 S1 1 δS1 δS2 ih¯ δ2 S2 + Gi jkl − Gi jkl +√ − 0 = Gi jkl . δh i j δh kl 2 δh i j δh kl 2 δh i j δh kl 2 δφ 2 h δφ δφ (2.15) In order to retrieve useful expressions, possibly with an ‘observational’ content, it is necessary to further manipulate the previous equation (see [9] and references therein, and also Chap. 4 of this volume): • Use (2.12) and replace S1 by F. ˘ i j , φ]. • Define S2 [h i j , φ] ≡ σ˘ 2 [h i j ] + η[h • Further require σ˘ 2 to satisfy Gi jkl

δS0 δσ˘ 2 δK δD δ2 K h¯ 2 h¯ 2 − 2 Gi jkl + =0. Gi jkl δh i j δh kl δh i j δh kl 2K δh i j δh kl K

This is a second-order WKB correction. • In the end, we can write5 Gi jkl

δS0 δη˘ h¯ 2 = δh i j δh kl 2F

−

δF δK δ2 F 2 + Gi jkl Gi jkl K δh i j δh kl δh i j δh kl

ih¯ δ2 η˘ ih¯ δη˘ δF . + √ +√ F h δφ δφ 2 h δφ 2

(2.16)

• The total wave functional is

iη˘ i 1 . Ψ = exp (MS0 + σ˘ 2 M−1 ) F exp K Mh¯ h¯ • The wave functional Θ ≡ F exp (iη/M ˘ h¯ ) then obeys the ‘corrected’ Schrödinger equation

5

Note that σ˘ 2 is a pure gravitational term.

18


δΘ h2 m ih¯ Θ+ = H⊥ Gi jkl δτ MF

1 δ2 F 1 δK δF − K δh i j δh kl 2 δh i j δh kl

Θ.

(2.17)

A more ‘computational’ form (from an ‘observational’ perspective) follows if we decompose the right-hand side of (2.17) into components tangential and normal to the integral curves of the vector field Gi jkl δS0 /δh i j . Decompose the first derivative as (see Exercise 2.3 for more details): Gi jkl

δF δS0 = α Gi jkl + akl , δh i j δh i j

(2.18)

where δS0 ai j = 0 . δh i j The coefficient α is determined by multiplying each side of (2.18) by δS0 /δh kl , summing over kl, and making use of the Hamilton–Jacobi equation, as well as the Schrödinger equation: α≡

i Hm F . 2h¯ U ⊥

As far as the second derivative terms on the right-hand side of (2.17) are concerned, a decomposition into its tangential and normal components produces, using the previous order equations and α, δΘ 4π G δ m m 2 ih¯ Θ+√ ) Θ + ih¯ 4π G = H⊥ (H⊥ δτ δτ h (3) R − 2Λ

m H⊥

√ Θ. h (3) R − 2Λ (2.19)

Note 2.4 Along with the Wheeler–DeWitt equation, momentum constraints must also be brought into this framework [see (B.6)]. This is straightforward enough. The wave functional remains unchanged under a spatial diffeomorphism at each order of this approximation scheme.

Note 2.5 As described, the proper quantum effects of the gravitational field (even on itself) require higher-order iterations of the relevant equations in inverse powers of the Planck mass. An expansion scheme restricted only to the quantum Hamiltonian constraint can be produced, but a broader and more complete picture (see Exercises 2.4 and 2.5) only emerges with the full set of

2.2

‘Predictions’ from Quantum Cosmology

19

constraint equations, implementing all constraints as interconnected by their commutator algebra. This is quite relevant when aiming at a larger description including all correction terms to the Schrödinger equation. The above expression for these terms only involved the ‘longitudinal’ derivatives (taken with respect to configuration space coordinates along the classical trajectory). This part of the correction terms follows solely from the previous order equations, while the remaining terms depend on boundary conditions. Explicit expressions for all correction terms (including the ‘transverse’ derivatives) for the full set of quantum constraints were presented in [3]. It is shown that this leads to the conventional Feynman diagram technique involving the graviton propagator, vertices, and loops, and thus provides a concrete physical interpretation for all terms. See also [9].

2.2 ‘Predictions’ from Quantum Cosmology Concerning the establishment of signatures from QC, thereby making a falsifiable formulation of the theory, let us note that this issue embodies the debate between the proponents of the ‘no-boundary’ wave function and proponents of the ‘tunneling’ wave function (see Sect. 2.7 of Vol. I). This is a significant point, because only one of them should constitute the real description of what actually happened in the quantum stages of the very early universe. The other must then be ruled out. This debate is still going on. There are plenty of opportunities for new ideas and approaches, and the reader may contribute if he or she is persistent enough.

2.2.1 Towards Structure Formation The formalism in the previous section corresponded to the full superspace description. In order to extract cosmological considerations, in particular tracing the origin of the primordial perturbations [19–22], we need to incorporate a consistent minisuperspace reduction. In either situation (superspace or minisuperspace QC), there are vast territories that remain to be charted, in particular, by venturing into the field of SQC (see Sects. 4.2 and 10.1) [16, 17]. A simplified setting to introduce this issue is to take a closed FRW minisuperspace as background, with small inhomogeneous perturbations to the metric and matter fields: h i j (x, t) ≡ a 2 (t) Ωi j + εi j , φ(x, t) ≡ φ0 (t) + δφ(x, t) ≡ φ0 (t) +

(2.20) 1 n (x) , Σnml f nlm (t)Q lm ς

(2.21)

20


N (x, t) ≡ N0 (t) + δN (x, t) ,

(2.22)

Ni (x, t) ≡ 0 + δNi (x, t) ,

(2.23)

n are spherwhere ς = 2π 2 , Ωi j is the unperturbed metric on the 3-sphere, the Q lm 3 ical harmonics on S , and the perturbations εi j are expanded in terms of spherical harmonics on the 3-sphere. Substituting (2.20), (2.21), (2.22), and (2.23) into the classical action, to quadratic order we have an action (see Sect. 2.8 of Vol. I and Appendix A for notation),

S ≡ S0 [q X , N0 ] + S2 [q X , N0 , εi j , δφ, δN , δNi ] ,

(2.24)

where S0 is the background minisuperspace action, S2 is quadratic in the perturbations, and the Hamiltonian constraint is now H⊥ = H0 + H2 .

(2.25)

The Wheeler–DeWitt equation is

1 2 H⊥ Ψ ≡ − ∇ + U (q) + H2 Ψ = 0 , 2

(2.26)

where ∇ 2 is the corresponding covariant Laplace–Beltrami operator of the minisupermetric, and H2 is a second order differential operator 1 1 ∂2 2 2 3 2 H2 ≡ Σnml − 3 2 + (n − 1)a + m a f nlm , 2 a ∂ f nlm

(2.27)

with m 2 φ 2 the potential for the scalar field. Allowed WKB solutions6 have the form Ψ = C(q, φ) eiS0 (q,φ) ,

(2.28)

which are peaked about classical trajectories, where S0 is of course a solution of the unperturbed Hamilton–Jacobi equation, and the functions satisfy the functional Schrödinger equation i

∂ = H2 . ∂t

(2.29)

If the various modes of the scalar perturbations do not interact, we get ˜ nlm (t, f nlm ) . = Πnml 6

(2.30)

Minisuperspace coordinates q X are treated as (semi)classical, while the perturbations constitute quantum mechanical quantitities.

2.2


21

Using the no-boundary or tunneling conditions, we can determine particular solutions of the functional Schrödinger equation, and consequently, a particular vacuum state for the matter modes, whence we may discuss what type of structure formation is induced, i.e., observable or otherwise. Both the no-boundary condition and the tunneling condition induce a de Sitter invariant state known as the Bunch–Davies vacuum [23] often used in cosmological calculations of density perturbations (see Exercise 2.6). From what was said in Vol. I, a potential question concerns the vacuum state in SQC. This is an open question (but see [16, 17]). The interested reader will find some useful aspects in Sects. 4.2 and 10.1 of this volume, with pointers to these and/or other related unexplored routes. In this section, we must speak of another issue that is intrinsically present. In the WKB limit, it is necessary to focus on one particular WKB component. The noboundary wave function is real, and the resulting current [see (2.44) in Sect. 2.2.4] is identically zero. However, ΨHH is the superposition of two WKB components, corresponding to contracting and expanding universes. In this discussion, a decoherence mechanism should be invoked, so that the interference between the two components is negligible [20, 9]. The difference between the outgoing WKB com ponent ΨHH and ΨV lies in the term exp ±1/3V (φ) , which nevertheless still plays a role, as indicated above. In more fundamental terms, the emergence of classical properties from the quantum mechanics formalism is still largely an open problem. However, considerable progress has been achieved through the so-called decoherence approach. In an operational way, the wave function evolves non-linearly and is led to its ‘collapse’.

2.2.2 Towards Spacetime Decoherence The reader may therefore be asking whether, by taking the gravitational part of the wave functional as K−1 exp(iMS0 /h¯ ), we were not already starting from the end, i.e., assuming an expression that induces classical-like correlations among the (mini)superspace variables. Let us therefore elaborate further on this feature. Take a solution of the Wheeler–DeWitt equation, at order M0 , in the superposition form Ψ =

1 1 exp(iMS0 /h¯ )F [h i j , φ] + exp(−iMS0 /h¯ )F ∗ [h i j , φ] . K K

(2.31)

A few notes are in order: • In the time-dependent Schrödinger equations obeyed by χ and χ ∗ , the time is defined through S0 and −S0 , respectively. • Equation (2.31) describes a quantum mechanical superposition and does not therefore ‘contain’ classical world correlations. There are interference terms between the two components. • It is interesting to consider the Born–Oppenheimer approximation for a useful analogy.

22


The route to follow is to consider (2.31) as representing an example of many possible such states for as many solutions of the Hamilton–Jacobi and Schrödinger equations, intertwining gravitational and non-gravitational degrees of freedom. Moreover, not all of these degrees of freedom are actually accessible for meaningful observation, while the others have to be traced or averaged. Note 2.6 We can be more precise through the following. ‘Global’ degrees of freedom, e.g., the scale factor a of a Friedmann universe, represent the semiclassical gravitational degrees of freedom, whereas ‘fluctuations’, e.g., matter density perturbations or gravitational waves, will be ‘environmental’ degrees of freedom, which are integrated out. Selecting the degrees of freedom of each type is an entirely arbitrary matter. A simple and general approach is to start from the semi-classical expansion in (2.31), to order M 0 , where the functionals F[h i j , φ] containing the nongravitational fields obey an approximate time-dependent Schrödinger equation. We can take these degrees of freedom as an ‘unobserved’ environment, and integrate them out to get the reduced density matrix for the gravitational degrees of freedom alone: ρ[h i j , h i j ] ≡ Trφ Ψ [h i j , φ]Ψ ∗ [h i j , φ] .

(2.32)

In more detail: • Divide the state which will eventually satisfy the Wheeler–DeWitt equation into subsystems. • One comprises the ‘relevant’ or observed degrees of freedom. • The other comprises the ‘environment’ or ‘behind the scenes’ degrees of freedom. • One standard framework is: – Take a in the FRW setting as the relevant function which will become (quasi)classical through continuous measurement interaction with the vast set of the remaining (e.g., fluctuation) degrees of freedom. – Technically, this means expanding the relevant degrees of freedom (3metric and, e.g., scalar field) into harmonics or higher multipoles on, e.g., S 3 . The perturbations are up to quadratic order, and interact with the relevant degrees of freedom, but are assumed to be mutually orthogonal [see (2.28), (2.29), and (2.30)]. – The density matrix would be, e.g., ρ(a1 , a2 ) =

Ψ0 (a1 )Ψ0∗ (a2 )

N n=1

˜ n (a1 , ϑn ) . ˜ n∗ (a2 , ϑn ) dϑn

2.2


23

– We are simplifying by using Ψ0 ≡ C(q, φ) eiS0 (q,φ) . – The Schrödinger equation is then ñ 1 2 4 1 1 ∂2 ∂ 2 ñ . + 3 − n a + m 2 a 6 f nlm i 2 ∂t 2 ∂ f nlm 2 a

(2.33)

– Note that t is the WKB time (defined through ∂/∂t ≡ ∇a S∇a , for example), which parametrizes the classical trajectories in the minisuperspace. – Typically, ρ becomes a narrow Gaussian (for the above setting, in a − a ) [10, 13, 14], implying a ‘localisation within one WKB component’ (representing a ‘measurement of the 3-geometry’), and it contains in addition a large suppression factor for the interference term between the WKB components. But in other cases it may not be as simple. If the F real components are of relevance, the two components in (2.31) further interfere (e.g., at a turning point of a recollapsing universe). It may not be possible to define classical time, and the back-reaction of non-gravitational fields on the Hamilton–Jacobi equation is significant.

In other words, instead of the full state (2.31), it is the reduced density matrix that has to be employed: • The density matrix is obtained by integrating out the ‘irrelevant’ degrees of freedom. • The density matrix obeys a non-unitary master equation, rather than a unitary (von Neumann) equation. • Observationally accessible states do not exhibit any interference terms locally. • This mechanism is referred to as decoherence.7 • Quantum interference effects among states of the system are suppressed by the interaction with the environment. This coarse-graining procedure leads to an effective action. • However, in addition to the notion of decoherence, a further condition that a system must satisfy in order to be regarded as classical is, of course, that it should be driven by classical laws, implying that a sharp correlation between configuration space coordinates and conjugate momenta should exist in the wave function (see Sect. 2.2.4). Before moving on, there is one other feature to introduce, which may be of relevance when dealing with realistic settings where the emergence of classical spacetime can occur [24]. 7

Compare with the Born–Oppenheimer approximation, where ignoring the off-diagonal terms amounts to assuming a decoherence process.

24


In models containing a single ‘classical’ degree of freedom, the Hamilton–Jacobi equation has only two solutions, generating the same trajectory in opposite directions. To be more precise, the semi-classical wave function has two WKB components, each of which may be called a WKB branch, of the form Ψ [O, E] =

C(n) [O]eiMP S(n) [O] (n) [O, E] . 2

(2.34)

(n)

Here the index (n) labels the WKB branches (taking only two values, say ±1, uniquely identifying the two possible solutions of the Hamilton–Jacobi equation), while O, E denote, respectively, the ‘classical’ physical observables and the extra degrees of freedom corresponding to the environment. To achieve proper decoherence, one not only requires the reduced density matrix to be diagonal, but in addition the different WKB components in (2.34) must have negligible interference among the diagonal terms. It is important to stress that the analysis of correlations should be done within each classical WKB branch (i.e., a diagonal term, n = n ). The interference effects between the two possibilities for moving along the one-dimensional classical trajectory (corresponding to the expanding and collapsing wave function components, respectively) have been shown to be effectively suppressed. This is interpreted as particle creation. A solution which would correspond to ‘classical’ behaviour of the a variable on some region of minisuperspace will have an oscillatory WKB form (2.34), viz., Ψ(n) [a, φ] = eiMP S(n) (a) C(n) (a)(n) (a, φ) . 2

(2.35)

If the different modes do not interact among themselves, we can solve the Wheeler– ˜ (n) (a, φ) that can be factorized as above. DeWitt equation for a wave function The wave function (2.35), say, is a functional only of the gravitational and matter physical degrees of freedom, and as a consequence the corresponding density matrix has no explicit time dependence. The full density matrix ρ is a biscalar on the full configuration space. From the pure minisuperspace-dependent part of (2.35), we can obtain a minisuperspace density matrix ρ0 . In the case where the background minisuperspace variables are equal (a = a in our present model), the relation between ρ and ρ0 is easily written. However, when a = a , say, the relation is not so straightforward. By dividing the full set of configuration variables into physical observables and small fluctuations from the environment, we can construct a reduced density matrix by tracing over the small ones. More precisely, the reduced density matrix takes the form ρred = ρ0 I(a, a ; . . .), which is written as 2 eiMP [S(n) (a1 )−S(n ) (a2 )] C(n) (a1 )C(n ) (a2 )In,n (a2 , a1 ) , (2.36) ρred = n,n

2.2


25

where In,n (a2 , a1 ) ≡

˜ (n) (a1 , φ)d[φ] , ˜ ∗ (a2 , φ) (n )

(2.37)

after the coarse-graining description of the full system has been performed as stated above and using the WKB wave function of the form (2.35). Note once again that the index (n) labels the WKB branches. The term In,n (a2 , a1 ) is sometimes referred to as the decoherence functional and describes the influence of the environment on the system. Notice that all modes must be included in (2.37).

2.2.3 Towards Spacetime Correlations It should be mentioned at this stage that the analysis of correlations between minisuperspace coordinates and momenta is, in quantum cosmology, usually discussed using the Wigner function criterion. A strong sharp peak is likely to be located close to a classical trajectory defined by the Hamiltonian–Jacobi equation plus quantum corrections. However, the Wigner function associated with the reduced density matrix (2.36) does not have a single sharp peak, even for a pure WKB function like (2.35) or a linear combination of such. Nevertheless, this problem can be overcome through the interaction of the environment with the ‘observed’ system. Such an interaction underlies the loss of quantum coherence, or decoherence, between different classical trajectories, i.e., WKB branches. More precisely, correlations between coordinates and momenta must be analysed within each classical branch (n = n ). This can be done by looking at the reduced density matrix associated with it, or the corresponding Wigner functional: FW(n) (a, πa ) =

−1/2 2

dΔ S(n) (a1 )S(n) (a2 ) e−2iπa Δ eiMP [S(n) (a1 )−S(n) (a2 )] In,n (a2 , a1 ) ,

+∞

−∞

(2.38) where Δ = (a1 − a2 )/2. A correlation among variables will correspond to a strong peak about a classical trajectory in the phase space. Thus there exists an important relation between correlation and decoherence, since one needs the latter, i.e., fairly small off-diagonal terms in (2.36) so that quantum interference between alternative histories is negligible (In,n ∝ δn,n ), in order to obtain the former. Hence, the decoherence process is rather crucial, as it is only when the decoherence between different WKB branches is successful that correlations may be properly predicted [8, 10, 13, 14]. Besides the decoherence between different WKB branches, the environment interaction also affects the correlations within a classically decohered branch. This is made explicit in the functional In,n (a2 , a1 ) in (2.38). The environmental degrees of freedom continuously measure the physical observables, and this interaction not only suppresses the off-diagonal (n = n ) terms in (2.36) and (2.37), but it also

26


induces a ‘localizing’ effect on the classical variables within each WKB branch. This corresponds to the back-reaction from the environment on the semi-classical evolution of the system. In particular, In,n (a2 , a1 ) will be damped for |a2 −a1 | 1, and the reduced density matrix associated with (2.38) will be diagonal with respect to a. The sharpness and position of the peak will be determined by the behaviour of In,n (a2 , a1 ). Furthermore, the localization effect inside a classical branch is much more efficient than the decoherence between diferent WKB branches. If the conditions for achieving an effective localization (and diagonalization) of (2.38) are met, then the interference between the different WKB branches is also highly suppressed. Actually, the functional In,n (a, a) has usually been identified as a measure of the decoherence between two different WKB histories, characterized by the parameters (n) and (n ). Before proceeding, we point out that the use of a Gaussian ansatz for the environment state with an analysis of the decoherence between different WKB histories and correlations via the Wigner function [with I(n,n)J (a, a )] requires the following conditions for successful diagonalization and ‘localization’, usually referred to as the adiabaticity, strong decoherence, and strong correlation conditions: • The adiabaticity condition warrants the validity of the zero order WKB evolution, since its violation implies that the semi-classical Einstein equations are not valid due to high-order contributions in the phase of In,n (a2 , a1 ). • On the other hand, a strong correlation reflects the fact that the peak in the Wigner function (shifted away from the expected classical trajectory by interaction with the environment) is sharp insofar as the center of the peak is large when compared to the spread. • Finally, the strong decoherence condition effectively corresponds to the requirement of diagonalization of the reduced density matrix associated with (2.38). It is important to mention that a compromise between decoherence and correlation is usually needed, since if the latter is too strong, then the peak in the Wigner function is actually broadened. But the picture from the previous paragraphs is not yet wholly realistic. Multidimensional minisuperspace models have, of course, a much richer structure, and are therefore far more interesting to consider with regard to the retrieval of classical behaviour. For a system with n degrees of freedom, the Hamilton–Jacobi equation is expected to have an n − 1 parameter family of solutions, each one generating an n − 1 parameter family of classical trajectories in the minisuperspace. In the multidimensional case, a general solution of the Wheeler–DeWitt equation may contain an infinite superposition of semi-classical solutions of the form (2.34), with the index (n) now corresponding to a set of parameters that uniquely identify a specific Hamilton–Jacobi solution. However, each WKB branch must actually be interpreted as describing a whole family of classical trajectories, i.e., a set of different universes, and not a single universe as for the n = 1 case. Furthermore, the n = 2 and n > 2 cases are rather different as far as the diagonalization of the reduced density matrix is concerned. An example of an n = 2 model is where the

2.2


27

two ‘classical’ degrees of freedom correspond to the scale factor and homogeneous mode of a minimally massless scalar field and the environment has been identified with the inhomogeneous perturbations of another minimally massless scalar field. The multidimensional cases also allow one to better address the relation between the reduced density matrix formalism and the Feynman–Vernon influence functional, and the Schwinger–Keldish or closed time path effective action, as pointed out in [25–27]. The analysis of multidimensional minisuperspace models is relevant as it provides a point at which to discuss several important issues. In particular, a specific solution of the Hamilton–Jacobi equation generates an n − 1 parameter family of trajectories, but there will be only one classical trajectory passing through each point of the minisuperspace generated by that solution of the Hamilton–Jacobi equation. In this sense, Inn could strengthen the suppression of interference between histories belonging to a given WKB branch, as it produces a more efficient diagonalization of the reduced density matrix. If then a minisuperspace is two-dimensional, the Hamilton–Jacobi equation is expected to have a one-parameter family of solutions, each one generating a family of classical trajectories in minisuperspace, with each WKB branch interpreted as describing a whole family of classical trajectories, i.e., a set of different universes (and not a single one as for the n = 1 case). From the Hamilton–Jacobi equation, it follows that the trajectories in such an n = 2 minisuperspace are far more complicated than those in the n = 1 case. A WKB time is now defined by ∂S ∂ d , = G XY dη ∂q X ∂qY

(2.39)

with X, Y = 1, 2, q1 = a, q2 = φ, and G X Y = diag(−1, 1). We shall have as many η-affine parameters as different values of the (n) parameter. Hence, different values of (n) will lead to different definitions of time for the Schrödinger equation. This implies that the influence functional in (2.36) and (2.37) is actually a functional ˜ (n) can be interpreted, not as being simply a function of two ‘histories’. A state of a point in minisuperspace, but instead as a function of the whole history, which corresponds to the only trajectory that belongs to the (n)-WKB branch and goes through that particular point (a, φ). Such a description is fairly similar to the one involving the Feynman–Vernon influence functional [25–27]. Finally, let us mention the issues of correlation and decoherence within each WKB branch and the relevant influence functional for such an N = 2 minisuperspace model. Correlations between each minisuperspace coordinate a and φ and their canonical momenta can be analysed by examining peaks in the reduced Wigner function FW1 (q X , π q1 ) =

dπ q2 FW(n) (q X ; π q X ) ,

(2.40)

28


where π q1 is the momentum conjugate to q A , X = 1. Let us nevertheless describe what becomes of the above framework when dealing with minisuperspaces where the relevant degrees of freedom are given by a and φ [8–15]. For a Gaussian ansatz regarding the ‘environment’ (higher multipoles), the reduced density matrix is

Nˇ 3

2 2 ˇ

2 − 2 (a − a ) − 3m N (aφ − a φ ) , ρ[a, φ; a , φ ] 6a (2.41) where Nˇ is a cutoff in the number of multipoles, which for Nˇ a brings (2.41) into the form a ρ[a, φ; a , φ ] Ψ0 (a, φ)Ψ0∗ (a , φ ) exp − (a − a )2 − 3m 2 a(aφ − a φ )2 . 6 (2.42) Let us mention a few points here:

Ψ0 (a, φ)Ψ0∗ (a , φ ) exp

• The ‘size’ of the universe, as indicated by a, is more classical, since the universe is larger. • A classical a leads to a classical φ, i.e., interference within φ is negligible if a ∼ a . • With fermions as environment, it is found that it is only

m2 N exp − fm (a − a)2 8

,

and less effective.

2.2.4 Inflation Although this topic could also have been discussed in Sect. 2.8 of Vol. I, we have chosen to present it here. The central issue has barely been considered within SQC, and for this reason we include here a brief description of the way it has been approached in QC. The main question for an oscillatory WKB wave function is to determine whether and to what extent an adequate inflationary period can be retrieved from a QC setting. Its applicability to current issues is pertinent, like the string landscape problem (see, e.g., [28, 29]). Basically, we must ask whether inflation can be generic or sufficient within the framework of string theory. Both WKB limits for the no-boundary and tunneling wave functions predict an inflationary stage for an FRW universe [20, 9, 22]. Moreover, the wave functions in the oscillatory region are strongly peaked about the set of classical solutions [30, 31], which constitute an adequate √ inflationary universe: a(t) ∼ e V t , φ(t) φ0 = c, with c a constant. Vilenkin’s approach [31–35], with a different choice of factor ordering in the Wheeler–DeWitt equation, leads to a WKB wave function

2.2


ΨV ∼ exp

3/2 iπ −1 1 2 + exp −i a V (φ) − 1 , 3V (φ) 3V (φ) 4

29

(2.43)

in the oscillatory region, also peaked about the inflationary classical trajectories. But is this inflation suitable? In other words, what is the probability that the universe inflates by 65 e-folds to solve the problems of standard cosmology mentioned in Sect. 2.1 of Vol. I? As the reader may now point out, in order to make predictions from a wave function, some probability measure must be defined. This is indeed the case, and it constitutes a rather delicate problem in quantum cosmology [36]. This in itself constituted one of the motivations for exploring beyond the borders of purely bosonic general relativistic minisuperspace and including fermions, preferably within a supersymmetric (SUGRA) background (see Chaps. 6, 7, 8, and 9 in either volume). Let us identify the essence of the problem. The minisuperspace Wheeler–DeWitt equation is a second order equation. From this equation, the current [20, 22] 1 J = − i Ψ ∇Ψ − Ψ ∇Ψ 2

(2.44)

is retrieved and shown to be conserved: ∇ ·J=0.

(2.45)

However, we should note the following: • The inner product constructed from J is not positive-definite, i.e., negative probabilities become possible. • There is no generic and well-defined definition of positive frequencies in full superspace, and we cannot decompose the wave function into positive and negative frequency components. • Some wave functions (e.g., the no-boundary wave function) are real and give J = 0. Some options we may consider are as follows: 1. Express Ψ in an operator representation, to create and annihilate universes. 2. Invoke SQC (!), since the constraints of SUGRA are like the Dirac square root of the constraints of general relativity, in the form of first-order equations. However, even so, the problem is not solved (see Chaps. 5 and 6 of Vol. I). 3. Use |Ψ |2 directly as a probability measure. For homogeneous minisuperspaces, this is a promising option since quantum cosmology is quantum mechanics with time reparametrization. However, there are examples where the wave function is not normalisable. 4. Whereas in quantum mechanics, |Ψ |2 assigns a probability density in the space of particle positions, in quantum cosmology we have a (mini)superspace where

30


time is implicit in the coordinates, so these cannot therefore be mere analogues of particle positions. 5. Consider Ψ associated instead with conditional probabilities [36–39], e.g., of eventually finding Ψ in a region A of minisuperspace given that Ψ is found also in another region B. When assessing the above possibilities, the WKB semi-classical limit is quite instructive [20, 22]. If we take a first-order WKB wave function

1 Ψn An exp iSn − 2

ds ∇ Sn 2

,

each Ψn has a conserved current Jn |An |2 ∇ Sn ⇐⇒ ∇ X JnX = 0 ,

(2.46)

along the direction of the classical trajectories, and dP = JnX d X

(2.47)

is a conserved probability measure on the set of trajectories with tangent ∇ Sn , with d X the element of a hypersurface in minisuperspace which cuts across the flow and intersects each curve in the congruence once and only once. For a set of trajectories near the classical trajectory, the probability density is positive-definite. On the one hand, this requires integrating the probability flux dP = JnX d X on the surface separating the tunneling and oscillatory regions [where a 2 V (φ) = 1]. In the WKB approximation, we have φ˙ 0, hence J ∼ ∇a, i.e., the minisuperspace hypersurface with a = constant. It then follows that ⎧ 2 ⎪ ⎪ ←− ΨHH , exp + ⎨ 3V (φ) dP = J · d ∼ 2 ⎪ ⎪ ⎩ exp − ←− ΨV . 3V (φ)

(2.48)

In the context of conditional rather than absolute probabilities, on an a = constant surface, we have [20, 9, 22]

2 dφ0 exp ± 3V (φ0 ) φ P (φ0 > φsuff | φ1 < φ0 < φ2 ) = suff , φ2 2 dφ0 exp ± 3V (φ0 ) φ1

φ2

(2.49)

where the (+) case refers to the no-boundary wave function and the (−) case to the tunneling wave function, and the values φ1 and φ2 are respectively lower and upper cutoffs for the allowed values of φ. The minimum value of the scalar field,

Problems

31

for which sufficient inflation is obtained, is given by φsuff , and φ = φ0 is the initial value. More details and plenty of reading can be found in the literature [36], so let us just indicate the most important elements: • Within the no-boundary camp, a probability of P 1, was obtained. • But the tunneling camp pointed out that, for some of the parameters employed, values go beyond the Planck scale, and the semi-classical approximation will no longer apply. • For φ2 (upper cutoff) at the Planck scale, the integral in the denominator becomes very large in the case of the no-boundary wave function (+ sign), leading to P 1, but not for the tunneling wave function (− sign), thereby predicting more inflation. • However, the calculations are in fact model dependent, and for specific potentials, both ΨHH and ΨV lead to P ∼ 1.

Summary and Review. As in Vol. I, before proceeding to further issues in SUSY and SUGRA in the next chapter, we briefly review the main elements in this chapter: 1. What semi-classical elements can be extracted [Sect. 2.1.1]? 2. How can we describe quantum gravitational corrections in a semi-classical background [Sect. 2.1.1]? 3. Can QC become ‘observational’ [Sect. 2.1]? 4. What is decoherence [Sect. 2.2.2]? 5. What tools can be employed to establish whether the universe behaves as a classical system [Sect. 2.2.3]? 6. What is the relevance of the Bunch–Davies vacuum [Sect. 2.2.1]? 7. Can inflation and the primordial seeds for structure formation be satisfactorily predicted from a wave function of the universe whatever boundary conditions are used, or with just some boundary conditions [Sects. 2.2.1 and 2.2.4]?

Problems 2.1 The Born–Oppenheimer approximation and Gravitation Summarize the main elements of the Born–Oppenheimer approximation in a cosmological setting (see [12, 18, 15]). 2.2 Hamilton–Jacobi Equation and the Emergence of ‘Classical’ Spacetime From the Hamilton–Jacobi equation for the gravitational field, demonstrate its equivalence to Einstein’s field equations and show how ‘classical’ spacetime can emerge.

32


2.3 Factor Ordering and Semi-Classical Gravity Factor ordering is a key problem in quantum mechanics, and of course in quantum gravity (see Chap. 2 of Vol. I). How useful is the semi-classical description here when factor ordering issues are raised? 2.4 Quantum Gravity Corrections and Unitarity Vs Non-Unitarity The second correction term in (2.19) is pure imaginary. Discuss non-unitary issues implied by this. 2.5 De Sitter Space and Quantum Gravity Corrections For the case of a minimally coupled scalar field in a flat de Sitter (FRW) space foliation, apply the framework from the Hamilton–Jacobi equation up to (2.19). 2.6 The Bunch–Davies Vacuum and Quantum Cosmology Derive the Bunch–Davies vacuum within the framework of Sect. 2.2.1.

References 1. Barvinsky, A.O., Kamenshchik, A.Yu., Kiefer, C., Mishakov, I.V.: Decoherence in quantum cosmology at the onset of inflation. Nucl. Phys. B 551, 374–396 (1999) 13 2. Barvinsky, A.O., Kamenshchik, A.Yu., Kiefer, C.: Effective action and decoherence by fermions in quantum cosmology. Nucl. Phys. B 552, 420–444 (1999) 13 3. Barvinsky, A.O., Kiefer, C.: Wheeler–DeWitt equation and Feynman diagrams. Nucl. Phys. B 526, 509–539 (1998) 13, 17, 19 4. Giulini, D., et al.: Decoherence and the Appearance of a Classical World in Quantum Theory, p. 366. Springer, Heidelberg (1996) 13 5. Giulini, D., Kiefer, C., Zeh, H.D.: Symmetries, superselection rules, and decoherence. Phys. Lett. A 199, 291–298 (1995) 13 6. Giulini, D., Kiefer, C.: Consistency of semiclassical gravity. Class. Quant. Grav. 12, 403–412 (1995) 13 7. Giulini, D., Kiefer, C., Lammerzahl, C. (eds.): Quantum Gravity: From Theory to Experimental Search, p. 400. Springer, Heidelberg (2003) 13 8. Kiefer, C.: Decoherence in quantum electrodynamics and quantum gravity. Phys. Rev. D 46, 1658–1670 (1992) 13, 25, 28 9. Kiefer, C.: Quantum Gravity. International Series of Monographs on Physics, vol. 136, 2nd edn., pp. 1–308. Clarendon Press, Oxford (2007) 13, 14, 17, 19, 21, 28, 30 10. Kiefer, C.: Continuous measurement of minisuperspace variables by higher multipoles. Class. Quant. Grav. 4, 1369 (1987) 13, 23, 25, 28 11. Kiefer, C.: Continuous measurement of intrinsic time by fermions. Class. Quant. Grav. 6, 561 (1989) 13, 28 12. Kiefer, C.: How does quantum gravity modify the Schrödinger equation for matter fields? Class. Quant. Grav. 9, S147–S156 (1992) 13, 14, 15, 28, 31 13. Kiefer, C., Polarski, D.: Emergence of classicality for primordial fluctuations: Concepts and analogies. Annalen Phys. 7, 137–158 (1998) 13, 23, 25, 28 14. Kiefer, C., Polarski, D., Starobinsky, A.A.: Quantum-to-classical transition for fluctuations in the early universe. Int. J. Mod. Phys. D 7, 455–462 (1998) 13, 23, 25, 28 15. Kiefer, C., Singh, T.P.: Quantum gravitational corrections to the functional Schrödinger equation. Phys. Rev. D 44, 1067–1076 (1991) 13, 14, 15, 17, 28, 31 16. Kiefer, C., Luck, T., Moniz, P.: The semiclassical approximation to supersymmetric quantum gravity. Phys. Rev. D 72, 045006 (2005) 13, 14, 19, 21

References

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17. Moniz, P.V.: Origin of structure in a supersymmetric quantum universe. Phys. Rev. D 57, 7071–7074 (1998) 13, 19, 21 18. Kiefer, C.: The semiclassical approximation to quantum gravity. gr-qc/9312015 (1993) 14, 31 19. Halliwell, J.J., Hawking, S.W.: The origin of structure in the universe. Phys. Rev. D 31, 1777 (1985) 19 20. Halliwell, J.J.: Introductory lectures on quantum cosmology. In: Proceedings of Jerusalem Winter School on Quantum Cosmology and Baby Universes, Jerusalem, Israel, 27 December 1989–4 January 1990 19, 21, 28, 29, 30 21. Padmanabhan, T., Singh, T.P.: On the semiclassical limit of the Wheeler–DeWitt equation. Class. Quant. Grav. 7, 411–426 (1990) 19 22. Wiltshire, D.L.: An introduction to quantum cosmology. In: Robson, B., Visvanathan, N., Woolcock, W.S. (eds.) Cosmology: The Physics of the Universe, Proceedings of the 8th Physics Summer School, A.N.U., January–February 1995, pp. 473–531. World Scientific, Singapore (1996). gr-qc/0101003 (1995) 19, 28, 29, 30 23. Birrell, N.D., Davies, P.C.W.: Quantum Fields in Curved Space, p. 340. Cambridge University Press, Cambridge (1982) 21 24. Bertolami, O., Moniz, P.V.: Decoherence of Friedmann–Robertson–Walker geometries in the presence of massive vector fields with U(1) or SO(3) global symmetries. Nucl. Phys. B 439, 259–290 (1995) 23 25. Hu, B.L., Verdaguer, E.: Stochastic gravity: Theory and applications. Living Rev. Rel. 11, 3 (2008) 27 26. Hu, B.L., Sinha, S.: A fluctuation–dissipation relation for semiclassical cosmology. Phys. Rev. D 51, 1587–1606 (1995) 27 27. Sinha, S., Hu, B.L.: Validity of the minisuperspace approximation: An example from interacting quantum field theory. Phys. Rev. D 44, 1028–1037 (1991) 27 28. Bouhmadi-López, M., Moniz, P.V.: Quantisation of parameters and the string landscape problem. hep-th/0612149 (2006) 28 29. Bousso, R.: Precision cosmology and the landscape. hep-th/0610211 (2006) 28 30. Hartle, J.B., Hawking, S.W.: Wave function of the universe. Phys. Rev. D 28, 2960–2975 (1983) 28 31. Vilenkin, A.: Quantum creation of universes. Phys. Rev. D 30, 509–511 (1984) 28 32. Vilenkin, A.: Creation of universes from nothing. Phys. Lett. B 117, 25 (1982) 28 33. Vilenkin, A.: Boundary conditions in quantum cosmology. Phys. Rev. D 33, 3560 (1986) 28 34. Vilenkin, A.: Quantum cosmology and the initial state of the universe. Phys. Rev. D 37, 888 (1988) 28 35. Vilenkin, A.: Approaches to quantum cosmology. Phys. Rev. D 50, 2581–2594 (1994) 28 36. Page, D.N.: Lectures on quantum cosmology. In: Mann, R.B., et al. (eds.) Proceedings of Banff Summer Institution on Gravitation, Banff, Canada, 12–15 August 1990, pp. 135–170. World Scientific, Singapore (1991) 29, 30, 31 37. Page, D.N.: Quantum cosmology lectures. gr-qc/9507028 (1994) 30 38. Page, D.N.: Aspects of quantum cosmology. gr-qc/9507025 (1995) 30 39. Page, D.N.: Quantum cosmology. hep-th/0610121 (2002) 30

Chapter 3

Additional SUSY and SUGRA Issues

Two frameworks associated with supersymmetry (SUSY) will be the focus of this chapter, with a view to their further application in probing SQC. These are SUSY breaking (see Sects. 3.1 and 3.2) [1–9] and supersymmetric quantum mechanics (SQM) (see Sect. 3.3) [10–70]. Neither of them has been much investigated in SQC, although the latter feature could be associated with the SQC approach introduced in Chap. 6 of Vol. I.

3.1 Fayet–Iliopoulos (FI) Potential In the context of Sect. 3.3 of Vol. I, another ingredient can be added [71, 72], namely the Fayet–Iliopoulos (FI) terms. Consider the case of a U(1) gauge group Gˆ [or one that contains U(1) factors]. If V denotes the Abelian vector superfield, with V → V + iδ − iδ † [see (3.96) of Vol. I], where δ is a chiral superfield, it follows that the corresponding d term (the term for θ θ θ θ ) is invariant up to a total derivative. The SUSY (and gauge) invariant Lagrangian can then be extended by LFI ≡

ξ

(a)

d2 θ d2 θ V(a) =

(a)∈Abelian factors

1 2

ξ (a) d(a) .

(3.1)

(a)∈Abelian factors

As a consequence, the full SUSY N = 1 Lagrangian together with the auxiliary field equations of motion f I† = −

∂W , ∂φ I

(3.2)

and1 d(a) −φI† T (a)IJ φJ − ξ (a) ,

1

(3.3)

ξ (a) = 0 if (a) does not take values in an Abelian factor of the gauge group.

Moniz, P.V.: Additional SUSY and SUGRA Issues. Lect. Notes Phys. 804, 35–60 (2010) c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-11570-7_3

36

3 Additional SUSY and SUGRA Issues

establishes the following expression [see (3.32), (3.108), and (3.146) of Vol. I]: 1 (a) (a)μν f − iλσ μ D˜ μ λ − (Dμ φ)† Dμ φ − iχ σ μ Dμ χ (3.4) L Tr − f μν 4

2 † ∂ W 1 ∂ 2W I I 1 χχ − χ I χ J − V (φ † , φ) + other terms , − 2 ∂φ I ∂φ J 2 ∂φ I ∂φ I

V (φ , φ) ≡ †

f I† f I

2 1 2 ∂W 2 1 † (a) (a) + d = + T φ + ξ φ . ∂φ I 2 2 I

(3.5)

(a)

But why this short section? A particularly relevant way of justifying the presence of the FI potential will be given next.

3.2 SUSY Breaking Whenever we discuss SUSY, we must face the reality that SUSY is a symmetry that we do not currently perceive in nature, so there must be some mechanism for breaking it. To discuss this issue, of potential relevance in SQC, since only a very few publications have addressed the question (see Chaps. 5 and 8), let us consider the following scalar potential, often used in the SUSY context [1, 2]: 1 V (φ, φ † ) ≡ f I† f I + d(a) d(a) , 2

(3.6)

retrieved with (3.2), (3.3), and the Fayet–Iliopoulos terms ξ (a) . The main point to notice is that the potential (3.6) is non-negative, with its global minimum (V = 0) given by f I (φ † ) = d(a) (φ, φ † ) = 0 .

(3.7)

This is an extremum, whose full set may include a local minimum (a ‘false’ vacuum) or a global minimum (a true vacuum). In either case ∂V ∂V (φ J , φJ† ) = (φ J , φJ† ) = 0 . † ∂φ I ∂φI

(3.8)

If we also require Lorentz invariance, this implies that only scalar fields φ I can have a non-vanishing vacuum expectation value φ I . In fact, we can express this by writing vμ(a) = λ(a) = χ I = ∂μ φ I = 0 ,

† V (φ0I , φ0I ) = minimum .

(3.9)

3.2

SUSY Breaking

37

Two types of extremum can then be discussed from a SUSY perspective [1, 2, 8, 71, 72]: • If the equations (3.7) have a solution V = 0, this solution is thus a global minimum of V . • If the equations (3.7) have no solutions, the scalar potential V can never vanish and its minimum V0 is strictly positive, i.e., V ≥ V0 > 0 (see Exercise 3.1): – But this vacuum necessarily breaks SUSY, because it is not invariant under all SUSY generators. – In short, SUSY is broken if either f I (φ † ) = 0 or d(a) (φ, φ † ) = 0. – More precisely, in terms of transformations for the fields (see Chap. 3 of Vol. I and [71]) with (3.9), for the unbroken setting: √ 2εψ I = 0 , √ √ δχ I −→ δχ I = 2i∂μ φ i σ μ ε + 2 f I ε = 0 √ 2 f I ε , = √ δ f I −→ δ f I = 2iσ μ ∂μ χ I = 0 , δφ I −→ δφ I =

(3.10) (3.11) (3.12)

and consistency requires f I ≡ f I (φ † ) = 0. For δλ(a) = 0, then d(a) ≡ d(a) (φ, φ † ) = 0 is a necessary condition for unbroken SUSY, so that the SUSY variations of the fermions must vanish in the vacuum.

3.2.1 Mechanisms for SUSY Breaking SUSY breaking2 may occur under some specific physical processes. In more detail, it will depend on the superpotential, W, and in particular the reader will appreciate here the importance of the Fayet–Iliopoulos parameters ξ (a) . O’Raifeartaigh Mechanism This is relevant when no U(1) factors are present or the ξ (a) vanish. SUSY will therefore be broken if ∂W/∂φ I = 0 and φJ† (T (a) )JI φ J = 0 have no solution. When there are no d(a) (φ, φ † ) terms, the process is reduced to the analysis of f I† = −∂W(φ)/∂φ I and referred to as f-term SUSY breaking. Fayet–Iliopoulos Mechanism This is also usually known as d-term SUSY breaking, and we will show here that it is relevant when a U(1) and non-vanishing ξ are present. Spontaneous SUSY breaking With a minimum of V such that f I = 0 or d(a) = 0, or in other words, no vacuum with f I = d(a) = 0, i.e., no solution φ I to these equations.

2

38


requires some field to acquire a non-zero vacuum expectation value. Within a purely chiral superfield description, only the f-term can provide this mechanism. But in the presence of a vector superfield, another possibility is the d-term, without breaking Lorentz invariance. This means analysing3 d = 0, as 0=

qI |φ I |2 + ξ .

(3.13)

I

In particular, note how important the FI terms become. In a simple U(1) situation, if the φ I are zero, then SUSY is unbroken. Note 3.1 A specific feature to note when SUSY is spontaneously broken is the appearance of the goldstino, a massless spin half particle. Goldstone’s theorem states that, whenever a continuous global symmetry is spontaneously broken (i.e., the vacuum is not invariant), there is a massless mode in the spectrum, i.e., a massless particle. The quantum numbers carried by the Goldstone particle are related to the corresponding broken symmetry. To obtain a more formal presentation, consider the requirements ∂V (φ J , φJ† ) = 0 ∂φ I and f I = 0 or d(a) = 0 (implying a vacuum that breaks SUSY), and use (3.6) and (3.7) to deduce 2 ∂V J ∂ W f + d(a) φJ† (T a )JI = 0 , ∂φ I ∂φ I ∂φ J

(3.14)

together with the gauge invariance of W (or its complex conjugate version), whence the whole framework can be assembled into a single matrix equation for a basis of the form {λ(a) , χI } [9]: !

⎡ ⎢ M ≡ ⎣!

0 " φL† (T (b) )L J

" ⎤ ( ) φL† (T (a) )L I d(a) # 2 $ ⎥ ⎦ ⇒ (M) · ( J ) = 0 , ∂ W f ∂φ I ∂φ J

(3.15)

This establishes that M has a zero eigenvalue. It is the fermion mass matrix, and the zero eigenvalue is the zero mass Goldstone fermion or goldstino.

3

Here, qi are the U(1) charges of φ I .

3.2

SUSY Breaking

39

Note 3.2 As a final comment, if SUSY is unbroken, all particles within a supermultiplet have the same mass. This will no longer be true if SUSY is (spontaneously) broken. The differences of the squared masses can be related to the SUSY breaking terms (parameters) f I and d(a) . The (super) Higgs effect will be described briefly in Sect. 3.2.2 where SUGRA is relevant.

3.2.2 Spontaneous SUSY Breaking (in SUGRA) In the context of global SUSY, the SUSY vacuum had zero energy and SUSY was (spontaneously) broken if V > 0. This breaking was associated with f I (the auxiliary field in the chiral supermultiplet) or d(a) (the auxiliary field in the vector supermultiplet), acquiring a non-zero vacuum expectation value, with the occurrence in either case of a Goldstone boson and induced goldstino. In the case of local SUSY (and local SUSY means considering the presence of an effective gravity theory induced using SUGRA), in contrast to the case of global SUSY, the vacuum energy is no longer positive semi-definite. In the single gauge-singlet chiral supermultiplet, with GIJ = δJI , from (3.146) in Vol. I [see also (3.154) and (3.155)] we get [contrast with (3.5)] φ∗ φ

V =e

2 ∂W 2 ∗ , ∂φ + φ W − 3 |W|

(3.16)

where it becomes possible to retrieve V < 0. Nevertheless, the analysis proceeds similarly by identifying the fields whose SUSY variation of the vacuum expectation value is not invariant. From the SUSY variations of χ or λ, invoking the fact that derivatives of fields do not receive vacuum expectation values (for Lorentz invariance), we will have δχ −→ 0 | f | 0 , δλ −→ 0 |d| 0 , and SUSY is (spontaneously) broken if the auxiliary fields acquire a nonvanishing expectation value. For the auxiliary fields, we can write4

4 The SUGRA action depends only on the functions G(φ ∗ , φ) and f (a)(b) (φ), whereas in the global SUSY case we had to define the input of K, W, and f. Hence, coupled to gravity, the kinetic function K, and the superpotential W lose their independent meaning and combine to G(φ ∗ , φ), which determines the Kähler manifold with G now the Kähler potential (see Sect. 3.5 of Vol. I).

40


1 1 (a) (b) −1 K JL J f I ∼ eG/2 (G−1 )JW GJ + f(a)(b)K (G−1 )K I λ λ − (G )I GK χJ χL− χI (GJ χ ) , 4 2

∗ i I iI 1 −1 (a) I (b)J (c) I d ∼iRe f(a)(b) −G TI φJ + f(b)(c) χI λ − χ λ(c) − λ(a) (GJ χJ ) . f 2 2 (b)(c) 2 We then note that a vacuum expectation value for the fermion terms therein is now relevant, possibly caused by a gauge interaction force, leading to a vacuum condensation of bilinear fermion–antifermion states. Of course, in the absence of this effect, SUSY breaking will depend on the behaviour of the function G (and f, if present), i.e., the function G j . As described in [1, 6], this is possible (without breaking Lorentz invariance) in the following situations: • No non-zero vacuum expectation values for fermions (but not their variations), i.e., with only scalars or vectors having non-zero vacuum expectation value and hence using δχi −eG/2 (G−1 )JI GJ ε ,

(3.17)

) i δλ(a) Re (f−1 (a)(b) )GI T(b)IJ φJ ε . 2

(3.18)

(

• Or with a gaugino condensate, where the non-zero vacuum expectation value of a product of two fermions (gauginos) is used. Then δχI · · · +

1 ∂ f (a)(b) −1 J λ(a) λ(b) + · · · , G I 8 ∂φ J∗

(3.19)

where an expectation value for λ(a) λ(b) can break SUSY, making δχI noninvariant under SUSY. In somewhat more detail, we examine several cases [1]. Single Gauge-Singlet Chiral Supermultiplet With (3.17) and (3.18), and with G = φ ∗ φ + ln |W |2 , in the single gauge-singlet chiral supermultiplet [see (3.16)], it is found that exp δχI ) δλ(a) = 0 ,

(

1 ∗ φ φ + ln |W|2

∂W∗ 2 ∗ + φW ε, W∗ ∂φ ∗

and SUSY breaking requires

(3.20) (3.21)

3.2

SUSY Breaking

41

∂W + φ ∗ W = 0 , ∂φ

(3.22)

generalizing the f-term SUSY breaking in Sect. 3.2.1. This implies that, in the SUSY vacuum, V −eφ∗φ 3 |W|2 ,

(3.23)

but V = 0 in the SUSY broken phase.5 Gauge Non-singlet Chiral Supermultiplet We now have G(φ † , φ) = φI φI∗ + ln |W|2 and f(a)(b) = δ(a)(b) . The potential is then 1 (a) (b) V ≡ V + GI GK TIJ TKL φJ φL , 2

(3.24)

with V as in (3.16). Equations (3.17) and (3.18) with no non-zero vacuum expectation values for fermions yield (3.20), and in addition ) i δλ(a) = GI T(a)IJ φJ ε . 2

(

(3.25)

The significant point is that we have two mechanisms ) which SUSY (in SUGRA) ( with can be broken, where at least one of δχI = 0 or δλ(a) = 0, i.e., not invariant under a SUSY transformation. Hence, ∂W + φI∗ W = 0 , ∂φI

(3.26)

GI T(a)IJ φJ = 0 ,

(3.27)

where (3.26) generalizes (3.24) and (3.27) generalizes the d-term SUSY breaking in SUGRA.

5

Hence, in SUGRA theories, one can find a mechanism to tune the cosmological constant to any chosen value: SUSY breaks if one or more auxiliary fields are non-zero at the vacuum. At the vacuum with unbroken SUSY, we have a cosmological constant given by 3 ! " Λ ∼ − 4 eG , k hence determining a Minkowski or anti-de Sitter space. Phenomenology requires SUSY to be broken, and also a small cosmological constant. These are non-trivial conditions on G.

42


Note 3.3

From the Lagrangian term in L F2 (see Sect. 3.5 of Vol. I), we have [a]

fI(a)(b) χ I σ μν λ(a) ψ ν γμ λ(b) + h.c. ,

(3.28)

( ) and a vacuum expectation value in Ξ ≡ fI(a)(b) λ(a) λ(b) χI which will mix [a]

with the gravitino in ψ ν . The gravitino can acquire a mass value from Ξ . This is the super Higgs effect. Given the relevance of this feature, let us add some more elements (see [1, 6, 73, 71] for more details).

Note 3.4 In proceeding from global SUSY, with f-term SUSY breaking, towards local SUSY, the goldstino (fermionic partner of the auxiliary field acquiring a vacuum expectation value) would be absorbed by the gauge field associated with this (SUSY) gauge symmetry. In other words, the gravitino becomes massive. A simple example is found as follows. For simplicity, we consider only f-term SUSY breaking. [d-term SUSY breaking (in SUGRA) is also possible. It involves different expressions, but the physical principle is essentially√the same, the goldstino being given by Ξ = GI χI − e−G/2 GI T(a)IJ φJ λ(a) / 2.] We take I (3.29) V = −eG 3 − GI G−1 GJ , J

GK + GJ GJK = 0 GIJ

=

∂V =0, ∂φk

(3.30)

GJ GJ = 3 ⇒ Vvacuum = 0 .

(3.31)

δJI

⇒

Rewriting the gravitino

[a] ψμ

≡

[a] ψμ

√ 2 −G/2 i − √ γμ Ξ − ∂μ Ξ , e 3 3 2

it follows for the mass terms that

1 i G/2 [a] μν [b] 1 G/2 e ψ μ σ ψν − e GI GJ + GIJ χ I χJ . 2 2 3

3.3

N =2 Supersymmetric Quantum Mechanics

43

The gravitino has a mass due to the goldstino fermion Ξ , viz., m 3/2 = eG/2 m P . See Exercise 3.2 for a discussion of the observable hidden sectors and SUSY breaking, e.g., where gaugino condensation occurs in the hidden sector if the gauge group is a product (as in string theory for E 8 × E 8 ) [9].

3.3 N = 2 Supersymmetric Quantum Mechanics This section constitutes a brief review of the essential ingredients of supersymmetric quantum mechanics (SQM), which may provide an interesting background to complement the content of Chap. 6 of Vol. I, Chap. 8 of this volume, and Chap. 5 in both volumes. Skimming through this may provide an enticing flavor of SUSY minisuperspace cosmology. Note 3.5 SQM [19, 22, 29] constitutes by itself a fascinating branch of research, a kind of avant-garde laboratory in which the investigation of some related or lateral features of broader theories (e.g., a field theory) can be probed. In fact, most of SQM, usually constituted by a (complex) bosonic coordinate φ and a two-component complex fermion χ A , can be obtained as a dimensional reduction of supersymmetric field theories in 2D, 4D, or 10D spacetime. The SQM models then describe the dynamics of the zero momentum modes of the field theory. As an example, the corresponding Wess–Zumino model possesses N = 4 supersymmetry (in the form of two complex or four real supersymmetry generators), obtained by dimensional reduction. It must also be said that the realm of application of SQM is quite vast (ranging from atomic and nuclear physics to statistical field theory), as the reader can confirm.

3.3.1 Main Features and Formalism Let us begin with a somewhat modest introduction (see [19, 74]), bearing in mind from the previous chapter the relevance of WKB wave functionals for the universe and the way they may induce the emergence of classical spacetime. At the lowest order in the WKB expansion, the quantization condition

xR xL

* dx 2m[E − V (x)] = (n + 1/2)h¯ π ,

(3.32)

44


with n = 0, 1, 2, . . . , is exact (since the higher order corrections vanish), but only for a few potentials. Could SUSY assist in extending this situation, by constructing a modified WKB scheme for which the quantization condition (in the lowest order approximation) will be exact for a wider class of potentials? The answer is affirmative if V (x) is replaced and a new potential V1 (x) is used to write explicitly

xR xL

+

h¯ (1) 2m E n − W2 (x) + √ W (x) dx = (n + 1/2)h¯ π . 2m

(3.33)

We will elaborate on this in the present section, but there is still much to do in SQM. In basic terms, in a language that comes closer to standard quantum mechanics, we can write −h¯ d A† ≡ √ + W(x) , 2m dx

h¯ d + W(x) , A≡ √ 2m dx

(3.34)

from which the ‘square root’ structure H1 ≡ A† A ,

H2 = AA† ,

(3.35)

emerges, where H1 = −

h¯ 2 d2 + V1 (x) , 2m dx 2

h¯ V1 (x) ≡ W2 (x) − √ W (x) , 2m

(3.36)

H2 = −

h¯ 2 d2 + V2 (x) , 2m dx 2

h¯ V2 (x) ≡ W2 (x) + √ W (x) , 2m

(3.37)

with W(x) the now well known superpotential in the context of SQM. V1,2 (x) are usually referred to as the supersymmetric partner potentials, and the equation above defining them in terms of W is the Ricatti equation (see Sect. 6.1 of Vol. I). Furthermore, we put this SUSY Hamiltonian in the matrix form

H1 0 H≡ 0 H2

,

(3.38)

together with the defined operators S=

0 0 A0

,

S† =

0 A† 00

,

(3.39)

which satisfy the closed (super)algebra [H, S] = [H, S † ] = 0 ,

{S, S † } = H ,

{S, S} = {S † , S † } = 0 . (3.40)

3.3


45

From the above we can identify whether SUSY is spontaneously broken or not. The ground state is written ψ0 ≡

(1)

ψ0

(2)

ψ0

.

(3.41)

Then we note the following: (1) (1) (1) • When Aψ0 = 0, this implies Sψ0 = 0, S † ψ0 = 0, so SUSY is spontaneously broken. (1) (1) (1) • If Aψ0 = 0, then Sψ0 = 0, S † ψ0 = 0 and SUSY is unbroken. (1) • The equation Aψ0 = 0 also provides the following: (1)

– If W(x) is known, e.g., the Ricatti equation can be solved, then Aψ0 = 0 is solved by using (3.34), and establishing the ground state wave function of H1 to be √ 2m x (1) W(y)dy . (3.42) ψ0 exp − h¯ (1)

– And if ψ0 is known, then it follows that

(1) h¯ ψ0 (x) W(x) = − √ . 2m ψ0(1) (x)

(3.43)

Let us elaborate a bit more on the SUSY breaking feature (as displayed above). In this case the operators S and S † must annihilate the vacuum, so that the ground state of the (super)Hamiltonian is zero [see (3.40) above]. Witten proposed a useful tool [67], generally called the Witten index, defined by Δ ≡ Tr(−1)F ,

(3.44)

where the trace is over all bound states of the Hamiltonian and F is the fermion number. The Witten index is also defined as the difference between the number of bosonic states, where (−1)F = +1 and fermionic states, where (−1)F = −1, at zero energy. Note 3.6 In order to understand and use the information that can be extracted from the Witten index, the following points are relevant: • The unique state in the SUSY spectrum is the ground state. Finite energy states are degenerate (they can be further classified). See Exercise 3.3.

46


• For a ‘bosonic’ state, written here in the SQM context as |b, we have F|b = 0, F|Sb = −1, F|S † b = +1, and |S † Sb = 0. • The contributions to (3.44) from all finite energy states cancel in pairs (see Exercise 3.4), leaving any imbalance in the ground state: – If Δ = 0, there are unpaired states, the ground state energy is zero, and SUSY is unbroken. – If Δ = 0, no conclusion can be extracted. SUSY can be broken if both the number of bosonic states and the number of fermionic states are zero, or not broken if both are different from zero. • To be more precise, the eigenspace with eigenvalue +1 (F = 1) is sometimes called the fermionic subspace, whereas the eigenspace with eigenvalue −1 (F = 0) is sometimes called the bosonic subspace. Some authors [75] call them subspaces of positive and negative (Witten) parity p, using −p ≡ (−1)F . Since p± = (1 ± p)/2, we can project from H into subspaces H± , which actually correspond to the H1,2 above [see (3.38) and Exercise 3.4]. – Then define states of (Witten) parity ψn =

(+)

ψn

(−)

ψn

,

(3.45)

where the ± correspond to eigenvalues of (−1)F with ±1. We can take the eigenvalue +1 associated with H1 and the bosonic sector, and the eigenvalue −1 associated with H2 and the fermionic sector. – All bound states are paired (in positive–negative Witten parity pairs) as explained here, except for unbroken SUSY, where there is an extra ground state in the bosonic sector with E = 0. This also illustrates that Δ = 0 for unbroken SUSY. • The points in the above list can be developed as follows (see Exercise 3.4). The energy eigenvalues and the wave functions for H1 and H2 are related. In particular, if E is an eigenvalue of the Hamiltonian H1 (H2 ) with eigenfunction ψ, then the same E is also the eigenvalue of the Hamiltonian H2 (H1 ) and the corresponding eigenfunction is Aψ (A† ψ). This argument is not applicable when Aψ0(1) = 0, i.e., when the ground state is annihilated by the operator A, i.e., when SUSY is unbroken. The exact relationship between the eigenstates of the two Hamiltonians will depend crucially on (1) whether Aψ0 is zero or nonzero, i.e., whether the ground state energy (1) E 0 is zero or nonzero.

3.3


47

• Moreover, if E > 0, as |Sb and |S † b cannot both vanish, the states must come in boson and fermion pairs. When this happens, we can ‘define’ that it is |Sb = 0.

Much more can be said within the SQM framework (the reader is referred to [76, 19, 77, 75]). An alternative way of presenting SQM is through superfields. This we explore in the following for N = 2 SQM (see Sect. 3.3 of Vol. I) [22]. The component form of the corresponding Lagrangian is obviously invariant under the desired SUSY transformations. The superspace for SQM6 is spanned by the coordinates t, η, η, where t is time, while η and its conjugate η are nilpotent Grassmann variables (see Chap. 3 of Vol. I) [26, 78–80]. The N = 2 supersymmetry transformations in this superspace, with the complex odd parameter ε, have the form7 δt ≡ iεη + iεη , δη ≡ ε ,

(3.46)

δη ≡ ε,

(3.47)

which are generated by the linear differential operators Q≡

∂ ∂ + iη ∂η ∂t

and

Q=

∂ ∂ + iη . ∂η ∂t

(3.48)

A real vector superfield Φ X is Φ X ≡ φ X + ηχ X − ηχ X + ηη f X ,

(3.49)

where φ X ≡ q X stands for all the bosonic degrees of freedom of the system (eventually a minisuperspace, in the context of Chap. 2, and also Sect. 2.8 of Vol. I) , χ X and χ X are their fermionic superpartners, and f X is an auxiliary bosonic field. Since the superfield Φ X transforms under the supersymmetry transformations as δΦ X ∼ (ε Q + ε Q)Φ X ,

(3.50)

the most general supersymmetric Lagrangian, with8 6

Recall that this is another superspace, not that of the full function space [i.e., superspace (see Chap. 2)] of all 3-metrics on a spacelike 3-surface embedded in spacetime.

7 Here we introduce the notation as given in [81–85], which is different from the usual notation of SUSY field theory in 2-spinor form. This choice is inherited from a 4-spinor notation (see Note A.6 in Appendix A), and is widely followed in most of the literature referring to the context in Chap. 8.

The Lagrangian can be obtained by dimensional reduction of the N = 1 sigma model in 1 + 1 dimensions.

8

48


D≡

∂ ∂ + iη ∂η ∂t

D≡−

and

∂ ∂ − iη , ∂η ∂t

(3.51)

anticommuting with Q and Q, is L=

1 dηdη − G X Y (DΦ X )(DΦ Y ) + W , 2

(3.52)

which is invariant under SUSY transformations. Note 3.7 This corresponds to the 1D (minisuperspace!) N = 2 SUSY (sigma) model [86, 87] (see Chap. 6 of Vol. I), characterized by the metric G X Y (X, Y = 1, . . . , n) of the target manifold M(Gi j ) and the (minisuperspace) superpotential W, both being functions of the superfield Φ X . It involves n real bosonic variables, which may be thought of as coordinates on an nY dimensional manifold M, and n complex fermions ψ X and ψ . In particular the Lagrangian can be obtained by dimensional reduction of the N = 1 sigma model in 1 + 1 dimensions.

Classical Formulation After integrating over the Grassmann variables and eliminating the auxiliary fields, the Lagrangian (3.52) can be written (see Sect. 6.1.1 of Vol. I and Sect. 8.1.1 of this volume) as L=

1 G X Y (q)q˙ X q˙ Y + iG X Y (q)χ I (χ˙ Y + YZ T q˙ Z χ T ) 2 1 1 + R X Y Z T χ X χ Y χ Z χ T − G X Y (q)∂ X W∂Y W − ∂ X ∂Y Wχ X χ Y , 2 2

(3.53)

where R X Y Z T and YX Z are the Riemann curvature and Christoffel connection, corresponding to the minisuperspace metric G X Y . Then the supercharges (or corresponding SUSY generators when the transformations are local, i.e., constraints) will be S = χ X (G X Y q˙ Y + i∂ X W) ,

(3.54)

S = χ (G X Y q˙ − i∂ X W) .

(3.55)

X

Y

Since the system has bosonic and fermionic degrees of freedom, with πq X , πχ Y , and πχ Y the momenta conjugate to q X , χ Y , and χ J , where πq X = G X Y q˙ X + i Y X Z χ Y χ Z ,

(3.56)

3.3


49

πχ X = −iG X Y χ Y ,

πχ Y = 0 ,

(3.57)

the system possesses second class fermionic constraints (see Appendix B and Sect. 4.1.3 of Vol. I) ζχ X ≡ πχ X + iG X Y χ Y ,

ζχ X ≡ πχ X ,

(3.58)

due to ζχ X , ζχ Y = −iG X Y . From the canonical Poisson brackets χ X , πχ Y = −δYX , q X , πq Y = δYX ,

(3.59)

χ X , πχ Y = −δYX ,

(3.60)

quantization has to be carried out using the Dirac brackets instead:

qa , qb D ≡ qa , qb − qa , ζc

1 ζd , qb . [ζc , ζd ]

(3.61)

Hence, q X , πq Y = δYX , D

χ X, χY

D

= −iG X Y ,

(3.62)

and replacing the Dirac brackets [ , ]D −→

±i[ , ] , ±i{ , } ,

(3.63)

{χ X , χ Y } = G X Y .

(3.64)

the following are obtained: [q X , πq Y ]− = iδYX ,

It is then convenient to introduce the projected (into the minisuperspace if required, see Chap. 6.1.1 of Vol. I for notation) fermionic operators χ x ≡ e xX χ X and χ x ≡ e xX χ X . The momentum conjugate to q X becomes π X ≡ πq X = G X Y q˙ X − iω X x y χ x χ y ,

(3.65)

ω X x y = −exY (∂ X eYy + YX Z e yZ )

(3.66)

where

50


is the corresponding (minisuperspace) spin connection. Regarding a choice of operator ordering, we follow [88, 78–80, 89]: S ≡ χ x exX (π X + iω X x y χ x χ y + i∂ X W ) ,

(3.67)

S = χ x exX (π X + iω X x y χ x χ y − i∂ X W ) ,

(3.68)

S = χ x exX (π˘ X + i∂ X W) ,

(3.69)

S ≡ χ x exX (π˘ x − i∂ X W) ,

(3.70)

π˘ X ≡ π X + iω X x y χ x χ y

(3.71)

or simply

where the operator

obeys, e.g., [q X , π˘ q Y ] = iδYX ,

[π˘ q X , π˘ q Y ] = −R X Y Z T χ Z χ T .

(3.72)

Now using (3.65), (3.66), (3.67), and (3.68), consider a classical Hamiltonian, which is actually a constraint: H −→ H0 =

1 XY 1 G π X πY + G X Y (q)∂ X W∂Y W = 0 . 2 2

(3.73)

In the quantum case this should be replaced by the condition on the quantum state Ψ , viz., HΨ = 0 ,

(3.74)

with the extended Hamiltonian H −→ H =

1 {S, S} , 2

(3.75)

giving H0 in the classical limit, i.e., when all fermionic fields are set equal to zero. In more detail, we also have, * 1 π˘ X det(G X Y )G X Y (q)π˘ Y H= √ 2 det(G X Y )

(3.76)

1 1 1 − R X Y Z T χ X χ Y χ Z χ T + G X Y (q)∂ X W∂Y W + ∂ X ∂Y W[χ X , χ Y ] . 2 2 2

3.3


51

Quantum States Two first order differential equations follow: SΨ = 0 ,

SΨ = 0 .

(3.77)

In order to describe the space of solutions, a useful framework (although not explored in this book) is that of a Fock space, spanned now by the fermionic creation and annihilation operators χ x and χ x , respectively, with {χ x , χ y } = η x y . The general state Ψ ≡| Ψ is then expanded as 1 x1 χ . . . χ xn Fx1 ...xn (q)|0 n! 1 = F(q)|0 + · · · + χ X 1 . . . χ X n F X 1 ...X n (q)|0 , n!

|Ψ = F(q)|0 + · · · +

(3.78) (3.79)

where the coefficients F(q) to Fx1 ...xn in the expansion of this series are p-forms defined on the manifold M(G X Y ) [90, 91], and due to the nilpotency of the fermionic creation operators, their number is finite.9 Since the fermion number operator F = χ x χx commutes with the Hamiltonian H and [F, S] = −S ,

[F , S] = S ,

(3.82)

one can consider states characterized by the different fermion numbers separately [88, 19, 67]. Let us now search for zero-energy solutions of H (3.73), (3.74), and (3.75) in the empty and filled fermionic sectors of a suitable Fock space. These states in the empty sector are like | f 0 = |0 f 0 (q), and in the filled sector we have | f n = |Ψn f n (q). So the corresponding solutions in empty and filled fermion sectors are simply expressed in terms of the superpotential W, with f 0,n = e∓W , or in more detail as follows: |Ψ0 e−W |0 , |Ψn

9

1 x1 χ . . . χ xn x1 ...xn e+W |0 . n!

(3.83) (3.84)

We define the fermion Fock vacuum by |0 and the completely filled fermion state by |Ψn , where χ x |0 = 0 ,

χ x |Ψn = 0 for all x ,

(3.80)

the former being trivially annihilated by S and the latter by S . Note also that [see (3.79) above] 1 x ...x χ x1 . . . χ xn |0 n! 1 n 1* = | det(G X Y )| X 1 ...X n χ X 1 . . . χ X n |0 . n!

|Ψn =

(3.81)

52


In fact, the effective bosonic Hamiltonians in the empty and filled sectors have the form 1 1 1 H± = − ∇ X ∇ X + ∇ X W∇ X W ∓ ∂ X ∂ X W 2 2 2 1 = − (∇ X ∓ i∇ X W) ∇ X ± i∇ X W . 2

(3.85)

These are just the Fokker–Planck Hamiltonians on the manifold M (see also Sect. 3.4), which have the zero-energy solutions e∓W . Note 3.8

In addition, note the following points:

• These are bosonic wave functions for supersymmetric ground states, constituting solutions that are normalizable and non-singular. • The existence of normalizable solutions of the system (3.77) means, in turn, that supersymmetry is unbroken quantum mechanically [78–80, 89]. This allows one to determine the physical states • In addition, it is interesting that there is a tendency for supersymmetric vacua to remain close to their semi-classical limits. The solutions e∓W are also the lowest order WKB approximations, being exact solutions of the zero-energy Hamilton–Jacobi equation (∇ X ln Ψ )2 = (∇ X W)2 ,

(3.86)

with all corrections to the state at high orders (when the Planck constant is introduced) vanishing. A normalizable solution of this Hamilton–Jacobi equation must exist in the semi-classical limit of any model with unbroken supersymmetry.

3.3.2 SQM, Topology, and Vacuum States We shall now turn to a few paragraphs of a more mathematical nature [88], but with significant potential implications regarding SQC in general, although much research remains to done (see Chap. 10): • When the manifold M is connected, the zero-energy solutions e∓W are the unique zero-energy states in the empty and filled sectors, providing supersymmetric ground states if normalizable and non-singular. • On non-compact manifolds or with a singular superpotential, only one or even neither may be present. • This remarkable link between the supersymmetry vacuum and topology is related to the choice of operator ordering in S, S.

3.3


53

In more detail, the states of the model are in a one-to-one correspondence with the differential p-forms on the manifold M. For any state with p fermions and wave function f X 1 ...X f (q), defining 1 f X ...X (q)χ X 1 . . . χ X f |0 , p! 1 f

|f =

(3.87)

with completely antisymmetric f X 1 ...X f , there is a unique p-form f ≡ dq X 1 ∧ . . . ∧ dq X p

1 f X ...X (q) . p! 1 p

(3.88)

In addition, we also have πμ |f ≡

1 X1 χ . . . χ X f |0(−i)f X 1 ...X f ;X , p!

(3.89)

and therefore 1 χ X 1 . . . χ X p−1 |0 (−ih¯ ∇ X + i∇ X W) f X X 1 ...X f −1 , ( p − 1)!

1 ∂ ˜ = S|f χ X 1 . . . χ X p+1 |0 −ih¯ X − i∇ X 1 W f X 2 ...X f +1 ( p + 1)! ∂q 1 +cyclic permutations .

S|f =

(3.90)

When the superpotential W is zero, these equations reduce to the actions of the co-exterior derivative δ and the exterior derivative d : 1 ∇ X f XX 1 ...X f −1 , ( p − 1)!

1 ∂ df = dq X 1 ∧ . . . ∧ dq X p+1 f + cyclic permutations , X ...X ( p + 1)! ∂q X 1 2 f +1 (3.91) in the sense that δf = −dq X 1 ∧ . . . ∧ dq X p−1

S|f ≡ i|δf ,

S|f = −i|df .

(3.92)

As a consequence, note the following points: • The supersymmetric vacua for W = 0 coincide with the harmonic p-forms defined by δf = df = 0. • The space of these harmonic forms is isomorphic to the p th de Rham cohomology group H p (M), i.e., the equivalence classes of closed but not exact p-forms (those where df = 0 but f = dg), differing by an exact form, e.g., dg, from a ( p − 1)-form g.

54


• A zero-energy state |f must have S|f = 0 and cannot be written as |f ≡ S|g. This would imply f|f = g|S|f = 0. • If |f is such that S|f = 0 and |f cannot be written as S|g, then either H|f = 0

or |˜f = (2H − SS)/2H |f is a well defined non-zero state with S|˜f = 0, S|˜f = 0, and |˜f, the unique non-zero energy state differing from |f by a state of the form S|g. Consequently, the number of supersymmetric vacua with p fermions is given by the p th Betti number b p = dim (H p (M)) of the manifold M, related to the Gauss– , Bonnet index theorem and Euler characteristic χ (M) = p (−1) p b p = Tr(−1)F . Note 3.9 The interesting point is that for any compact connected manifold b0 = bn = 1, and therefore supersymmetric vacua always exist (when W eventually vanishes). Then H 0 (M) is a constant and in H n (M) they are proportional to the volume form dq X 1 ∧ . . . ∧ dq X n

1* det(G X Y ) X 1 ...X n (q) , n!

(3.93)

corresponding to the vacua |0, |Ψn previously established when W goes to zero. Solutions |0e−W , |Ψn e+W when W = 0 follow from S = e−W S 0 eW and S = eW S0 e−W , with S (S) annihilating every state in the empty (filled) sector. By way of summarizing this important feature, where the search for supersymmetric vacua is translated into a problem of topology, let us go into a little more detail: 1. For solutions in other fermion sectors, in the case of a vanishing superpotential, the corresponding operators S 0 and S0 act on the p-forms f as exterior and co-exterior derivatives, respectively. 2. Thus, a solution of the equation S(W = 0) ≡ S 0 |Ψ = 0 cannot be written as |Ψ p = S 0 |Ψ p−1 ,

(3.94)

except in the case where the corresponding p th cohomology group H p (M) of the manifold M(G X Y ) is nontrivial. 3. Noting that S = e−W S 0 eW ,

S = eW S0 e−W ,

(3.95)

and using (3.94) and (3.95), it can be proved that the general solution in pfermion sectors ( p = 1, . . . , n − 1) for the case of the trivial cohomology group H p (M) is |Ψ p = S|Ψ p−1 .

(3.96)

3.4

Nicolai Maps and SQM

55

4. However, because S and S are Hermitian adjoints of one another, the second equation in (3.77) indicates that this state has zero norm, and so is unphysical. 5. Therefore, the possible existence of supersymmetric ground states, i.e., solutions of a zero-energy Schrödinger-type equation, is directly related to the topology of the considered (minisuperspace) manifold M(G X Y ), since all states except those in purely bosonic and filled fermion sectors can be excluded even without solving the system (3.77), if the topology of the manifold M(G X Y ) is trivial.

3.4 Nicolai Maps and SQM Throughout this book the reader will encounter plenty of SUSY features. However, in close relation with the topological invariants as well as application of Witten index properties and identification of vacuum states in SQM (and then in SQC), it is crucially important to mention that any supersymmetric theory can be transformed into a free bosonic theory by a transformation whose Jacobian determinant is exactly cancelled by the integral of the fermionic variables [92–96]. This transformation is called a Nicolai map. It allows one to ‘map’ a supersymmetric theory into an (equivalent) simpler theory with only non-interacting bosonic fields. The integration over the fermionic variables is somewhat the same as in the computation of the Witten index where one traces out the bound states of the Hamiltonian. Of course, when SUSY is broken, one cannot use the framework of the Nicolai map. However, when supersymmetry is broken by the presence of boundary terms in the action (e.g., with the variation of the supersymmetric action producing a boundary term), if we restore SUSY, or better still the invariance of the action under a non-trivial subalgebra of SUSY generators, it is still possible to ensure the existence of a Nicolai map. This can be done by adding a suitable boundary correction term. In other words, a Nicolai map will exist provided that the boundary conditions we impose are invariant under this subalgebra.10 It is important to refer that the strict applicability of a Nicolai map is only valid in the Euclidean formulation of a supersymmetric theory. It becomes a stochastic differential equation, namely a Fokker–Planck equation in Euclidean time [see (3.85)]. In order to allow for Lorentzian metrics, a Wick rotation of the time axis by replacing t by −iτ , where τ is the Euclidean time, is necessary. The interest of Nicolai maps in QC and SQC is that, when specific boundary conditions are imposed, only certain quantum states can actually be generated by Nicolai maps. In QC (and in SQC it would come naturally), we can form a wave function of the Universe consisting of linearly independent components corresponding to quantum states, each satisfying different boundary conditions. And because Nicolai maps are interpreted as stochastic processes, this suggests a probabilistic interpretation of certain components of the wave function.

10 Boundaries (e.g., 3-manifolds) are important when dealing with quantum amplitudes, and so cannot be neglected in off -shell situations (see Note 2.3 in Vol. I).

56


Nicolai maps for N = 2 supersymmetric extensions of (cosmological) minisuperspace were constructed in [93]. The Nicolai maps exist for only a very restricted set of states and these are the two states corresponding to the empty and the filled fermion sectors retrieved in Chap. 6 of Vol. I. Only one is normalizable and the reader should recall the comments above for the application of the Witten index in SQM. Since it has barely been explored in SQC [93, 97, 98, 96, 92, 99–102], the relation of SQM to Nicolai maps [92, 99–102] deserves to be described. A Nicolai map works in the following way. Consider a system with n bosonic variables bi ˙ f, f˙) = L 1 (b, b) ˙ + and m Grassmann variables f j , with a Lagrangian L(b, b; ˙ ˙ L 2 (b, b; f, f ). Supersymmetry of L means there is a transformation of variables ˙ b to (b) reducing L 1 to a quadratic form (apart from a total derivative) L 0 (b, b). SQM constitutes a particular case of the Nicolai map, when it is local. Moreover, when it is local in time, the map can be considered as a local stochastic differential equation relating a Gaussian stochastic process B to B. In particular, a Fokker– Planck equation (3.85) can be obtained for functions of the form e∓W ϑ(q, τ ) (where τ is the Euclidean time), establishing a scalar probability density for a Markov process. Summary and Review. Here is a list of review points (within the context of this book) in order for the reader to assess his or her progress: 1. How can (spontaneous) SUSY breaking be achieved? That is, what are the conditions on the fields and what possible mechanisms exist [Sects. 3.1 and 3.2]? 2. And if SUGRA is present [Sect. 3.2.2]? 3. Are there features common to SQM and SQC [Sect. 3.3.1]? 4. And how does SQM benefit from a topological perspective [Sect. 3.3.2]? 5. What exactly is a Nicolai map and why can it prove useful in SQC [94– 96] [Sect. 3.4]?

Problems 3.1 SUSY Breaking Conditions Describe how the minimum of a potential is positive when SUSY is broken. 3.2 No-Scale SUGRA Investigate (see [1]) the Kähler class of potentials for which the effective potential for the hidden sector is flat, i.e., constant. 3.3 Degeneracy of Finite Energy States Establish that finite energy states are degenerate in SQM.

References

57

3.4 Pair States Show that if E is an eigenvalue of the Hamiltonian H1 (H2 ) with eigenfunction ψ, then the same E is also the eigenvalue of the Hamiltonian H2 (H1 ) and the corresponding eigenfunction is Aψ (A† ψ).

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25. Deotto, E., Gozzi, E., Mauro, D.: Supersymmetry in classical mechanics. hep-th/0101124 (2001) 35 26. Donets, E.E., Pashnev, A., Rivelles, V.O., Sorokin, D.P., Tsulaia, M.: N = 4 superconformal mechanics and the potential structure of AdS spaces. Phys. Lett. B 484, 337–346 (2000) 35, 47 27. Faux, M., Spector, D.: Duality and central charges in supersymmetric quantum mechanics. Phys. Rev. D 70, 085014 (2004) 35 28. Faux, M., Kagan, D., Spector, D.: Central charges and extra dimensions in supersymmetric quantum mechanics. hep-th/0406152 (2004) 35 29. Freedman, B., Cooper, F.: Fun with supersymmetric quantum mechanics. Physica D 15, 138 (1985) 35, 43 30. Gamboa, J., Ramirez, C.: Hamiltonian approach to 2D supergravity. Phys. Lett. B 301, 20–24 (1993) 35 31. Gamboa, J., Zanelli, J.: Supersymmetric nonrelativistic quantum mechanics. Phys. Lett. B 165, 91–93 (1985) 35 32. Gamboa, J., Zanelli, J.: Supersymmetric quantum mechanics of fermions minimally coupled to gauge fields. J. Phys. A 21, L283–L286 (1988) 35 33. Gamboa, J., Zanelli, J., Ruiz-Altaba, M.: Supersymmetrization of scalar field theories. Phys. Lett. B 206, 252 (1988) 35 34. Gamboa, J., Zanelli, J.: Ground state and supersymmetry of generally covariant systems. Ann. Phys. 188, 239 (1988) 35 35. Gozzi, E.: Ground state wave function ‘representation’. Phys. Lett. B 129, 432 (1983) 35 36. Gozzi, E.: The Onsager principle of microscopic reversibility and supersymmetry. Phys. Rev. D 30, 1218 (1984) 35 37. Gozzi, E.: On the nodal structure of supersymmetric wave functions. Phys. Rev. D 33, 3665 (1986) 35 38. Gozzi, E.: Comment on ‘on the hidden supersymmetry in stochastic quantization’. Phys. Rev. D 44, 3994–3996 (1991) 35 39. Gozzi, E.: Universal hidden supersymmetry in classical mechanics and its local extension. hep-th/9703203 (1997) 35 40. Grosse, H.: Supersymmetric quantum mechanics. Lectures given at Brasov International School on Recent Developments in Quantum Mechanics, Poiana Brasov, Romania, 29 August–9 September 1989 35 41. Henneaux, M., Teitelboim, C.: Relativistic quantum mechanics of supersymmetric particles. Ann. Phys. 143, 127 (1982) 35 42. Hernandez, R., Sfetsos, K.: Supersymmetric quantum mechanics from wrapped branes. Phys. Lett. B 582, 102–112 (2004) 35 43. Heumann, R., Manton, N.S.: Classical supersymmetric mechanics. Ann. Phys. 284, 52–88 (2000) 35 44. Macias, A., Obregon, O., Socorro, J.: Supersymmetric quantum cosmology. Int. J. Mod. Phys. A 8, 4291–4317 (1993) 35 45. Ivanov, E., Lechtenfeld, O.: N = 4 supersymmetric mechanics in harmonic superspace. JHEP 09, 073 (2003) 35 46. Ivanov, E.A., Krivonos, S.O., Pashnev, A.I.: Partial supersymmetry breaking in N = 4 supersymmetric quantum mechanics. Class. Quant. Grav. 8, 19–40 (1991) 35 47. Jackiw, R., Polychronakos, A.P.: Supersymmetric fluid mechanics. Phys. Rev. D 62, 085019 (2000) 35 48. Jaffe, A., Lesniewski, A., Lewenstein, M.: Ground state structure in supersymmetric quantum mechanics. Ann. Phys. 178, 313 (1987) 35 49. Kihlberg, A., Salomonson, P., Skagerstam, B.S.: Witten’s index and supersymmetric quantum mechanics. Z. Phys. C 28, 203–209 (1985) 35 50. Kostelecky, V.A., Nieto, M.M.: Evidence for a phenomenological supersymmetry in atomic physics. Phys. Rev. Lett. 53, 2285 (1984) 35

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51. Kostelecky, V.A., Nieto, M.M., Truax, D.R.: Supersymmetry and the relationship between the Coulomb and oscillator problems in arbitrary dimensions. Phys. Rev. D 32, 2627 (1985) 35 52. Kotanski, J., Wosiek, J.: Hamiltonian study of supersymmetric Yang–Mills quantum mechanics. Nucl. Phys. Proc. Suppl. 119, 932–934 (2003) 35 53. Lancaster, D.: Supersymmetry breakdown in supersymmetric quantum mechanics. Nuovo Cim. A 79, 28 (1984) 35 54. Quesne, C., Tkachuk, V.M.: More on a SUSY QM approach to the harmonic oscillator with nonzero minimal uncertainties in position and/or momentum. math-ph/0312029 (2003) 35 55. Rietdijk, R.H.: Supersymmetric quantum mechanics and the index theorem. J. Geom. Phys. 11, 545–551 (1993) 35 56. Salomonson, P., van Holten, J.W.: Fermionic coordinates and supersymmetry in quantum mechanics. Nucl. Phys. B 196, 509 (1982) 35 57. Shabanov, S.V.: On minisuperspace gauge models with fermions. In: Proceedings on Classical and Quantum Gravity, Evora, 1992, pp. 322–331. World Scientific, River Edge, NJ (1993) 35 58. Soroka, D.V., Soroka, V.A., Wess, J.: Supersymmetric D = 1, N = 1 model with Grassmann-odd Lagrangian. Phys. Lett. B 512, 197–202 (2001) 35 59. Soroka, V.A.: On square root of Hamilton’s equations for supersymmetric systems. J. Experiment. Theor. Phys. Lett. 60, 387–392 (1994) 35 60. Soroka, V.A.: Supersymmetry and the odd Poisson bracket. Nucl. Phys. Proc. Suppl. 101, 26–42 (2001) 35 61. Szwed, J.: The ‘square root’ of the Dirac equation within supersymmetry. Phys. Lett. B 181, 305 (1986) 35 62. Szwed, J.: The ‘square root’ of the Dirac equation and solutions on superspace. hepth/0403036 (2004) 35 63. Tanimoto, M.: The role of Killing–Yano tensors in supersymmetric mechanics on a curved manifold. Nucl. Phys. B 442, 549–562 (1995) 35 64. Urrutia, L.F., Zanelli, J.: Local supersymmetry in nonrelativistic systems. J. Math. Phys. 31, 2271–2277 (1990) 35 65. Volkov, D.V., Pashnev, A.I., Soroka, V.A., Tkach, V.I.: Hamiltonian systems with even and odd Poisson brackets. Duality of their conservation laws. JETP Lett. 44, 70–72 (1986) 35 66. Witten, E.: Dynamical breaking of supersymmetry. Nucl. Phys. B 188, 513 (1981) 35 67. Witten, E.: Constraints on supersymmetry breaking. Nucl. Phys. B 202, 253 (1982) 35, 45, 51 68. Wosiek, J.: Spectra of supersymmetric Yang–Mills quantum mechanics. Nucl. Phys. B 644, 85–112 (2002) 35 69. Wosiek, J.: Recent progress in supersymmetric Yang–Mills quantum mechanics in various dimensions. Acta Phys. Polon. B 34, 5103–5118 (2003) 35 70. Znojil, M.: Relativistic supersymmetric quantum mechanics based on Klein–Gordon equation. J. Phys. A 37, 9557–9572 (2004) 35 71. Wess, J., Bagger, J.: Supersymmetry and Supergravity, 259pp. Princeton University Press, Princeton, NJ (1992) 35, 37, 42 72. West, P.C.: Introduction to Supersymmetry and Supergravity, 425pp. World Scientific, Singapore (1990) 35, 37 73. Van Nieuwenhuizen, P.: Supergravity. Phys. Rep. 68, 189–398 (1981) 42 74. Khare, A., Maharana, J.: Supersymmetric quantum mechanics in one dimension, two dimensions, and three dimensions. Nucl. Phys. B 244, 409 (1984) 43 75. Junker, G.: Supersymmetric Methods in Quantum and Statistical Physics. Texts and Monographs in Physics, 172pp. Springer, Heidelberg (1996) 46, 47 76. Cooper, F., Khare, A., Sukhatme, U.: Supersymmetry in Quantum Mechanics, 210pp. World Scientific, Singapore (2001) 47 77. Cooper, F., Khare, A., Sukhatme, U.: Supersymmetry and quantum mechanics. Phys. Rep. 251, 267–385 (1995) 47 78. Donets, E.E., Pashnev, A., Rosales, J.J., Tsulaia, M.: Partial supersymmetry breaking in multidimensional N = 4 SUSY QM. hep-th/0001194 (1999) 47, 50, 52

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79. Donets, E.E., Pashnev, A., Juan Rosales, J., Tsulaia, M.M.: N = 4 supersymmetric multidimensional quantum mechanics, partial SUSY breaking and superconformal quantum mechanics. Phys. Rev. D 61, 043512 (2000) 47, 50, 52 80. Donets, E.E., Pashnev, A.I., Tsulaia, M.M., Sorokin, D.P., Rivelles, V.O.: Potential structure of AdS spaces. Czech. J. Phys. 50, 1215–1220 (2000) 47, 50, 52 81. Obregon, O., Rosales, J.J., Tkach, V.I.: Superfield description of the FRW universe. Phys. Rev. D 53, 1750–1753 (1996) 47 82. Tkach, V.I., Obregon, O., Rosales, J.J.: FRW model and spontaneous breaking of supersymmetry. Class. Quant. Grav. 14, 339–350 (1997) 47 83. Tkach, V.I., Rosales, J.J.: Supersymmetric action for FRW model with complex matter field. gr-qc/9705062 (1997) 47 84. Tkach, V.I., Rosales, J.J., Martinez, J.: Action for the FRW model and complex matter field with local supersymmetry. Class. Quant. Grav. 15, 3755–3762 (1998) 47 85. Tkach, V.I., Rosales, J.J., Obregon, O.: Supersymmetric action for Bianchi type models. Class. Quant. Grav. 13, 2349–2356 (1996) 47 86. Duplij, S., Siegel, W., Bagger, J. (eds.): Concise Encyclopedia of Supersymmetry and Noncommutative Structures in Mathematics and Physics. Kluwer Academic Publishers, Dordrecht (2003) 48 87. Lidsey, J.E., Wands, D., Copeland, E.J.: Superstring cosmology. Phys. Rep. 337, 343–492 (2000) 48 88. Claudson, M., Halpern, M.B.: Supersymmetric ground state wave functions. Nucl. Phys. B 250, 689 (1985) 50, 51, 52 89. Donets, E.E., Tentyukov, M.N., Tsulaia, M.M.: Towards N = 2 SUSY homogeneous quantum cosmology: Einstein–Yang–Mills systems. Phys. Rev. D 59, 023515 (1999) 50, 52 90. DeWitt, B.S.: Supermanifolds. Cambridge Monographs on Mathematical Physics, 2nd edn., pp. 1–407. Cambridge University Press, Cambridge (1992) 51 91. Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation, 1279pp. Freeman, San Francisco (1973) 51 92. Farajollahi, H., Luckock, H.: Stochastic quantisation of locally supersymmetric models. grqc/0406022 (2004) 55, 56 93. Graham, R., Luckock, H.: Nicolai maps for quantum cosmology. Phys. Rev. D 49, 2786–2791 (1994) 55, 56 94. Luckock, H.: Boundary conditions for Nicolai maps. J. Phys. A 24, L1057–L1064 (1991) 55, 56 95. Luckock, H.: Boundary terms for globally supersymmetric actions. Int. J. Theor. Phys. 36, 501–508 (1997) 55, 56 96. Luckock, H., Oliwa, C.: The cosmological probability density function for Bianchi class A models in quantum supergravity. Phys. Rev. D 51, 5483–5490 (1995) 55, 56 97. Graham, R., Roekaerts, D.: Stochastic representation of arbitrary fermion sectors in supersymmetric quantum mechanics in flat space. Phys. Lett. A 120, 223 (1987) 56 98. Luckock, D., Oliwa, C.: Quantisation of Bianchi class A models in supergravity and the probability density function of the universe. In: 7th Marcel Grossmann Meeting on General Relativity (MG 7), Stanford, CA, 24–30 July 1994 56 99. Graham, C., Roekaerts, D.: A Nicolai map for supersymmetric quantum mechanics on Riemannian manifolds. In: Proceedings on Stochastic Processes—Mathematics and Physics, Bielefeld, pp. 98–105 and BIBOS 082 (86, REC.AUG.), 8pp. Bielefeld University, Bielefeld (1985) 56 100. Graham, R., Roekaerts, D.: Supersymmetric quantum mechanics and stochastic processes in curved configuration space. Phys. Lett. A 109, 436–440 (1985) 56 101. Graham, R., Roekaerts, D.: Stochastic representation of arbitrary fermion sectors in supersymmetric quantum mechanics on Riemannian manifolds. Phys. Rev. D 34, 2312 (1986) 56 102. Graham, R., Roekaerts, D.: Stochastic representation of arbitrary fermion sectors in supersymmetric quantum mechanics in flat space. Phys. Lett. A 120, 223 (1987) 56

Chapter 4

Semiclassical N=1 Supergravity

A promising line of investigation, following the framework of Chap. 4 in Vol. I, is to try to retrieve (quantum) physical states in canonical quantum N = 1 SUGRA. We will discuss this in Sect. 4.1, where the Carroll–Freedman–Ortiz–Page (CFOP) argument and demonstration are presented [1–3]. In Sect. 4.2, a semiclassical framework for SUGRA is then extracted from the canonical quantization presented in Vol. I (see [4]). It is at this point that we must first face the challenge of determining whether there is an imprint of an early stage of quantum and SUSY dominance on the observable features of the universe. That is, we are asking whether and how a transition might be possible from a universe wholly described by quantum mechanics and SUSY elements to today’s observed structure. In a broader context, the aim is to carry a supersymmetric theory of quantum gravity, present within the framework of superstrings, into a potential testing ground. The main feature is that the canonical quantization of N = 1 SUGRA still offers plenty to explore, in order to further clarify the subject. The following sections constitute starting points from which to continue the investigation.

4.1 Physical States in N=1 SUGRA The starting point here is as follows. A quantum state of SUSY quantum gravity is a wave functional Ψ defined on the space of all tetrad and gravitino fields (and possibly other fields) on a spatial hypersurface Σ. As introduced in [4], we shall call this space SuperRiem(Σ), extending from the Riem(Σ) space in canonical quantum gravity (see Sects. 2.2, 2.5, and 2.6 of Vol. I). Early attempts suggested that bosonic and finite fermion number states were possible [5], but subsequent analysis has suggested otherwise [6, 10] (see also [11, 12]). Overall, the discussion has been beneficial, identifying several problems regarding some attempts which advanced purely bosonic solutions [13–15, 2, 16]. For an additional appraisal, where investigations probed another direction, see [17, 19]. In this section, we outline the arguments of [6, 7, 10] that physical quantum states in pure N = 1 SUGRA can only have an infinite number of fermions. The Moniz, P.V.: Semiclassical N =1 Supergravity. Lect. Notes Phys. 804, 61–83 (2010) c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-11570-7_4

62

4 Semiclassical N=1 Supergravity

form of the supersymmetry constraints is the central issue. Since these constraints are homogeneous in the gravitino field ψ Ai (x), it is consistent to look for solutions involving homogeneous functionals of order (ψ)n , e.g., n = 2 means quadratic and n = 4 quartic. The essential result is the CFOP argument: • For n = 0, i.e., pure bosonic state Ψ ≡ Ψ [e], a simple scaling argument assuredly excludes the purely bosonic states. This constitutes the first part of the result. • For states with a finite, non-zero number of fermions, the argument is less strong. However, an extension of the formulation of the CFOP theorem was recently presented in [4], although in another physical context (see Sect. 4.2). Here it was proved in a complete (and possibly stronger version) that there can only be an infinite number of fermions in a physical state that satisfies the quantum constraints J AB , J A B , S A , and S A (see Exercise 4.1).

4.1.1 Bosonic (n=0) States Let us consider Lorentz invariant states. , An arbitrary state can be expanded in a finite power series as Ψ [e A A i , ψ Ak ] = n Ψ (n) [e A A i , ψ Ak ]. Note that odd n states need not be considered because they are not local Lorentz invariant. The constraint equations must be satisfied independently by each term Ψ (n) [e A A i , ψ Ak ]. Then take the case where Ψ [e A A i , ψ Ai ] ≡ Ψ (0) [e A A i ] is a functional only of the tetrad, that is, δΨ (0) /δψ Ai = 0. We shall refer to states of this type as bosonic

states. In this case, any Lorentz invariant state satisfying S A Ψ (0) = 0 automatically satisfies all other constraints, since S A Ψ (0) vanishes identically. But no such solution can exist. In fact, the supersymmetry constraint can be written in the form (see Chap. 4 of Vol. I)

Ψ

(0)

−1

(0)

SΨ

≡ −ε

i jk i

e

A A (

D j ψkA A ) +

3s

δ ln Ψ (0) h¯ k2 =0. ψi A A

2 δeiA A

(4.1)

A contradiction occurs by using an integrated form of (4.1) with an arbitrary continuous spinor test function ε(x):

d xε(x) −ε 3

i jk i

e

A A (

D j ψkA A ) +

3s

δ ln Ψ (0) h¯ k2 ψi A A

=0,

2 δeiA A

(4.2)

for all ε(x), e A A i (x), and ψkA (x). Denote the integral in (4.2) by I , and let I = I + "I be the integral when ε(x) is replaced by ε(x)e−φ(x) and ψiA (x) is replaced by ψiA (x)eφ(x) , where φ(x) is a

4.1

Physical States in N =1 SUGRA

63

scalar function. Since ε(x)ψiA (x) is unchanged, the second term (with the functional derivative) cancels in the difference between I and I , so that "I = −

d3 xεi jk eiA A (x)ε A (x)ψk A (x)∂ j φ(x) .

(4.3)

Notice that "I is independent of the state Ψ (0) . Clearly, it is possible to choose the

arbitrary fields ε(x), φ(x), eiA A (x), and ψk A (x) such that (4.3) is nonvanishing. If "I = 0, we cannot have both I = 0 and I = 0, so (4.1) cannot be satisfied for all

ε(x), e A A i (x), and ψkA (x). We may conclude that bosonic wave functionals are inconsistent with the supersymmetry constraints of pure N = 1 supergravity.

4.1.2 States with Finite Fermion Number In [6, 7], it is also suggested that there may be no states involving a finite nonzero number of fermion fields. The argument is based on the mode expansion of the gravitino field on the quantization surface. However, explicit solutions of the mode equations are only found when the hypersurface is flat. A general proof remains to be established. The assumptions and claims are satisfied in the free spin 3/2 field case, where it has been shown that all wave functionals necessarily contain an infinite number of fermion fields. Hence, all quantum wave functionals of N = 1 supergravity with a finite (zero or nonzero) fermion number would be inconsistent with the supersymmetry constraints. From the free spin 3/2 field case, it has been shown that all wave functionals necessarily contain an infinite number of fermion fields. In essence, the conclusions in [6, 7, 10] suggest that the claims in [13–15, 2, 5, 16], namely that N = 1 SUGRA is finite to all orders, should be revised. Moreover, recent results in [17–19] seem to further establish that N = 1 SUGRA with boundary terms is fully divergent even at one-loop order.

4.1.3 Solutions with Infinite Fermion Number To end this section, we review the method adopted in [8], which allowed for an exact quantum solution of the full theory, in agreement with the CFOP argument. The new elements are the commutators

(4.4) H A A (x), S B (y) = −ih¯ δ(x − y)ε AB D A B C (x)J B C (x) ,

H A A (x), S B (y) = ih¯ δ(x − y)ε A B D A BC (x)J BC (x) (4.5) .

+ih¯ δ(0) E A C D (x)J C D (x) − n AC (x)h −1/2 (x)S C (x) ,

64


and the functionals of ei a and ψ p A :

D A BC ≡ n b ei c ε B D (σb σ c )C D h −1/2 δ i j ε AE ψ j E + εi jk σa A A ClAj F ε F E ψk E D(e, p) , (4.6)

E AC

D

i

n a eib + n b eia ε AC σ a C C σ b D B ψi B , 2h 1/2

(4.7)

where D(e, p) is a functional of the tetrad and the canonical momentum, and D BA C and E A C D are the (matrix) adjoints of these expressions. The physical state has the form Ψ = Π(x) S A (x)S A (x)υ(ei B B ) .

(4.8)

It contains a formal product over all space points, with a bosonic functional υ satisfying the Lorentz constraints. The S B constraint and the J AB , J A B constraints are automatically satisfied. Using the generator algebra and the properties of υ, the S B

constraint is satisfied if

S A S A υ eiB B = 0 . (4.9)

It is important to note that the operators eiA A S A S A , and n A A S A S A are Lorentz invariant. Remarkably, (4.9) does have a solution in the inhomogeneous case, given through the functional υ 0 :

1 i A A

3 i jk A A

) = exp − d x ε ei ∂ j ek A A . (4.10) υ 0 (e 2h¯ There is, however, a crucial step. When J A B υ 0 = 0 is satisfied, we also have J AB υ 0 = 0. A fully Lorentz invariant amplitude υ is obtained from υ 0 with (4.11) g = Dμ[ω] exp iω AB J AB g0 , where Dμ[ω] is an appropriate measure. The infinite product in (4.8) is written as a Grassmannian path integral over a Grassmann field ε A (x). Applying the factors S A (x) explicitly to the functional υ, an exact physical state Ψ is found to be A

Ψ [h i j , ψ i ] =

A

Dε1 Dε2 exp − d3 xεi jk ε A (X)∂ j ei A A (X)ψ k (X)

1

+ ei A A (X)∂ j ek A A (X) 2h¯

× F[oCB , ei A A , ψ i A ] ,

(4.12)

4.1

Physical States in N =1 SUGRA

65

where oCB (x) are rotation matrices. This describes an exact quantum state of the full supergravity field theory.1 The reader may be wondering whether the CFOP argument [7] was the final word regarding the claims in, e.g., [14]. Several publications followed, namely in [2, 5, 25]. It seemed (and still seems) to the vast majority of the scientific community that the CFOP line is the appropriate vantage point. Note that studies of the semiclassical approximation to simple supergravity in Riemannian four-manifolds with boundary suggest that simple supergravity is not even one-loop finite in the presence of boundaries [20, 18, 19]. Furthermore, [21] indicated that the Gibbons–Hawking boundary conditions (usable in quantum gravity) break local SUSY if one imposes local boundary conditions on all fields. However, nonlocal boundary conditions are not ruled out. All this is quite pertinent because, when N = 1 SUGRA was discovered, partial integrations in the proof of local supersymmetry were performed without taking boundary terms into consideration. However, in the presence of boundaries, local supersymmetry (and other local symmetries such as local coordinate invariance and local Lorentz symmetry) can only remain unbroken if one imposes certain boundary conditions on the fields and on the parameters.2 Hence the question: Are there boundary conditions for SUGRA theories which maintain all local symmetries, even at the boundaries? For models with auxiliary fields and models without them, it was found that this extended local SUSY in supergravity requires vanishing extrinsic curvature K ji |∂ M = 0. Another pertinent point, although requiring considerable assessment, is that the algebra of constraints arising in the canonical quantization of N = 1 supergravity in four dimensions, using the holomorphic action, does not close formally for two chosen operator orderings [24]. The main element is that the gravitino spinor corresponding to the complex conjugate of ψμA in the real theory is considered there to be independent, since the complex conjugate of a holomorphic function is not holomorphic. To highlight some of the details involved in this discussion, it may prove useful to indicate the following. The main point under discussion concerns the use of physical wave functionals established by a superposition from the amplitude K to go from

prescribed data (eiA A , ψ iA )I on an initial surface to data (eiA A , ψ Ai )F on a final surface, given globally by a Feynman path integral. It is claimed that, in a semiclassical expansion of this amplitude around a classical solution, the SUSY constraints The amplitude υ reduces to exp(−(3) Vm pq h pq /2h¯ ) in the spatially homogeneous case, where indicating that (4.12) should be interpreted as a ‘wormhole’ (Hawking–Page) state [22] (see Sect. 5.2 of Vol. I).

1

(3) V is the 3-volume,

2 In general relativity, [23], we can cancel several boundary terms which occur when varying the metric in the Einstein–Hilbert action [see equation (2.17) in Vol. I] √ by adding a boundary / term which contains the extrinsic curvature of the boundary, viz., ∂ M h K . The proposal was to impose the boundary condition that the variations of the metric in the surface vanish, i.e., δh i j |∂ M = 0. But this violates local SUSY if one only admits local boundary conditions (in particular no boundary conditions on curvature components, but only on the fields themselves).

66


of N = 1 SUGRA imply that the one- and higher-loop terms are invariant under left-handed supersymmetry transformations at the final surface, and under righthanded transformations at the initial surface. It is this resummation of the standard semiclassical expansion that has been the subject of debate, since claims [2, 5, 25] were advanced that it would help in improving the convergence of the theory. All in all, this means including boundary terms in the SUGRA action. For this reason, the reader is referred to Exercise 4.1, where some features of the variation of the SUGRA action are brought out, and the question of invariance is raised whenever boundaries are present.

4.2 Semiclassical Expansion From the content of the previous section, the ‘obvious’ route to explore would be to check if (and how) our physical (observed) universe can be retrieved from the physical states just described. In this section we will therefore explore and extract a semiclassical approximation scheme for canonical N = 1 SUGRA [4], following in the pioneering footsteps of the non-supersymmetric case (see also Chap. 2). Although it may look like a mere application, the content of this discussion should not be underestimated. The reader will soon discover that it provides a framework and guiding directions in the search for an imprint of an SQC stage in the very early universe. Furthermore, it will indicate just how a physical classical 4D spacetime (in fact, our own!) can be retrieved from N = 1 SUGRA. In other words, we are asking whether a full SUSY background (from within any related theory, such as superstrings) can be cosmologically realistic? Although many questions will nevertheless remain open, it will contribute additional strong arguments to the discussion about which type of physical states are indeed admissible in quantum N = 1 SUGRA (see Sect. 4.1 and Exercise 4.1). The semiclassical approximation explored here3 will be based on [4]. This has the advantage that the (formal) closure of the algebra is automatically implemented (see Sect. 4.2.2 of Vol. I). To be more precise, we will use the following expression, obtained from (4.101)

of Vol. I by means of a projection n A A H A A (see Exercise 4.3): H⊥ = −n

A A

2 A A

H A A = − 4π iG h¯ n

ψiB

0 2 A A

+ 4π Gih¯ n 0

3

δ δe AB j

δe AB j 12

D BBm j D CA kl

12 (i)

δ

ε

ilm

D

B B

(ii)

We henceforth drop the suffix 2 introduced in Sect. 4.3 of Vol. I.

ij

δ δeiB A

3

δ δψkC

3

4.2

Semiclassical Expansion

67

δ h¯

+ √ εi jk eiBC e A C l (3s) D j ψ Ak δψlB h 0 12 3 (iii)

+ ih¯ n A A

(3s)

0

Di

12

δ

δeiA A

3

(iv)

−

1 U [e] . 0G 12 3

(4.13)

(v)

C B BB The functional derivative δ/δe AB j on D B m j D A kl and Di j gives (see Appendix A)

n A A εilm

3i i C B

C ij C B l D D = − ε − 2h ε n e δ √ jkl

B mj k B A kl B B δ(0) , δe AB h j (4.14) δ

and n AA

δ

δe AB j

i

DiBj B = √ eiB A δ(0) . h

(4.15)

Note 4.1 The ‘divergence’ δ(0) is due to the functional derivative acting at the same space point. To fully address this issue, a thorough regularization is required, but we will not probe this feature further. In a semiclassical description, it will simply correspond to a factor-ordering ambiguity, and we will not therefore include δ(0) in the remainder of this section.

Starting with (4.13), we shall thus explore the SUSY version of the Wheeler–DeWitt equation written in the form m )Ψ = 0 , (H⊥ + H⊥

(4.16)

where H⊥ corresponds to the SUSY contribution to the complete Hamiltonian conm is the matter contribution. Generally, we can include scalar and straint, and H⊥ vector bosonic fields and their corresponding SUSY partners, but in this section we restrict ourselves [also to be able to compare the results with the bosonic scenario m corresponding to the Hamiltonian density of a minimally (see Sect. 2.1)] to a H⊥ coupled scalar field φ: m H⊥

√ ij √ 2 2 1 h¯ 2 δ2 = + hh φ,i φ, j + h m φ + V (φ) , −√ 2 h δφ 2

(4.17)

68


where the self-coupling potential V (φ) is left unspecified. The extension of the semiclassical analysis given here to more realistic situations with supermatter (see Chap. 3) remains an open problem.

4.2.1 The Hamilton–Jacobi Equation As in Chap. 2 (see also [23]), we propose to use an ansatz of the form

Ψ [e, ψ, φ] ≡ exp

i S[e, ψ, φ] h¯

,

(4.18)

for the quantum state and expand S in a power series with respect to G as follows: S[e, ψ, φ] ≡

∞

Sn [e, ψ, φ]G n−1 .

(4.19)

n=0

By examining (4.18) and (4.19) with (4.16), and taking into account (4.13), we then obtain a suitable expansion in powers of G. To Order G −2 The lowest order G −2 implies (as in the bosonic case) that S0 is independent of the matter field φ. In more formal terms, S0 ≡ S0 [e, ψ]. To Order G −1 Let us now proceed to the next order, G −1 . The Hamilton–Jacobi equation of N = 1 SUGRA is determined at this stage. In more detail, we obtain 0 = 4πi

ψiB

δS0

C AB i

E B jk AB δe j

δS0 δψkC

−n

A A

DiBj B

δS0

δS0

δe AB δeiB A j

δS δS0 i

0 − n A A (3s) Di A A − U , + √ εi jk eiBC e A C l (3s) D j ψ Ak B δψl δei h

(4.20)

with

AB i ≡ n A A εilm D B B E CB jk

C m j D A m j

.

(4.21)

The key question at this point is: Can the Hamilton–Jacobi equation of the bosonic case (see Chap. 2), viz., δS0 δS0 1 + Ug = 0 , Gi jkl 2 δh i j δh kl

(4.22)

4.2


69

be included in (4.20) above? To address this question, let us proceed in two steps. First, we ignore all terms involving the gravitino. Second, we transform the terms that involve the tetrad into terms depending on the three-metric h i j . Assuming then that S0 [e] can be rewritten as S0 [h i j ] [see Appendix A and in particular, (A.122)], and noting that the expansion mentioned in Chap. 2 includes an additional factor of 32π , viz., U [e] ≡ −

1 U bos [h i j ] , 32π

(4.23)

1 bos S , 32π 0

(4.24)

and S0 ≡ this, finally, allows us to write δS bos δS0bos δS bos 1

+ 64πn A A ∂i e A A j 0 + V bos = 0 . Gi jkl 0 2 δh i j δh kl δh i j

(4.25)

From (4.22), (4.23), (4.24), and (4.25), there are grounds for claiming an equivalence. The only (apparently) ‘weak’ feature is that the second term in (4.25) has no counterpart. Here the reader should be wondering what its origin could be. To address this question requires further discussion. It has been proposed [23] to associate this term with the momentum constraints Hi . If we had directly applied the approximation scheme to a quantum version of 1 H⊥ (see Sect. 4.3 of Vol. I), this second term in (4.25) would be absent. As a consequence, a redefined version of the Hamiltonian constraint (4.13), which we identify as H˘ ⊥ , without the term (iv), ih¯ n A A

(3s)

Di

δ δeiA A

,

(4.26)

seems more appropriate.

Assuming then that the term 64πn A A ∂i e A A j δS0bos /δh i j is indeed not properly relevant and has the origin described above, the Hamilton–Jacobi equation (4.22) for the pure bosonic sector, i.e., general relativity, is contained in the full equation (4.20). In fact, we can obtain a decomposition of (4.20) into a part depending only on the tetrad plus other terms with both bosonic and fermionic variables. A decomposition such as (see Exercise 4.4) S0 [e, ψ] = B0 [e] + F0 [e, ψ]

(4.27)

will provide the standard classical spacetime background from the solutions of the Hamilton–Jacobi equation obtained by this procedure.

70


But then what about the fermions in (4.20)? At a more fundamental level, what are the implications of the enlarged (supersymmetric!) Hamilton–Jacobi equation with regard to issues such as the emergence of classical spacetime (see Chap. 2)? The present author does indeed believe this to be a really crucial issue! In other words how should we interpret the solutions of the Hamilton–Jacobi equation (4.20)? SUSY ‘Spacetime’ A solution of the Hamilton–Jacobi equation would provide a spacetime which could serve as the appropriate background for the higher orders. In general relativity (a solution equivalent to the field equations originating from the Einstein–Hilbert action [26]), every solution S0 describes a family of solutions to the classical field equations. This was DeWitt’s own interpretation. For every three-geometry, there is one member of this family with a spacelike hypersurface with this three-geometry. An intriguing feature of the Hamilton–Jacobi equation for N = 1 SUGRA in (4.20) is the presence of the gravitino in the first and third terms: ψ, ψ would thus be explicitly present in the solutions S0 . [Indeed, it can be proved (see Exercise 4.1) that S0 must depend on the gravitino, with similar conclusions extending to all orders in the expansion (4.19).] The key implication of (4.20) is that the Hamilton–Jacobi equation and the ‘background’ spacetime thereby retrieved must necessarily involve the gravitino. This important additional element corresponds to dealing with a configuration defined on the space of all possible spatial tetrads and gravitino fields which we called SuperRiem(Σ). A standard (classical) background can be established from (4.20) through B0 [e] in (4.27) and (B.80) without a gravitino, and the remaining terms in F0 [e, ψ] would then provide contributory expressions involving gravitinos, within the perspective of SuperRiem(Σ). To be more precise, through its solutions, the SUSY Hamilton–Jacobi equation (4.20) induces a spacetime background with both tetrad (graviton) and fermionic (gravitino) terms. This supersymmetric spacetime configuration will indeed be a solution of the equations of motion of the theory, with a metric represented as g = gB + gS ,

(4.28)

where the term gB denotes the body and gS the soul, adopting the definitions and nomenclature introduced by B. DeWitt in [27]. gB and gS correspond to the purely bosonic and fermionic sectors, respectively. In other words, the Hamilton–Jacobi equation would lead to a spacetime configuration constituting a Grassmann-algebravalued field that could be decomposed into the body, which takes values in the domain of real or complex numbers, and a soul which is nilpotent. A solution of the SUSY Hamilton–Jacobi equation (4.20) will thus correspond to a spacetime (yielding an appropriate background for the higher orders), whose metric includes the standard classical bosonic sector plus contributions in the form of gravitino terms.

4.2


71

Every such solution S0 describes a family of solutions to the classical field equations. For every 3-geometry, there is one member of this family with an appropriate spacelike hypersurface. But it ought to be said that such an interpretation does not close the discussion of this issue, and further analysis will certainly be required (see Chap. 9) Before proceeding, two interesting directions require further discussion: • On the one hand, SUGRA (and superstring) backgrounds have to satisfy the equations of motion, i.e., in particular, they must be invariant under SUSY transformations. This implies conditions on the parameters of the SUSY transformation: they must satisfy a Killing spinor equation [28–30]. But there have been some notable exceptions, namely, solutions of the field equations of SUGRA constituting a supersymmetric generalization of a black hole geometry. The full metric solution consists of a supermultiplet, formed by supertranslated partners to the purely bosonic configuration. This overall description corresponds to the scenario of supermanifolds (and therefore superRiemannian geometries) described in detail in [31–37]. • On the other hand, the reader may quite rightfully ask whether the presence of the gravitino, even at higher orders of approximation, is in conflict with observation? In this context, it is curious to note that W. Pauli [38] performed a WKB approximation for a Dirac electron and found that spin effects do not occur at the higher orders, but only spatial effects. So we are left wondering whether the spin 3/2 nature of the gravitino would play a significant role at higher orders, e.g., the order of the Hamilton–Jacobi equation.

4.2.2 The DeWitt Supermetric Assuming that the supersymmetric Hamilton–Jacobi equation provides a window on the physical reality of the complete spacetime properties, i.e., assuming that the classical spacetime we commonly perceive should emerge from the space of all possible spatial tetrad and gravitino fields SuperRiem(Σ), let us see where this can lead [4]. We introduce a metric G s on the space SuperRiem(Σ), which we shall call the DeWitt supermetric, being the supersymmetric analogue of the DeWitt metric Gi jkl . Note that SuperRiem(Σ) is the direct sum of the tetrad space and the gravitino

space, therefore acting on vectors of the form (e A A i , ψ jB ) ≡ qa . Henceforth, the indices starting with a are ‘condensed’ superindices which run through all bosonic and fermionic degrees of freedom. The metric G s reads in block form (the reader is invited to consult [4] for additional technical details)

s ≡ Gab

B M1 M2 Fˇ

.

(4.29)

The blocks B and Fˇ correspond to the pure bosonic and pure fermionic parts, respectively, while M1 and M2 are the mixed off-diagonal parts. In the next section,

72


we determine the specific form of these blocks explicitly as functionals of the tetrad and the gravitino. For current purposes, it is sufficient to point out that the upper diagonal part contains the DeWitt metric (indeed, a tetrad version of it). In fact, for quantities that can be written in terms of the three-metric h i j and the gravitino, the block B is the DeWitt metric (see Exercise 4.5). Using the DeWitt supermetric in (4.29), the Hamilton–Jacobi equation (4.20) [omitting the term (4.26)] takes the condensed form 1 s δS0 [e, ψ] δS0 [e, ψ] + A(S0 [e, ψ]) − V = 0 , G 2 ab δqa δqb

(4.30)

where we have introduced the operator δ i

. A ≡ √ εi jk eiBC e A C l (3s) D j ψ Ak δψlB h

(4.31)

4.2.3 The Schrödinger Equation In this section, we will discuss how a time direction can be defined in the canonical formulation of N = 1 SUGRA, comparing with the purely bosonic case, i.e., general relativity, as indicated in [4] (see also Sect. 2.1). Moreover, the retrieval of a Schrödinger-like equation is most relevant, as it will describe from a quantum mechanical view, by means of functionals, an equation of motion for the matter field variables within a classical background previously determined by the Hamilton– Jacobi equation. It therefore constitutes another crucial step in describing a consistent view of the early universe within SQC. So we start by considering the semiclassical expansion of (4.13) and (4.16). To Order G0 As explained in Sect. 4.2.1, we do not include the contribution of term (iv) in (4.13), and hence we find √ 2 2 ih¯ δS1 δS1 √ i j 1 1 δ2 S1 −√ + hh φ,i φ, j + h m φ + V (φ) 0= √ 2 h δφ 2 h δφ δφ δS(0 δ2 S0 AB i δS1) AB i +4π 2iψiB AB E CB jk − h¯ ψiB E CB jk

C AB δψk δe j δe j δψkC

−h¯

3i

√ ψkC + ψ B j ε jkl n C B elB B

h

δS1 δψkC

− 2in A A

δS(0

δe AB j

DiBj B

δS1) δeiB A

4.2


73

δ2 S0 ih¯

δS0

− √ eiB A B A − h¯ n A A DiBj B AB B A

δei δe j δei h δS i

1 + √ εi jk eiBC e A C l (3s) D j ψ Ak . δψlB h

(4.32)

Equation (4.32) can be further simplified using the wave functional [see (2.7)]

i F = K[e, ψ] exp S1 [e, ψ, Φ] h¯

,

(4.33)

i.e., expressing terms in S1 above as F and then, with the WKB prefactor K restricted to satisfy (see Exercise 4.6) 0 = ψiB

δS0

δe AB j

−n A A

δK

AB i E CB jk

δS0

δe AB j

δψkC

DiBj B

+ ψiB

δK

δeiB A

δK

− n AA

AB i E CB jk

δK

δS0

δe AB j

δe AB j

δψkC

DiBj B

δS0 δeiB A

δK 1

√ εi jk eiBC e A C l (3s) D j ψ Ak δψlB 4π h

δ2 S0 3i C δS0 B C AB i Bj C B l − ψi E B jk + √ ψk + ψ ε jkl n e B B

C AB δψkC δe j δψk h +

+n

A A

DiBj B

δ2 S0

BA δe AB j δei

i

δS0 + √ eiB A B A

δei h

K,

(4.34)

we introduce (4.33) and (4.34) into (4.32) and therefore retrieve the Tomonaga– Schwinger equation, i.e., a local4 Schrödinger equation: m F H⊥

δS0 δ δF AB i δ AB i δS0 = ih¯ + ψiB E CB jk ≡ −4π h¯ ψiB AB E CB jk

C C δτ δψk δψk δe AB δe j j −n A A

δS0

δe AB j

DiBj B

δ

δeiB A

− n AA

δS0

δeiB A

DiBj B

δ 1

+ √ εi jk eiBC e A C l (3s) D j ψ Ak δψlB 4π h 4

δ δe AB j

F,

(4.35)

The usual functional Schrödinger equation is found from (4.35) after integration over space.

74


associated with definition of the time functional τ (x; e, ψ], given by δ(x − y) = 4π ψiB −n A A

δS0 AB i E CB jk

AB δe j (y) δS0

δe AB j (y)

DiBj B

δ

δψkC (y) δ

δeiB A (y)

AB i + ψiB E CB jk

− n AA

δS0

δeiB A (y)

δS0

δ

δψkC (y)

δe AB j (y)

DiBj B

δ

δe AB j (y)

1 δ

+ √ εi jk eiBC e A C l (3s) D j ψ Ak τ (x; e, ψ] . δψlB (y) 4π h (4.36) The crucial conclusion seems to be that the presence of the gravitino is mandatory for the definition of the time functional as well as for the Schrödinger equation. The question thus arises: How can we interpret a classical background containing the gravitino and thereby inducing this type of time functional? Perhaps within the proposed description of a classical spacetime emerging from the space of all possible spatial tetrad and gravitino fields SuperRiem(Σ), i.e., from the time functional (4.36) we might also define a local many-fingered time parameter. Of course, further discussions are still required as this is a largely uncharted territory [see (2.9) and (2.10)]. Note 4.2 It should also be noted that the functional Schrödinger equation can only be recovered in this way if a real solution S0 to the Hamilton–Jacobi equation is found. But a superposition ∝ exp(iS0 ) + exp(−iS0 ) may not allow it, as the various semiclassical components may interfere with each other. In other words, the issue of decoherence [23] (see also Sect. 2.2.2) but now in SQC, is surely fundamental to this investigation. Components in such a superposition must effectively become independent, e.g., due to additional degrees of freedom.

4.2.4 The DeWitt Supermetric and the Schrödinger Equation Let us now proceed to use the formalism of the DeWitt supermetric (4.29) to present the Schrödinger equation in a more compact form and in a context similar to that of the Hamilton–Jacobi equation (4.30), and therefore formally close to the expression used in the pure bosonic case (i.e., general relativity, as in Sect. 2.1). The framework of the DeWitt supermetric (4.29) and of the space of all possible spatial tetrads and gravitino fields SuperRiem(Σ), means that the blocks in5 The reader should note the use of F in (4.33) and Fˇ in (4.29) and (4.37), which may appear together in a few expressions.

5

4.2


75

s Gab =

B M1 M2 Fˇ

(4.37)

are determined by the requirement that this metric applied to the vectors, e.g.,

δS0 /δqa ≡ (δS0 /δeiA A , δS0 /δψ jB ) and δF /δqb , provide all terms containing two derivatives in the local Schrödinger equation (4.35). For B one gets

B = −4πi(n A A DiBj B + n B B D Aji A ) .

(4.38)

Since (4.35) contains no terms with a double derivative with respect to ψiA , the lower diagonal block Fˇ vanishes: Fˇ = 0 .

(4.39)

For M1 and M2 we get

S1 = 4π in B B ψ Cj ε jkm DC A mi D D B lk , S2 = 4π in

A A

ψiB εilm D B

B

C m j D A kl

.

(4.40) (4.41)

The local Schrödinger equation (4.35) can therefore be set in the form ih¯ Gab

δS0 δF δF m + ih¯ AF = ih¯ F, = H⊥ δqa δqb δτ

(4.42)

s and the operator A, the condition (4.34) becomes and, in terms of the metric Gab

δS0 δK δ2 S0 K − A(K) − Gab δqa δqb 2 δqa δqb

i B A δS0 3i C δS0 Bj C B l −4πiK √ ei . √ ψk + ψ ε jkl n e B B

+ δψkC δeiB A h h

s 0 = Gab

(4.43)

4.2.5 Quantum N=1 SUGRA Corrections This section brings the reader to an interesting and promising area for investigation (and investment!). This is where we will be able to seek and establish a link between SQC (or at a more fundamental level, N = 1 SUGRA and superstrings) and observational data. From the description of the supersymmetric Hamilton–Jacobi equation (see Sect. 4.2.1), where the emergence of the classical spacetime background was introduced, and from the discussion of a supersymmetric local Schrödinger equation for the quantum mechanical matter fields (see Sect. 4.2.3), we can identify the quantum gravitational elements (within N = 1 SUGRA) that influence the quantum description of matter. To do this, our semiclassical expansion scheme must be extended.

76


To Order G 1 The corresponding equation is ⎡ Ψ

−1

O(G 1 ) m (H⊥ +H⊥ )Ψ =

⎢ ⎢ B δS1 C AB i δS1 δ2 S1 B C AB i 4π G ⎢ E + h ψ E iψ ¯

i i B jk B jk ⎢ C δψkC δe AB δe AB ⎣ j j δψk 0 12 3 0 12 3

− h¯ 0

(i)

3i

√ ψkC + ψ B j ε jkl n C B elB B

h 12

0

0

AB i E CB jk

12

(iv)

− in A A

− h¯ n

δS(0 δe AB j δS1

δe AB j

DiBj B

12

(vi)

A A

δS2) δψkC 3

− 2in A A 0

(ii)

δS1 δψkC 3

(iii)

+ 2iψiB

0

δS(0 B B δS2)

Di j

δe AB δeiB A j 12 3 (v)

δS1

δeiB A 3 ⎤

⎥ ih¯ B A δS1 ⎥ ⎥ − √ ei

B A

B A ⎥ δe AB δe δe h ⎦ j i 12 i 3 12 3 0

DiBj B

δ2 S1

(viii)

(vii)

δS iG

2 + √ εi jk eiBC e A C l (3s) D j ψ Ak δψlB h 0 12 3 (ix)

G δ2 S2 δS1 δS2 + √ =0. − ih¯ 2 2 δφ δφ δφ (x) 2 h 12 3 0

(4.44)

(x)

It will be from the above that we can obtain the Schrödinger equation with corrections of order G, in a straightforward but lengthy manner. The reader can jump directly to (4.50) for the final result. Briefly, the analysis goes as follows: 1. Rewrite the expressions containing S1 in (4.44) in terms of F and K. 2. Make the decomposition ˘ ψ, φ] , S2 [e, ψ, φ] = σ˘ 2 [e, ψ] + η[e,

(4.45)

4.2


77

to separate the pure gravitational parts of (4.44) from those containing the matter field. 3. Use the condition on σ˘ 2 [e, ψ], viz., δ2 K 4π h¯ 2 2 δK δK δσ˘ 2 B C AB i − AB C =− ψi E B jk

δτ iK K δe AB δψkC δe j δψk j

3i δK

+ √ ψkC + ψ B j ε jkl n C B elB B

δψkC h −

2 A A B B δK δK n Di j

K δe AB δeiB A j

+n

A A

DiBj B

δ2 K

BA δe AB j δei

i

δK − √ eiB A B A

δei h

,

(4.46)

in an analogous way to standard quantum mechanical WKB expansions. 4. Rewrite (4.44) as 1 δF δK AB δF − + AB

Ke j

C K δψkC δe AB δe j δψkC j δψk

3i C δF

− √ ψk + ψ B j ε jkl n C B elB B

δψkC h 4π h¯ 2 A A B B

δ2 F 1 A A B B δK δF δF δK + − n Di j n Di j

+

B A

iF K δe AB δe AB δeiB A δe AB δeiB A j δei j j

i B A δF δ2 η˘ iG h¯ 2 δF δη˘ + √ ei + . (4.47) + √

δφ 2 δeiB A h 2 h F δ# δφ

4π h¯ 2 δη˘ =− δτ iF

AB i ψiB E CB jk

δ2 F

5. Note that, up to the current order, the wave functional has the form

i i 1 −1 Ψ = exp S0 G + σ˘ 2 G F exp ηG ˘ . K h¯ h¯

(4.48)

6. Hence investigate the matter part using

i ηG ˘ Θ ≡ F exp h¯

.

7. Multiply the Schrödinger (4.35) by exp (iηG/ ˘ h¯ ). 8. Add it to equation (4.47) multiplied by −GF exp (iηG/ ˘ h¯ ).

(4.49)

78


In essence, and with the assistance of Appendix A, the local Schrödinger equation with corrections up to order G 1 takes the form: δΘ 4π G h¯ 2 δF δK 1 B C AB i δK δF m ih¯ + AB

= H⊥ Θ + − ψi E B jk

δτ iF K δψkC δe AB δe j δψkC j

AB i +ψiB E CB jk

δ2 F

C δe AB j δψk

3i C δF

− √ ψk + ψ B j ε jkl n C B elB B

Θ δψkC h δ2 F 1 4π G h¯ 2

n A A DiBj B AB B A − n A A DiBj B − F K δe j δei δK δF δF δK i B A δF × + √ ei Θ.

+

δe AB δeiB A δe AB δeiB A δeiB A h j j

(4.50)

4.2.6 The DeWitt Supermetric and Quantum Gravity Corrections for the Schrödinger Equation For the same reasons as those presented in Sects. 4.2.1 and 4.2.3, it will prove worthwhile [4] to use the DeWitt supermetric G s (4.29) on the space SuperRiem(Σ). The corrected local Schrödinger equation (4.50) then reads δ2 F G h¯ 2 1 s δF δK 1 δΘ m − Gab = H⊥ Θ + Gab ih¯ δτ F K δqa δqb 2 δqa δqb

3i δF

+4πi √ ψkC + ψ B j ε jkl n C B elB B

δψkC h 4π

δF − √ eiB A B A Θ . δei h

(4.51)

For the case U = 0, no further expression can be obtained and the subsequent analysis [see (4.56)] would proceed from (4.51) alone. But for the case of nonvanishing U , it is of interest to decompose it into normal and tangential parts. These directions are defined with respect to hypersurfaces SuperRiem(Σ) in which S0 = constant holds. The normal part is given by a vector parallel to δS0 /δqa and the tangential part by a vector orthogonal to δS0 /δqa . To be more precise, normal means normal to hypersurfaces S0 = constant (directed along the classical spacetimes defined by S0 ). Tangential is consequently tangential to S0 = constant. We thus take a trajectory of the classical spacetime

4.2


79

in configuration space, i.e., the space SuperRiem(Σ), and investigate in the direction of the evolution and in the direction transverse to it. The idea is to rewrite the correction terms by means of a factorization as

1 s δ2 F 4π 1 s δF δK

δF − Gab − √ eiB A B A

Gab K δqa δqb 2 δqa δqb δei h

3i δF

+4πi √ ψkC + ψ B j ε jkl n C B elB B

Θ. δψkC h

G h¯ 2 Cn ⊕ Ct ≡ F

(4.52)

Let us therefore introduce the decomposition s Gab

δF s δS0 = γ Gab + Tb , δqa δqa

(4.53)

where γ represents a factor. The tangential part Tb , has to satisfy Tb

δS0 =0. δqb

(4.54)

Then, multiplying (4.53) by δS0 /δqb leads to s Gab

δF δS0 s δS0 δS0 = γ Gab , δqa δqb δqa δqb

(4.55)

and from the Hamilton–Jacobi equation (4.30) and the Schrödinger equation (4.42), we thus obtain γ ≡

m − ih A)F (H⊥ ¯ , 2ih¯ U˜

(4.56)

where U˜ = (U − AS0 ) is a modified potential. Furthermore, differentiating (4.53) with respect to qb , we can use s Gab

δ2 F δ2 S0 δγ s δS0 s = Gab + γ Gab + T˜ , δqa δqb δqa δqa δqa δqb

(4.57)

where T˜ denotes the sum of the tangential parts. The tangential part Ct obtained from (4.52) remains an open issue (for the nonsupersymmetric case, see [23] and references therein). Regarding the normal part Cn , to obtain an explicit form for it from (4.52), we need a further decomposition. We thus define i 3i

(4.58) wa ≡ √ eiB A , √ ψkC + ψ B j ε jkl n C B elB B , h h

80


and write wa

δF δS0 = γ wa + wa T˜ a , δqa δqa

(4.59)

where T˜ a is the tangential part. In essence, G h¯ 2 (AK) 1 s δS0 δγ −γ − Gab F K 2 δqa δqb m − ih A) m 2 δ(H⊥ ih¯ δU˜ G ¯ m − ih¯ A) − − (AU˜ ) (H⊥ = H⊥ + ih¯ δτ U˜ δτ 4V˜ F m 3AH⊥ 2(AK) m 2 − (H⊥ − ih¯ A) + − 2A F . (4.60) K ih¯

Cn =

Finally, rewriting all expressions containing F in (4.60) in terms of Θ and with δn Θ δn F Θ = + O(G) , δq n δq n F

(4.61)

we find the normal part of the corrected local Schrödinger equation: ih¯

δΘ m Θ = H⊥ δτ

m − ih A) δ(H⊥ ih¯ δU˜ ¯ m − ih¯ A) − − (AU˜ ) (H⊥ H⊥ + ih¯ δτ U˜ δτ m 3AH⊥ 2(AK) m − (H⊥ − ih¯ A) + − 2A2 Θ . (4.62) K ih¯ G + 4U˜ F

m 2

If the operator A [see (4.31)] is negligible [4], ih¯

δΘ m Θ = H⊥ δτ

(4.63)

√ ⎫ ⎧ (3s)R ⎬ ⎨ m δ h m 2 δH⊥ 4π G ih¯ m +√ −√ H⊥ Θ. H⊥ + ih¯ ⎭ δτ δτ h (3s)R ⎩ h (3s)R

On a formal level, it is similar to the purely bosonic case as obtained from the expansion of the Wheeler–DeWitt equation in the general relativity case. However, there is an important difference in that the definition of the time functional involves the gravitino. Nevertheless, it seems to indicate that the bosonic limit of the canonical quantization of N = 1 SUGRA yields bosonic canonical quantum gravity up to

Problems

81

the first order correction terms. This constitutes a rather satisfying argument for the overall consistency of the supersymmetric theory.

4.2.7 Towards SQC with ‘Observational’ Insights? At this point the attentive reader will be wondering about the potential use of this semiclassical framework to provide SQC with an ‘observational’ viewpont as indim in the above corrections cated at the beginning of Sect. 4.2.5? The presence of H⊥ (4.63) allows us to estimate their importance with regard to potential observational effects. For a Friedmann universe with scale factor a, we can roughly estimate the ratio of the second (and third) to the first correction term in (4.63): m m h¯ a˙ dH⊥ h¯ H0 h¯ δH⊥ ∼ ∼ , m m E (H⊥ )2 δτ (H⊥ )2 da

(4.64)

where E is a typical energy associated with the matter field. For E = 700 GeV and H0 = 70 km/(s Mpc) we obtain approximately 10−44 . Therefore the quadratic matter Hamiltonian is the most important correction. But this is not an observable effect for our current standard of measurements. Hence, it still remains to initiate future investigations, which will deal with the application of (4.62) in the context of observational consequences within SQC, such as structure formation (see, however, [4, 39, 40] for a preliminary attempt, and also Sect. 10.1 of this volume). Summary and Review. Before moving into the core of SQC (Part III of this book), let us have a look at the main points of the semiclassical perspective on the canonical quantization of the full theory of N = 1 SUGRA: 1. What type of physical states can be expected? What are the consequences of the CFOP argument [Sect. 4.1]? 2. How does the DeWitt supermetric relate to the usual DeWitt metric in (general relativity) geometrodynamics [Sect. 4.2.2]? 3. What are the main features of SUGRA (quantum) corrections to the (Schrödinger equation for the) matter fields [Sects. 4.2.5 and 4.2.6]? 4. Can SQC become ‘observational’ [Sects. 4.2.6 and 4.2.7]?

Problems 4.1 S0 Must Depend on the Gravitino ψ A i Prove that S0 , along with other terms Sn , n ≥ 1 in (4.19), must depend on the gravitino ψ A i .

82


4.2 N=1 SUGRA Action, SUSY Transformations, and Time Gauge Discuss how the action for N = 1 SUGRA changes for arbitrary SUSY transformations, as well those that preserve the time gauge. 4.3 Obtaining (4.13) By writing the expression for 2 H A A in a less symmetric form, but allowing for the terms containing (3s) D j (ψ A k ) to cancel out, find (4.13). 4.4 Decomposition of the Hamilton–Jacobi Equation into Bosonic and Fermionic Sectors Using the ansatz S0 [e, ψ] ≡ B0 [e] + F0 [e, ψ], obtain a factorization of the Hamilton–Jacobi equation in terms of a purely bosonic part, corresponding to B0 [e], and a mixed part including B0 [e] and F0 [e, ψ]. s Should Contain the Usual DeWitt Metric 4.5 The DeWitt Supermetric Gab Show that the DeWitt supermetric (4.29) includes the usual DeWitt metric Gi jkl in the block sector B.

4.6 Towards a Conservation Law? Find out whether the condition (4.34) can be rewritten as a conservation law.

References 1. D’Eath, P.D.: Quantum supergravity via canonical quantization. In: Penrose, R., Isham, C. J. (eds.) Proceedings on Quantum Concepts in Space and Time, Oxford, pp. 341–350 (1984) 61 2. D’Eath, P.D.: Supersymmetric Quantum Cosmology, 252pp. Cambridge University Press, Cambridge (1996) 61, 63, 65, 66 3. D’Eath, P.D.: The canonical quantization of supergravity. Phys. Rev. D 29, 2199 (1984) 61 4. Kiefer, C., Luck, T., Moniz, P.: The semiclassical approximation to supersymmetric quantum gravity. Phys. Rev. D 72, 045006 (2005) 61, 62, 66, 71, 72, 78, 80, 81 5. D’Eath, P.D.: Physical states in N = 1 supergravity. Phys. Lett. B 321, 368–371 (1994) 61, 63, 65, 66 6. Carroll, S.M., Freedman, D.Z., Ortiz, M.E., Page, D.N.: Bosonic physical states in N = 1 supergravity? gr-qc/941005 (1994) 61, 63 7. Carroll, S.M., Freedman, D.Z., Ortiz, M.E., Page, D.N.: Physical states in canonically quantized supergravity. Nucl. Phys. B 423, 661–687 (1994) 61, 63, 65 8. Csordas, A., Graham, R.: Exact quantum state for N = 1 supergravity. Phys. Rev. D 52, 6656–6659 (1995) 63 9. Moniz, P.V.: Supersymmetric quantum cosmology – shaken not stirred. Int. J. Mod. Phys. A 11, 4321–4382 (1996) 10. Page, D.N.: Inconsistency of canonically quantized N = 1 supergravity? hep-th/9306032 (1993) 61, 63 11. DeWitt, B., Matschull, H.J., Nicolai, H.: Physical states in D = 3, N = 2 supergravity. Phys. Lett. B 318, 115–121 (1993) 61 12. Matschull, H.-J.: About loop states in supergravity. Class. Quant. Grav. 11, 2395–2410 (1994) 61 13. D’Eath, P.D.: What local supersymmetry can do for quantum cosmology. In: Workshop on Conference on the Future of Theoretical Physics and Cosmology in Honor of Steven Hawking’s 60th Birthday, Cambridge, England, 7–10 January 2002 61, 63 14. D’Eath, P.D.: Canonical formulation and finiteness of N = 1 supergravity with supermatter. In: 7th Marcel Grossmann Meeting on General Relativity (MG 7), Stanford, CA, 24–30 July 1994 61, 63, 65

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15. D’Eath, P.D.: Finite N = 1 supergravity. In: 7th Marcel Grossmann Meeting on General Relativity (MG 7), Stanford, CA, 24–30 July 1994 61, 63 16. D’Eath, P.D.: Loop amplitudes in supergravity by canonical quantization. hep-th/9807028 (1998) 61, 63 17. Esposito, G., Kamenshchik, A.Yu.: Axial gauge in quantum supergravity. hep-th/9604194 (1996) 61, 63 18. Esposito, G., Kamenshchik, A.Yu.: One-loop divergences in simple supergravity: Boundary effects. Phys. Rev. D 54, 3869–3881 (1996) 63, 65 19. Esposito, G., Pollifrone, G.: Noncovariant gauges in simple supergravity. Int. J. Mod. Phys. D 6, 479–490 (1997) 61, 63, 65 20. Esposito, G.: Local boundary conditions in quantum supergravity. Phys. Lett. B 389, 510–514 (1996) 65 21. Van Nieuwenhuizen, P., Vassilevich, D.V.: Consistent boundary conditions for supergravity. Class. Quant. Grav. 22, 5029–5051 (2005) 65 22. Hawking, S.W., Page, D.N.: The spectrum of wormholes. Phys. Rev. D 42, 2655–2663 (1990) 65 23. Kiefer, C.: Quantum Gravity. International Series of Monographs on Physics vol. 136, 2nd edn., pp. 1–308. Clarendon Press, Oxford (2007) 65, 68, 69, 74, 79 24. Wulf, M.: Non-closure of constraint algebra in N = 1 supergravity. Int. J. Mod. Phys. D 6, 107–118 (1997) 65 25. D’Eath, P.D.: Supergravity and canonical quantization. Int. J. Mod. Phys. D 5, 609–628 (1996) 65, 66 26. Gerlach, U.H.: Derivation of the ten Einstein field equations from the semiclassical approximation to quantum geometrodynamics. Phys. Rev. 177, 1929–1941 (1969) 70 27. DeWitt, B.S.: Supermanifolds. Cambridge Monographs on Mathematical Physics, 2nd edn., pp. 1–407. Cambridge University Press, Cambridge (1992) 70 28. Alonso-Alberca, N., Lozano-Tellechea, E., Ortin, T.: Geometric construction of Killing spinors and supersymmetry algebras in homogeneous spacetimes. Class. Quant. Grav. 19, 6009–6024 (2002) 71 29. Ortin, T.: The supersymmetric vistas of the supergravity landscape. Annalen Phys. 15, 251–263 (2006) 71 30. Ortin, T.: Gravity and Strings. Cambridge University Press, Cambridge (2004) 71 31. Aichelburg, P.C., Dereli, T.: Exact plane wave solutions of supergravity field equations. Phys. Rev. D 18, 1754 (1978) 71 32. Aichelburg, P.C., Dereli, T.: First nontrivial exact solution of supergravity. Czech. J. Phys. B 29, 252–255 (1979) 71 33. Aichelburg, P.C., Embacher, F.: Supercharge and background perturbations of multi-black hole systems. Class. Quant. Grav. 2, 65 (1985) 71 34. Aichelburg, P.C., Embacher, F.: The exact superpartners of N = 2 supergravity solitons. Phys. Rev. D 34, 3006 (1986) 71 35. Aichelburg, P.C., Embacher, F.: Supergravity solitons. IV. Effective soliton interaction. Phys. Rev. D 37, 2132 (1988) 71 36. Aichelburg, P.C., Gueven, R.: Supersymmetric black holes in N = 2 supergravity theory. Phys. Rev. Lett. 51, 1613 (1983) 71 37. Dereli, T., Aichelburg, P.C.: Exact plane wave solutions of O(2) extended supergravity. Phys. Lett. B 80, 357 (1979) 71 38. Pauli, W., Fierz, M.: On relativistic field equations of particles with arbitrary spin in an electromagnetic field. Helv. Phys. Acta 12, 297–300 (1939) 71 39. Moniz, P.V.: Can the imprint of an early supersymmetric quantum cosmological epoch be present in our cosmological observations? In: COSMO 97: 1st International Workshop on Particle Physics and the Early Universe, Ambleside, England, 15–19 September 1997 81 40. Moniz, P.V.: Origin of structure in a supersymmetric quantum universe. Phys. Rev. D 57, 7071–7074 (1998) 81

Chapter 5

Further Explorations in SQC N=1 SUGRA

In Chap. 5 of Vol. I, we discussed the basic framework and associated results from the methodology followed there (see [1] and also [2–4]). Recall that this uses the reduction of canonical quantum N = 1 SUGRA in four spacetime dimensions (see Sects. 4.1 and 4.2 of Vol. I) to a cosmological minisuperspace, with the essential feature that all momenta are represented by differential operators of the canonical conjugate variables [5–7]. In this volume, we will present and apply another methodology in SQC, entailing specific differences in results. Choosing a matrix representation for the fermions complying with the Dirac bracket algebra (see Chap. 7). There is no compelling reason from fundamental theory for believing these approaches to quantization with fermionic variables to be physically equivalent in the framework of SQC, and no indication of which should be preferred. In fact, the equivalence or any clear relation between them is yet to be fully established. The aim would be to explain clearly whether they are connected, and if so, to what extent and in what way. The reader should recall the difference between ADM (Ω-time) and DWD (t-time) backgrounds, and the way the fermionic differential operator representation uses the latter while the matrix formulation employs the former (see Sects. 2.3, 2.4, and 2.5 and Exercise 2.2 in Vol. I). In this chapter we will thus extend the canonical quantization of supersymmetric minisuperspaces extracted directly from N = 1 SUGRA. In Sect. 5.1, we investigate how the issue of SUSY breaking might be discussed, employing an FRW background model. Additional (richer) structures are assembled in Sect. 5.2. In particular, we extend our suppermatter setting to an FRW model with scalar and vector superfields and then move to a Bianchi IX model with scalar superfield or vector superfield (the latter in N = 2 SUGRA).

5.1 The Issue of Supersymmetry Breaking The N = 4 SUSY FRW minisuperspace analysed in Vol. I provides the opportunity to discuss (from a somewhat lateral perspective) the important issue of supersymmetry breaking in SQC. This is a crucial problem in current theories such as SUGRA, superstrings, and M-theory. In fact, it is fundamental to identify clearly Moniz, P.V.: Further Explorations in SQC N=1 SUGRA. Lect. Notes Phys. 804, 87–109 (2010) c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-11570-7_5

88

5 Further Explorations in SQC N=1 SUGRA

the precise conditions under which it can be broken [8–11], thereby leading to a universe like the one we observe. Several different proposals for (spontaneous) supersymmetry breaking have been advanced, and stimulating scenarios have been investigated in supersymmetric quantum mechanics (see Sects. 3.2 and 8.2.1). It should also be stressed that, in spite of the extensive literature in SQC models from N = 1 SUGRA (see, e.g., [2]), not much has been presented concerning the issue of SUSY breaking (see, however, Sect. 6.1.5 of Vol. I). In SQC, much of the related work can also be found in [12–15]. The point is that this is a very rich topic, where much remains to be investigated. The approach in this section is somewhat different from what was done in Sect. 3.3. Nevertheless, it aims to introduce the reader to an open topic of SQC research, where a lateral approach may be beneficial and interesting to explore. Within the framework of cosmological minisuperspaces with N = 4 SUSY, retrieved from 4D N = 1 SUGRA, (5.123) and (5.124) of Vol. I imply important consequences regarding the physical states of an FRW minisuperspace. In particular, concerning the possibility of establishing whether the existence or otherwise of supersymmetric FRW quantum states relates to the form of the superpotential P. Taking into account the fact that A 0 and E 0 have different a dependences as seen from (5.104) and (5.105) of Vol. I, then consistency with (6.123) and (6.124) requires us to take ˇ φ P(1) = 0 , ˇ φ P(1) = D D

(5.1)

A 0 = E 0 = 0 .

(5.2)

or instead use

Let us analyse in detail the physical consequences of either conditions (5.1) or (5.2). The first expression (5.1) enables us to determine A0 , . . . , E0 as well as A1 , . . . , E1 . As far as the orders $2 or higher are concerned, the equations for A2 , . . . , E2 allow us to write ˇ φ P(2) = D ˇ φ P(2) = 0 , D

(5.3)

ˇ φ P(n) = D ˇ φ P(n) = 0 , D

(5.4)

which seems to point to

at all orders. Then we could in principle proceed to calculate the sets (An , . . . , En ) by inspecting the corresponding equations. Moreover, ˇ φ P(1) = 0 ⇒ P(1) ∼ e−φφ , ˇ φ P(1) = D D and similarly at other orders in $.

(5.5)

5.1

The Issue of Supersymmetry Breaking

89

If we had chosen the possibility (5.2), i.e., A 0 = E 0 = 0, then at order $0 and we could proceed to analyse the equations (5.100), (5.101), (5.102), (5.103), (5.104), (5.105), (5.106), and (5.107) and (5.108), (5.109), (5.110), (5.111), (5.112), ˇ φ P(1) and its Hermi(5.113), (5.114), and (5.115) of Vol. I, with an undetermined D tian conjugate. This would allow for any other type of superpotential different from (5.5). However, problems seem to occur at order $2 . In fact,1 at this order, we will be forced to choose between

$1 ,

ˇ φ P(1) = 0 , ˇ φ P(1) = D D

(5.6)

A 1 = E1 = 0 .

(5.7)

i.e., (5.1) again, or

If we follow (5.7), then we get a set of equations for B 0 , D 0 , and C 0 involving P(1) , ˇ φ P(1) , and Hermitian conjugates that imply that we must now use D ˇ φ P(1) = 0 P(1) = D

(5.8)

and their Hermitian conjugates, or alternatively we must have B 0 = D 0 = C 0 = 0 .

(5.9)

If we choose the latter, then since A0 and E0 are already set to zero, this means that, as we go to higher orders, the wave function will be zero. This means that no nontrivial physical states will be found. If we choose instead option (5.8) above, then we get nonzero values for the remainder of the bosonic functionals. Nevertheless, this means that we will have a SUSY FRW model without any superpotential terms. This is basically the case of a k = +1 supersymmetric FRW model where a particular set of wormhole states or a restricted Hartle–Hawking state can be found [16–18]. The overall situation is illustrated in Table 5.1. Table 5.1 Quantum FRW SUSY solutions with a superpotential ˇ φ P(1) = 0 D ⇓ ˇ φ P(n) = 0 D ⇓ P ∼ e−φφ ⇓ FRW(SPot) ΨSUSY

or ˇ φ P(1) = 0 D

A0 = E0 = 0 ⇓ or B0 = C0 = D0 = 0 ⇓ FRW = 0 ΨSUSY

A1 = E1 = 0 ⇓ or

ˇ φ P(1) = 0 P(1) = D ⇓ FRW ΨSUSY

1 Extrapolating to higher orders, this situation seems to go through without any formal modifications.

90


FRW = 0 means It is therefore tempting to consider the following idea. If ΨSUSY that no non-trivial supersymmetric wave function can be obtained (satisfying the supersymmetry constraints), this does not prevent a class of non-trivial states from being found to solve the Hamiltonian constraint. In fact, a state satisfying the supersymmetry constraints S A and S A , i.e., S A |Ψ = 0 and S A |Ψ = 0, will satisfy the Hamiltonian constraint H|Ψ ∼ S A S A |Ψ + S A S A |Ψ = 0. Now another state |Ψ´ may comply with H|Ψ´ = 0, but ∼ S A S A |Ψ´ + S A S A |Ψ´ = 0 does not necessarily imply that S A |Ψ´ = 0 and S A |Ψ´ = 0. FRW = 0 may indicate that supersymmetry breakHence, let us conjecture that ΨSUSY ing has taken place. The only wave function satisfying the SUSY constraints is the trivial state, a direct consequence of the form of the superpotential and its derivatives. Of course, this proposal is merely a hypothesis. In order to be substantiated, non-trivial solutions must be found for the Hamiltonian constraint, while the superFRW = 0. symmetry constraints would have solutions ΨSUSY Our FRW model with a scalar supermultiplet having N = 4 local SUSY and time reparametrization invariance revealed that supersymmetric wave functions could only be found when the superpotential had either an exponential behavior, an effective cosmological constant form, or was zero (φ −→ φ0 ). Moreover, if the superpotential has distinct behavior during the evolution of the universe, the wave FRW = 0. This suggests an interesting connection between function is trivial, i.e., ΨSUSY the geometry of the spacetime, the Kähler manifold (see Sect. 3.2.3 of Vol. I), and the dependence of the superpotential on φ, φ regarding the existence of non-trivial FRW = 0 could be interpreted as meaning that supersymmetric states. Moreover, ΨSUSY supersymmetry breaking has taken place, as long as non-trivial quantum solutions could be found for the constraint H = 0. This does constitute a particular difference regarding standard supersymmetric quantum mechanics (see, e.g., Sect. 3.3). It is just the positivity of the potential (and of the Hamiltonian expressed through a spectrum of energy levels) that establishes whether supersymmetry is broken. We should stress again that models like ours have N = 4 supersymmetry and time reparametrization invariance (associFRW = H ated with HSUSY free + HSPot = 0), endorsed through the N = 1 supergravity action. These combined properties are absent in standard supersymmetric quantum mechanics. All such aspects and hypotheses could be further investigated with regard to the problem of supersymmetry breaking within SUGRA, superstring theory, and M-theory, by means of simple (minisuperspace) quantum mechanical models with N = 2 or even N = 4 SUSY.

5.2 Supermatter Although the framework and subsequent calculations in this section may seem a straightforward extension of those in Sect. 5.1 of Vol. I, the fact is that some unexpected problems and obstacles do occur, and this is the main reason for displaying them in this volume.

5.2

Supermatter

91

5.2.1 Generic Gauge Supermatter in FRW For the FRW model with Yang–Mills fields obtained from the more general theory of N = 1 SUGRA with gauged supermatter (see Sect. 5.1.5 of Vol. I), associated with a gauge group Gˆ = SU(2), we have put all scalar fields and corresponding supersymmetric partners equal to zero.2 It should be noted that Yang–Mills fields coupled to N = 1 SUGRA can also be found in [19–25]. We are using the ansatz (a) (a) (a) (5.163) of Vol. I for Aμ ≡ vμ . This implies that Aμ is parametrized by a single effective scalar function f (t). Ordinary FRW cosmologies with this Yang–Mills field ansatz are totally equivalent to an FRW minisuperspace with an effective conformally coupled scalar field, but with a quartic potential instead of a quadratic one. In this section we consider a larger supermatter sector, including a (complex) scalar field and a vector (gauge) field, together with their supersymmetric partners, providing an FRW minisuperspace with a considerably richer content. However, a significant problem occurs, which has yet to be dealt with. In fact, the results were quite unexpected [26, 2]. The only allowed physical state was Ψ = 0! Overall, the results mentioned above strongly suggest that the treatments of N = 4 FRW SQC here and in Chap. 5 of Vol. I may benefit from a fresh and critical approach to the consideration of supermatter. Ansätze and Quantum Constraints Once again we take the group Gˆ = SU(2) as the gauge group in a k = +1 FRW model. In this section, however, in contrast with Sect. 5.1.5 of Vol. I, we do have to deal with the Killing potentials (see Appendix A and also Sect. 4.2.3 of Vol. I): D (1)

1 ≡ 2

φ+φ 1 + φφ

D (2)

,

i ≡− 2

φ−φ 1 + φφ

,

D (3)

1 ≡− 2

1 − φφ

. 1 + φφ (5.10)

The Kähler potential is K = ln(1 + φφ), while the corresponding Kähler metric is Gφφ =

1 (1 + φφ)2

,

Gφφ = (1 + φφ)2 .

The Levi-Civita connections of the Kähler manifold are just φ

%φφ = Gφφ

∂Gφφ ∂φ

= −2

φ 1 + φφ

,

and its complex conjugate, the other components being zero. It was shown in [27] for the case of the gauge group SO(3) ∼ SU(2) that invariance under homogeneity and isotropy as well as gauge transformations require all components of φ to be zero. Only for SO(N ), N > 3, can we have φ = (0,0,0, φ1 , . . . , φ N −3 ).

2

92


The ansätze are those of Sects. 6.1.1, 6.1.4, and 6.1.5, i.e., we take the scalar supermultiplet consisting of a complex scalar field φ, φ and a spin 1/2 field χ A , χ A

to be spatially homogeneous, depending only on time. Similarly, for the spin 1/2 (a) field λ(a) A = λ A (t), (a) = 1, 2, 3, using the ansätze (5.1) and (5.2) together with expression (5.165) of Vol. I. The action of the full theory reduces to one with a finite number of degrees of freedom. Again, we bring together the procedure and features of Sects. 5.1.1, 5.1.4, and (a) 5.1.5 of Vol. I, i.e., redefining the fermionic fields χ A , ψ A , and λ A in order to simplify the Dirac brackets. We take these to be [a, πa ]D = 1 ,

[φ, πφ ]D = 1 ,

[φ, πφ ]D = 1 ,

[ f, π f ] D = 1 ,

(5.11)

and with the unprimed spinors, [χ A , χ B ]D = −iε AB ,

[ψ A , ψ B ]D = iε AB ,

λ(a) A , λ(a) A

D

= −iδ ab ε AB , (5.12)

from which the only nonzero (anti)commutator relations are now {λ(a) A , λ(b) B } = δ ab ε AB ,

{χ A , χ B } = ε AB ,

{ψ A , ψ B } = −ε AB ,

[a, πa ] = [φ, πφ ] = [φ, πφ ] = [ f, π f ] = i .

(5.13)

(5.14)

Finally, we choose {λ(a) A , χ A , ψ A , a, φ, φ, f } to be the coordinates of the configuration space, and {λ(a) A , χ A , ψ A , πa , πφ , πφ , π f } to be the momentum operators in this representation. Hence, λ(a) A −→ − ∂ πa −→ − , ∂a

∂ ∂λ(a)A

,

χ A −→ −

∂ πφ −→ −i , ∂φ

∂ , ∂χ A

ψ A −→

∂ πφ −→ −i , ∂φ

∂ , ∂ψ A

∂ π f −→ −i . ∂f

(5.15)

When matter fields are taken into account, the generalization of the J AB constraint is

(a)

J AB = ψ(A ψ B n B)B − χ(A χ B n B)B − λ(A λ(a)B n B)B = 0 .

(5.16)

The contributions from the complex scalar fields φ and φ to the SUSY S A constraint are found to be3

3

X a ia the Killing vector of the associated Kähler geometry.

5.2

Supermatter

93

1 iς 2 a 3 5i φ ς 2a3φ

n B B χ B χ B n B B λ(a)B λ(a)B − √ √ χ A πφ + √ 3 2 2 2 (1 + φφ) 2 2 (1 + φφ) 3i ς 2 a 3 φ 3 ς 2a3

B B B B − √ n B B χ ψ n B B ψ ψ − √ 2 2 (1 + φφ) 2 (1 + φφ)2 ς 2a2 g f

+√ σ a A A n AB χ B X a . 2 2(1 + φφ)

(5.17)

The contributions to the SUSY S A constraint from the spin 1 field are √

2 ς 2a3 a

π f σ a B A λ(a)B + σ B A λ(a)B n C B σ bCC ψ C λ(b)B + σ b AB ψ A λ(b)C 3 6 1 1 + √ ς 2 a 4 σ (a)C A 1 − ( f − 1)2 λ(a) C + ς 2 a 3 λ(a)A (5.18) 2 8 2 1 1

(a)B (a)B (a)B (a)B × −n AB ψ A λ + n B A ψ A λ − n A A ψ B λ + n A A ψ B λ . 2 2 −i

The contributions to the S A constraint from the spin 2 and spin 3/2 fields are i 3 3

√ aπa ψ A − √ ς 2 a 2 ψ A + ς 2 a 3 n B A ψ B ψ B ψ B . 8 2 2 2

(5.19)

The following terms are also present in the S A supersymmetry constraint: ς 2a3 1 − √ ς 2 a 3 g D (a) n A A λ(a)A + 2 2 1 + φφ

−n

B A

ψ

B

1

+ nBB ψ A χ Bχ B 2

1

− ς 2 a 3 n AB λ(a)A λ(a) A ψ B + n A A λ(a)A λ(a) B ψ B 4 1

− ς 2 a 3 n AB χ A χ A ψ B + n A A χ A χ B ψ B . 4(1 + φφ)2

(5.20)

The supersymmetry constraint S A is then the sum of the above expressions. The supersymmetry constraint S A is just the complex conjugate of S A . Notice that the above expressions correspond to a gauge group SU(2) and hence a compact Kähler manifold, which implies that the analytical potential P(φ I ) is zero. Following the ordering used in [2], we put all the fermionic derivatives in S A on the right. In S A , all the fermionic derivatives are on the left. Implementing all these redefinitions, the supersymmetry constraints have the differential operator form

94


∂ ∂ i 1 S A = − √ (1 + φφ)χ A − √ aψ A − ∂φ ∂a 2 6 2

7

3 2 2 ∂ 5i ς a ψ A − √ φχ A χ B 2 ∂χ B 4 2 √ ∂ ∂ ∂ 3 ∂ 1 i 5 − √ ψ B ψ B A − √ φχ A ψ B B − √ χ A ψ B + √ χ B ψB B ∂ψ ∂ψ ∂χ ∂χ A 8 6 4 6 4 2 4 2 ∂ ∂ 1 1

+ √ ψAχ B + √ σ a AB σ bCC n DB n B C λ(a)D ψC ∂χ B ∂λ(b)B 2 6 3 6 ∂ ∂ 1 1

+ √ σ a AB σ bB A n D B n E A λ(a)D λ(b) B − √ ψ A λ(a)C E (a)C ∂ψ λ 6 6 2 6 ∂ 3 a (a)C ∂ 1 2 3 (a) a 1 + √ λ Aλ + √ ς a g D λ A − √ ψ C λ(a) C C ∂ψ ∂λ(a)A 8 6 4 6 2 2 ς 2a2 g f

+√ σ a A A n B A X (a) χ B 2(1 + φφ) √ 2 1 ∂

+σ a A A n B A λ(a) B − + √ 1 − ( f − 1)2 ς 2 , 3 ∂f 8 2

(5.21)

and S A is just the Hermitian conjugate of (5.21). (No) Quantum States The Lorentz constraint J AB implies that a physical wave function should be a Lorentz scalar. We can easily see that the most general form of the wave function is Ψ = A + iBψ C ψC + Cψ C χC + iDχ C χC + Eψ C ψC χ C χC +ca λ(a)C χC + da λ(a)C χC + cab λ(a)C λ(b) C + ea λ(a)C χC ψ D ψD +fa λ(a)C ψC χ D χD + dab λ(a)C χC λ(a)D χD + eab λ(a)C λ(b) C ψ D ψD +fab λ(a)C λ(b) C χ D χD + gab λ(a)C λ(b) C χ D ψD + cabc λ(a)C λ(b) C λ(c)D ψD +dabc λ(a)C λ(b) C λ(c)D χD + cabcd λ(a)C λ(b) C λ(c)D λ(d) D +hab λ(a)C λ(b) C ψ D ψD χ E χ E + eabc λ(a)C λ(b) C λ(c)D χD ψ E ψ E +fabc λ(a)C λ(b) C λ(c)D ψD χ E χ E + dabcd λ(a)C λ(b) C λ(c)D λ(d) D ψ E ψ E +ν3 λ(1)C λ(1) C λ(2)D λ(2) D λ(3)E χ E + eabcd λ(a)C λ(b) C λ(c)D λ(d) D χ E χ E +fabcd λ(a)C λ(b) C λ(c)D λ(d) D ψ E χ E + gabcd λ(a)C λ(b) C λ(c)D ψD λ(d)E χ E +μ1 λ(2)C λ(2) C λ(3)D λ(3) D λ(1)E ψ E + μ2 λ(1)C λ(1) C λ(3)D λ(3) D λ(2)E ψ E +μ3 λ(1)C λ(1) C λ(2)D λ(2) D λ(3)E ψ E + ν1 λ(2)C λ(2) C λ(3)D λ(3) D λ(1)E χ E +ν2 λ(1)C λ(1) C λ(3)D λ(3) D λ(2)E χ E + Fλ(1)C λ(1) C λ(2)D λ(2) D λ(3)E λ(3) E +habcd λ(a)C λ(b) C λ(c)D λ(d) D ψ E ψ E χ F χ F +δ1 λ(2)C λ(2) C λ(3)D λ(3) D λ(1)E ψ E χ F χ F

5.2

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95

+δ1 λ(2)C λ(2) C λ(3)D λ(3) D λ(1)E ψ E χ F χ F +δ2 λ(1)C λ(1) C λ(3)D λ(3) D λ(2)E ψ E χ F χ F +δ3 λ(1)C λ(1) C λ(2)D λ(2) D λ(3)E ψ E χ F χ F +γ1 λ(2)C λ(2) C λ(3)D λ(3) D λ(1)E χ E ψ F ψ F +γ2 λ(1)C λ(1) C λ(3)D λ(3) D λ(2)E χ E ψ F ψ F +γ3 λ(1)C λ(1) C λ(2)D λ(2) D λ(3)E χ E ψ F ψ F +Gλ(1)C λ(1) C λ(2)D λ(2) D λ(3)E λ(3) E ψ F ψ F +Hλ(1)C λ(1) C λ(2)D λ(2) D λ(3)E λ(3) E χ F χ F +Iλ(1)C λ(1) C λ(2)D λ(2) D λ(3)E λ(3) E χ F ψ F +Kλ(1)C λ(1) C λ(2)D λ(2) D λ(3)E λ(3) E ψ F ψ F χ G χG ,

(5.22)

where A, B, etc., are functions of a, φ ,φ, f alone. This ansatz contains all allowed combinations of the fermionic fields and is the most general Lorentz invariant function we can write down (see Note 5.5 of Vol. I). Let us now solve the supersymmetry constraints S A Ψ = 0 and S A Ψ = 0 for the more general case where all supermatter fields are present. The number of constraint equations will be considerable and we present some examples of the calculations involving the S A Ψ = 0 constraint (see [28] for more details). Consider the terms linear in χ A : ∂A ς 2a2 f i

χA + √ σ a A A n B A X (a) Aχ B = 0 . − √ (1 + φφ) ∂φ 2 2(1 + φφ) Since this is true for all χ A , the above equation becomes

∂A ς 2a2 f i

εA B + √ σ a A A n B A X (a) A = 0 . − √ (1 + φφ) ∂φ 2 2(1 + φφ)

(5.23)

We now multiply the whole equation by n B B and use the relation n B B n B A =

ε B A /2. We can see that the two terms in (5.23) are independent of each other since the σ matrices are orthogonal to the n matrix. Thus, we conclude that A = 0. Now consider, e.g., the terms linear in χ B ψ C ψC . We have

√ 3 2 2 ∂B 1 a ∂C 7 0 = (1 + φφ) + φB + i √ −i √ C+i ς a C χ A ψ C ψC ∂φ 2 2 4 3 ∂a 4 3 ς 2a2 g f

+i √ σ a A A n B A X (a) Bχ B ψ C ψC . 2(1 + φφ)

(5.24)

By the same argument as above, the first term is independent of the second one, and we have the result B = 0.

96


As we proceed, this pattern keeps repeating itself. Some equations show that the coefficients have certain symmetry properties. For example, let us take d(ab) = 2g(ab) . But when these two terms are combined with each other, they become zero. This can be seen as follows: dab λ(a)C χC λ(a)D χ D +gab λ(a)C λ(b) C χ D ψ D (a)C (b)

= −gab λ

λ

Cχ

D

(a)C (b)

ψ D + gab λ

λ

(5.25) Cχ

D

ψD ,

using the property that gab = gba and the spinor identity θ AB = θC C ε AB /2, where θ AB is antisymmetric in the two indices. The same property applies to the terms with coefficients fabcd and gabcd . Other equations imply that the coefficients cabc , dabc , cabcd , eabc , fabc , dabcd , eabcd , and habcd are totally symmetric in their indices. This then leads to terms cancelling each other, as can easily be shown. In the end, considering both the S A Ψ = 0 and S A Ψ = 0 constraints, we are left with the surprising result that the wave function (5.22) must be zero in order to satisfy the quantum constraints!

5.2.2 Scalar Supermultiplets in Bianchi IX In this section and the next, we extend the scope of SQC to discuss models with supermatter (see Sect. 5.2 of Vol. I). There is, however, an important point. We employ the methodology of [29, 30] and are still far from importing the setting of [31–34] or even of [35]. We leave this as a suggestion for the fellow explorer. Although it may seem like a retreat, the point is that, at this level, our model bears important differences with regard to FRW models with supermatter. In particular, besides the anisotropic gravitational degrees of freedom that are present, additional gravitino spin 3/2 modes can now be included. As a consequence, their presence may play an important role, revealing some of the features of the full theory of N = 1 SUGRA with supermatter. Moreover, this ‘reduced’ Lorentz invariant construction is still of some use, namely in obtaining and investigating solutions in the Bianchi SQC backgrounds with (super)matter fields. It will also help us to investigate how a transition from an SQC k = +1 FRW model with a scalar supermultiplet to a Bianchi IX brings about severe complications. Including vector (gauge) fields will be even more strenuous (see next section). The issue is by no means closed and much remains to be discussed. Let us therefore take a Bianchi type IX model coupled to a scalar supermultiplet, as studied in [36, 37]. This model has a spatial metric in diagonal form, for which we will be using the contribution from supersymmetry constraints in the form (4.80), A

(4.81) of Vol. I. Here we require the components ψ A 0 , ψ 0 to be functions of time

alone, and also ψ A i and ψ A i to be spatially homogeneous in the basis ea i . We restrict our case to a supermatter model comprising only a scalar field and its spin 1/2 partner with a 2D flat Kähler geometry. The scalar supermultiplet is chosen to be spatially homogeneous.

5.2

Supermatter

97

wave functions of the form The quantum description can be made by studying

A A A A

Ψ e i , ψ i , χ A , φ, φ . The choice χ A ≡ n A χ A rather than χ A is designed so that the quantum constraint S A will be of first order in momenta. The momenta are represented4 by p A A i −→ −ih¯

πφ −→ −ih¯

δ

δe A A

i

1 − √ εi jk ψ A j ψ A k , 2

∂ , ∂φ

∂ 1

ψ A i −→ √ ih¯ D A A ji h 1/2 , ∂ψ A j 2

πφ −→ −ih¯

∂ ∂φ

,

√ ∂ χ A −→ − 2h¯ . ∂χ A

(5.26)

(5.27)

(5.28)

With (see Appendix A) δ δ = −2eCC i ,

δh i j δeCC j the supersymmetry constraints become [see Sect. 4.2.3 of Vol. I and in particular (4.90)] √ δ δ 1 1

i

j A CC A CC S A = −i 2 −ih¯ e A A i ψ j e e A A i ψ j e ¯

− ih

4 4 δeCC j δeCC i √ ∂ i + 2εi jk e A A i 3(s)ω A B j ψ B j − √ h¯ n C A χ C ∂φ 2 √ √ ∂

ˇ φ Pn A A ∂ −i 2h¯ heK/2 P(φ)n A A ei AB D C B ji 2h¯ heK/2 D − i C ∂χ A ∂ψ j ∂ i

− √ h 1/2 h¯ φn B B n C B χ C n D A χ D B ∂χ 2 2 ∂ i

− √ h¯ h 1/2 φεi jk e B B j ψk B D C B li n D A χ D ∂ψlC 4 2 √ ∂

−h¯ 2h 1/2 e B B m n C B ψmC n D A χ D B ∂χ ∂ i

− √ h¯ εi jk e A A j ψiA n D B χ D e B B k B ∂χ 2 ∂ 1

+ √ h¯ h 1/2 ψi A (e B A i n AC − e AC i n B A )n DC χ D B , ∂χ 2 2

(5.29)

4 It is interesting to note that, for the χ, χ fields, no powers of h seem to be needed to establish the equations for the coefficients in Ψ .

98


and S A is the Hermitian conjugate. Notice that: • Taking the analytical potential P(φ) to be constant is similar to including a cosmological constant term. • The last four lines in (5.29) correspond to a mixing between ψ and χ. The integrated form of the constraints is employed throughout. Our particular Lorentz-invariant wave function is then taken to be a polynomial of degree eight in Grassmann variables: Ψ (a1 , a2 , a3 , φ, φ) = A + B1 β A β A + B2 χ A χ A + C1 γ ABC γ ABC +D1 β A β A γ E BC γ E BC + D2 χ A χ A γ E BC γ E BC +F1 (β A β A γ E BC γ E BC )2 + F2 χ A χ A (γ E BC γ E BC )2 +G1 β A β A χ B χ B + H1 β A β A χ B χ B γ E DC γ E DC +E1 (γ ABC γ ABC )2 + I1 β A β A χ B χ B (γ E DC γ E DC )2 +Z1 χ A β A + Z2 χ A β A γ E DC γ E DC +Z3 χ A β A (γ E DC γ E DC )2 .

(5.30)

Its limitations will now be obvious to the attentive reader. As far as the middle sectors are concerned, the new Lorentz invariants built with gravitational degrees of freedom are not present. The complete ansatz method described in Sect. 5.2.3 of Vol. I to construct the correct middle fermionic sectors would give the correct spectrum of solutions. The action of the constraint operators S A , S A on Ψ leads to a system of coupled first order differential equations which the bosonic amplitude coefficients of Ψ must satisfy. These coefficients are functions of a1 , a2 , a3 , φ, and φ. The equations are

obtained after eliminating the eiA A and n A A resulting in S A Ψ = 0, S A Ψ = 0. We

contract them with combinations of e Bj B and n CC , followed by integration over S 3 . These equations correspond essentially to expressions in front of terms such as χ , β, γ , β A χ χ, γ ABC χ χ, etc., after the fermionic derivatives in S A , S A have been performed. The number of equations obtained will be very large indeed. Actually, there will be 44 × 3 equations, taking into account cyclic permutations on a1 , a2 , a3 . The full analysis is thus rather long-winded. We will therefore limit ourselves here to some of the steps involved in the calculations, and the interested reader is referred to [36, 37] for more details. The supersymmetry constraint S A has fermionic terms of the type βA ,

γ ABC ,

χA ,

∂ , ∂ψiA

∂ , ∂χ A

χχ

∂ , ∂χ

ψχ

∂ , ∂ψ

ψχ

∂ , ∂χ

while S A is of second order in fermionic derivatives and includes terms such as

5.2

Supermatter

∂ , ∂ψiA

∂ , ∂χ A

99

βA ,

γ ABC ,

χA ,

χ

∂ ∂ , ∂χ ∂χ

ψ

∂ ∂ , ∂χ ∂ψ

χ

∂ ∂ . ∂χ ∂ψ

Some of these fermionic terms applied to Ψ increase the fermionic order by a factor of one (e.g, χ ), while others like χ (∂/∂χ)(∂/∂ψ) decrease it by the same amount. It is worthwhile stressing the following result, which holds regardless of whether

we put P(φ) = 0 or not. Using the symmetry properties of eiA A , n A A , γ ABC , and ε AB , we can check that all equations which correspond to the terms γ , γββ, χ χγ , γ γ γ , γββγ γ , γ χ χ γ γ , γββχ χ , and γββχ χγ γ in S A Ψ = 0 and S A Ψ = 0 will give a similar expression for the the coefficients A, B1 , B2 , C1 , D1 , D2 , E1 , F1 , F2 , G1 , H1 , and I1 , in fact of the form 2 2 2 a1 +a2 +a3

P(a1 , a2 , a3 ; φ, φ)e±

.

(5.31)

This does not apply to the Z1 , Z2 , Z3 coefficients, since the βχ γ and βχ γ (γ γ ) terms from the two supersymmetry constraints just mix them with other coefficients in Ψ . This can be seen, e.g., from the equations corresponding to the γ D E F (ββ) term in S A Ψ = 0:

0 = 2εi jk e A A i ω A B j n D B eC B k B1 − h¯ n D B eC B

i

δB1

δe B A i

+(BCD −→ CDB) + (BCD −→ DBC) .

(5.32)

In the following we will describe two cases separately: when the analytic potential K(φ) is arbitrary and when it is identically zero. We begin by the former: • Consider the equations obtained from S A Ψ = 0 with terms linear in β and γ .

After contraction with expressions in eiA A and n A A and integrating over S 3 , we get C1 = 0. From the linear terms in β and γ from S A Ψ = 0, we get a relation between Z1 and

8π 2 2 2 2 , B1 Y exp a1 + a2 + a3 h¯ viz.,

8π 2 2 ∂Z1 2 2 =0. a1 + a2 + a3 + 2φZ1 − 8i Y(a1 a2 a3 ; φ, φ) exp ∂φ h¯

(5.33)

• For the particular case B1 = 0, i.e., Y = 0, it follows from the remaining equations that the only possible solution is Ψ = 0. • For an arbitrary B1 , (5.33) allows us to write an expression for Z1 in terms of functions of φ, φ, and a1 , a2 , a3 . If we use that expression in the other equations, we get formulas for the other bosonic coefficients.

100


From this procedure, we would get the general solution of this complicated set of differential equations. Although apparently possible, we could not establish a definite result in the end due to the complexity of the equations involved. As in [36, 37], there is apparently no easy way of obtaining an analytical solution to this set of equations. Moreover, the exponential terms eK/2 lead to some difficulties. A possible route to explore might be to extend the methods in [38] to a Bianchi IX model. Let us now consider the case where the analytical potential is chosen to be identically zero. From the equations obtained directly from S A Ψ = 0 and S A Ψ = 0, we have self-contained groups of equations relating the 15 wave function coefficients. This applies to 3 groups involving (A, B1 , B2 , C1 , Z1 ), (G1 , D1 , D2 , E1 , Z2 ), and (H1 , F1 , F2 , I1 , Z3 ). Note also that: • The equations corresponding to the terms linear in β, γ , χ in S A Ψ = 0 and βχ χ (γ γ )2 , ββχ (γ γ )2 in S A Ψ = 0 completely determine the coefficients A and I1 . • In addition A and I1 do not appear in any other equation. We thus have 8π 2 2 A = f (φ) exp − , a1 + a22 + a32 h¯ 8π 2 2 2 I1 = k(φ) exp − a1 + a22 + a32 e−2π φφ . h¯

(5.34)

• The equations involving B1 , B2 , C1 , and Z1 can also be said to be self-contained in the same sense. They involve only these coefficients and no others. Moreover, these coefficients do not occur in any other equations. This can be checked by examining the equations for the terms linear in β and χ in S A Ψ = 0, ββχ and βχ χ in S A Ψ = 0, ββγ in S A Ψ = 0, χ γ γ , βγ γ , and γ γ γ in S A Ψ = 0, and γ in S A Ψ = 0. The previous terms in χ γ γ , βγ γ , γ γ γ , and γ just involve C1 . All the other equations involve B1 , B2 , and Z1 . • However, the βχ γ equation in S A Ψ = 0 mixes B1 , C1 , and Z1 . Actually, it is the only equation which mixes C1 with the remaining bosonic coefficients in the corresponding group. The same structure of equations and relations between coefficients also occurs, in particular, for the subsets involving D1 , D2 , E1 , Z2 , and F1 , F2 , I1 , Z3 . • From the analysis of the group of equations which includes B1 , B2 , C1 , Z1 (ββχ , βχ χ equations from S A Ψ = 0 and β, χ equations from S A Ψ = 0), we get Z1 = 0. Consequently, the equations corresponding to only β and γ involve just B1 and C1 . These equations are then like those of a Bianchi IX model with Λ = 0 and no supermatter. • The only possible solution of these equations with respect to a1 , a2 , a3 is the trivial one, i.e., B1 = C1 = 0. The equations corresponding to χ and combinations of it with β or γ would give, with B1 = C1 = Z1 = 0, the dependence of B2 on a1 , a2 , a3 , φ. This corresponds to

5.2

Supermatter

101

8π 2 2 2 2 2 B2 = h(φ)a1 a2 a3 exp − a1 + a2 + a3 e−2π φφ . h¯

(5.35)

• This pattern repeats itself in a similar way when we consider the two groups involving D1 , D2 , E1 , Z2 , and F1 , F2 , Z3 . We get E1 = G1 = H1 = 0 from Z2 = D2 = 0, Z2 = D1 = 0, and Z3 = F1 = 0. Hence, besides A and I1 , only B2 and F2 will be different from zero. For the solution of the constraints, we can then write 2 2 2 8π 2 2 2 2 2 Ψ = f (φ) exp − a1 +a2 +a3 +h(φ)a1 a2 a3 e −a1 −a2 −a3 e−2π φφ χ A χ A h¯ 2 8π 2 2 2 2 +g(φ)a1 a2 a3 exp a1 + a2 + a3 e−2π φφ β A β A (γ BC D γ BC D )2 h¯ 2 8π +k(φ) exp (5.36) a12 + a22 + a32 χ A χ A β E β E (γ BC D γ BC D )2 . h¯

5.2.3 Vector Fields in Bianchi I (N = 2 SUGRA) On the one hand, this section will acquaint the reader with Bianchi SQC when the matter fields have a spatial (vectorial) presence. On the other hand, this will be carried out within a richer framework. In fact, models with a richer structure can be found from extended supergravity theories [39]. These are theories with more gravitinos, which have additional symmetries coupling several physical variables. N = 2 SUGRA couples a graviton–gravitino pair with another pair constituted by another gravitino and a Maxwell field. It contains a manifest O(2) invariance which rotates the two gravitinos into one another. We consider the two cases when the internal O(2) symmetry is global or local. Here, we mention the canonical quantization of Bianchi class A models in N = 2 SUGRA, as addressed in [40–42] (see also [43–46]). It was found that the presence of the Maxwell field in the supersymmetry constraints leads to a non-conservation of the fermion number. This then implies a mixing between Lorentz invariant fermionic sectors in the wave function. It should be stressed that the intertwining of the different fermionic sectors in minisuperspaces obtained in N = 1 supergravity with supermatter is different from the mixing now caused by the Maxwell field. The action for the general theory of N = 2 SUGRA in the global symmetry case can be written in the form e e a a (a) R e , ω e , ψ εμνρσ γ5 γν Dρ (ω)ψσ(a) − ψ (a) μ μ ν 2 μ 2k2 e 2 k 1 ˜ˆμν ) ψ (b) εab , (5.37) μν μν μν ˆ ˜ − f μγ + √ ψ (a) + f ) + ( f + f γ e( f 5 μ ν 4 2 4 2

L≡−

102


where k (a) fˆμν = ∂μ Aν − √ ψ μ ψν(b) εab − (μ ↔ ν) , 2 2

(5.38)

(a)

and f˜μν is εμνρσ f ρσ . The gravitinos ψμ are expressed here in the 4-component representation, (a) = 1, 2 are O(2) group indices, Aμ ≡ vμ is a Maxwell field, ω is the connection, γμ are Dirac matrices, and now γ5 = γ0 γ1 γ2 γ3 . Furthermore, ε12 = 1, ε21 = −1 (see Appendix A). The canonical quantization of Bianchi class A models requires the following two steps. On the one hand, we have to rewrite the action (5.37) in 2-component spinor notation. On the other hand, we impose a consistent Bianchi anzätze for all fields, similar to the previous section. But let us be more precise. Choosing a symmetric basis, the tetrad components eia are time dependent only, like the spatial components of the gravitino fields ψiA(a) and the Maxwell field Ai . We also require the other components to be time dependent only. Then we note the following: • The momentum conjugate to the vector field Ai is ek (a)A (b)i (b)i A

ψ + ψ ψ π i = eh i j ∂0 A j − √ ∂0 Ai εab ψ (a)

A 0 0A 2 iek (a) (b)A

(a)A (b) − ψ j ψk A . + √ ∂0 Ai εi jk εab ψ j A ψ k 2 2 • The Dirac bracket relations are

j j ei A A , pˆ B B = δi δ BA δ BA ,

(5.40)

D

(a)A

ψi

(b)A

,ψ j

Ai , π j

D

D

(5.39)

= −δ ab D A A i j , j

= δi ,

(5.41) (5.42)

where we have defined 1 (a) (a) pˆ iA A ≡ piA A − εi jk ψ j A ψ k A . 2

(a)

(a)i

• The Lorentz constraints are J AB = pi(A A eiB)A + ψi(A π B) and its conjugate. Notice that, as a consequence of choosing a symmetric basis, we have f μν = Aν,μ − Aμ,ν + Aσ Cσμν ,

5.2

Supermatter

103

where the structure constants characterizing the Bianchi model are Cσμν = 0 if one or more indices are equal to zero. This simply means that, according to the chosen Bianchi type, we can have a pure electric, pure magnetic, or both fields. Quantum mechanically, the bracket relations become -

j j ei A A , pˆ B B = ih¯ δi δ B A δ B A ,

(a)A

ψi

(b)A

,ψ j

.

(5.43)

= −ih¯ δ ab D A A i j ,

(5.44)

j Ai , π j = ih¯ δi .

(5.45)

. Choosing Ai , ei A A , ψi(a)A as the coordinate variables in our minisuperspace, we then have ∂

j

pˆ A A −→ −ih¯

∂e Aj A

,

ψ (a)A −→ −ih¯ D A A i j j

∂ (a)A ∂ψi

,

π i −→ −ih¯

∂ . ∂ Ai (5.46)

We now consider the physical states which are solutions of the above constraints. (a) The quantum states may be described by the wave function Ψ e A A i , A j , ψ Ai . (a) From the Lorentz constraint, Ψ must be expanded in even powers of ψ Ai , symbolically represented by (ψ)0 , (ψ)2 up to (ψ)12 . This is due to the anticommutation relations of the six spatial components of the two types of gravitino.

Global O(2) Notice that we do not get any gauge (or central charge) constraint term of the form A0 Q (see Chap. 3 of Vol. I). This is due to our specific choices of homogeneous ansätze mentioned above, and also due to the choice of global invariance. Multiplying the Lorentz constraints by ω0AB and ω0A B and adding them in this way to the supersymmetry constraints, the latter take the form (a) (a)A i ab i 1/2 k imn cd (c)B (d) 1/2 1 i jk S A =−i p A A ψi +kε ψ i(b) ε ε ψm ψn B + ih f jk ε π − ih A

4 8 (5.47) and its Hermitian conjugate. After all the simplifications, we finally obtain the quantum supersymmetry constraints in the form (a)

∂ 1 (a)A (b) − h¯ εi jk ' ab ψi ψ j A D B A lk (5.48) (b)B 2 ∂ψl ∂ ∂ ab c 1/2 k i jk cd (c)B (d) i 1/2 i jk −h , ε ε ψ j ψk B + h ε f jk +h¯ kε D A mi h¯ (b)C ∂ Ai 4 8 ∂ψm (a)A

S A = −h¯ ψi

∂

∂eiA A

104


12 = ' 21 = plus the terms in (5.52), while S (a) A is the Hermitian conjugate, with ' 1 and the remaining zero. An important consequence of (5.48) is that fermion number is not conserved by (a) (a) either of the supersymmetry constraints S A and S A . In fact, a mixing between fermionic modes occurs for Ψ . This is due to terms involving

∂ ∂ ∂ Ai ∂ψm(b)C

and

∂ (b) ψ , ∂ Ai i A (a)

or the ones associated with f jk . While the remaining fermionic terms in S A as given by (5.48) act on Ψ by increasing the fermionic order by a factor of one, ∂ ∂ ∂ Ai ∂ψm(b)C (a)

decreases it by the same amount. Concerning the S A constraint, the situation is precisely the reverse. The nature of this problem can also be better understood as follows. Let us consider the 2-fermion level. Since we have 12 degrees of freedom associated with the gravitinos, we may expect to have up to 66 terms in this fermionic sector. Thus, the 2-fermion level of the more general ansatz for the wave function can be written as Ψ2 ≡ Ci jab + Ei jab ψ (a)i B ψ (b) j

B

+ Ui jkab + Vi jkab ei A A n BA ψ (a) j A ψ (b)k B , (5.49)

where Ci jab =C(i j)(ab) ,

Ei jab =E[i j][ab] ,

Ui jkab =Ui( jk)[ab] ,

Vi jkab =Vi[ jk](ab) .

When Ψ is truncated to the second fermionic order, we obtain a set of equations, relating ∂Ψ0 ∂ai

with

∂Ci jab , ∂ Ai

∂Ei jab , ∂ Ai

∂Ui jkab , ∂ Ai

∂Vi jkab , ∂ Ai

(a)

from S A , and relating ∂Ci jab , ∂ai (a)

∂Ei jab , ∂ai

∂Ui jkab , ∂ai

∂Vi jkab , ∂ai

with

∂Ψ0 ∂ Ai

from S A . Here ai , i = 1, 2, 3, stand for scale factors in a Bianchi class A model, Ψ0 denotes the bosonic sector, and Ψ0 , Ci jab , Ei jab , Ui jkab , and Vi jkab are functions of A j , ai alone. Moving to the equations corresponding to higher fermionic terms, this pattern keeps repeating itself, with algebraic terms added to it.

5.2

Supermatter

105

However, the present situation is rather different from the one in FRW models in N = 1 supergravity with supermatter (see Sect. 5.1.5 of Vol. I), where the mixing occurs only within each fermionic level and decoupled from other Lorentz invariant fermionic sectors of different order. In the present case, the mixing is between fermionic sectors of the same and any different adjacent order. This situation is quite similar to the one caused by a cosmological constant. It is then found [40, 41] that the presence of the Maxwell field in the supersymmetry constraints leads to nonconservation of the fermion number. This in turn implies mixing between Lorentz invariant fermionic sectors in the wave function. It should be stressed that the intertwining between different fermionic sectors in minisuperspaces obtained in N = 1 supergravity with supermatter is different from the mixing now caused by the Maxwell field. At this stage, it is tempting to see the relations between gradient terms such as (a) (a) ∂Ψ0 /∂ai with ∂Ψ2 /∂ Ai (from S A ) and ∂Ψ2 /∂ai with ∂Ψ0 /∂ Ai (from S A ) as a consequence of N = 2 SUGRA [39], realizing Einstein’s dream of unifying gravity with electromagnetism. These relations establish a duality between the coefficients of Ψ in fermionic sectors of adjacent order relatively to the intertwining of the derivatives ∂/∂ Ai and ∂/∂ai . Local O(2) Had we considered a mininal coupling, i.e., gauging the O(2) transformations, then a term A0 Q could be present in the Hamiltonian. Let us spell this out. In fact, the promotion of the O(2) internal symmetry to a gauge transformation implies that the following terms (already written in 2-component spinor notation) should be added to the Lagrangian in (5.37): 7

2Λ A(a) μ B ν B(a)

(a) + ψ A μ e B A μ eνB B ψ νB (a) ψμ e AB e B ψν −eΛ − e − 3 7 1 Λ μνρσ A(a)

(a) (b) − e − ε ψμ eν A A Aρ ψ σA (b) − ψ μA eνA A Aρ ψσ A εab , 2 6

(5.50)

where the cosmological constant Λ < 0 is related to a gauge coupling constant, as a consequence of coupling the Maxwell field minimally to the fermions. From local invariance, we now get a gauge (central charge) constraint term A0 Q in the action. For the case of our Bianchi models, this takes the form 7 Q=h (a)

1/2

−

Λ ab i jk (a)A (b)A

. ε ε ψi e j A A ψ k 6

(5.51)

Furthermore, the S A supersymmetry constraint (5.47) gets the additional contributions 7 7 Λ 2Λ A i (b)A (a)B

− h 1/2 − . (5.52) e A n AB ψ i h 1/2 − εab εi jk ei A A A j ψk 6 3

106


In spite of the additional difficulties caused now by the cosmological constant and gravitino mass terms, equation (5.51) allows us to extract some information concerning the form of the wave function. Quantum mechanically, the gauge constraint takes the form Q ∼ εab ψm(a)A

∂ (b)A

∂ψm

.

(5.53)

Notice that the gauge constraint has no factor ordering problem due to the presence of εab . It is then a straightforward matter to check that the following fermionic expressions satisfy the gauge constraint operator above: (c)

QS mn ψm A ψn(c)A = 0 ,

QTi[ jk] eiA A n B A ψ (a) j A ψ (a)k B = 0 ,

(5.54)

where S mn = S (mn) . Hence, the most general solution of the quantum gauge constraint can be written as

(c) (5.55) Ψ = Ψ eiA A , A j , S mn ψm A ψn(c)A , Ti[ jk] eiA A n B A ψ (a) j A ψ (a)k B . However, obtaining non-trivial solutions of the supersymmetry constraints in the metric representation for Bianchi class A models with Λ = 0 has so far proved to be difficult. A simplified adaptation of the method outlined in [34] was nevertheless employed in [40, 41]. We will extend a solution previously obtained in [34] for the case of the general theory of N = 2 SUGRA. Regarding Bianchi class A models, this becomes Ψ = ei(F+G)+H ,

(5.56)

with 1 i j (a)A (a) B m ψi ψ j A + εi jk ψi(a)A ψk(a) F ∼ −√ B AjA , Λ

2 i m i j AiAB A j AB + AiAB ACjB AkC A , G∼ Λ 3 (a)i H ∼ − εi jk ψ A ψ (a)Ak A j + εi jk Ai f jk ,

(5.57) (5.58) (5.59)

where Ai AB are the complexified spin connections (see Chap. 6). In the metric representation, and for the case of a Bianchi IX model, the 12 th fermion level term, with half for each gravitino type, involves a bosonic coefficient Ψ12 (e A A i , Ai ) ∼ (A2 + A4 + A6 )e−m

i j eCC e jCC

i

eε

i jk A

i

f jk

,

(5.60)

Problems

107

where A2 = Ai Ai . Other bosonic coefficients for the 8 th fermionic order would include Ψ8 ∼ A4 e−(a1 +a2 +a3 )+(a1 a2 +a2 a3 +a3 a1 ) e 2

i jk A

i

f jk

,

(5.61)

Ψ8 ∼ A4 e−(a1 +a2 +a3 )+(a1 a2 −a2 a3 −a3 a1 ) e

i jk A

i

f jk

,

(5.62)

2

2

2

2

2

half for each gravitino type. The Chern–Simons functional constitutes an exact solution to the Ashtekar–Hamilton–Jacobi equations of general relativity with nonzero cosmological constant (see Chap. 6). Furthermore, the exponential of the Chern– Simons functional provides a semiclassical approximation to the no-boundary wave function in some minisuperspaces [48, 49]. However, the exponential of the Chern– Simons functional has also been shown not to be a proper quantum state, because it is non-normalizable (see [50] and references therein for related discussions). Summary and Review. The following is a list of items to assist and guide the reader through this chapter: 1. 2. 3.

Can (spontaneous) SUSY breaking be (further) discussed within SQC [Sect. 5.1]? Is there a problem when both scalar and vector supermultiplets are used? What are the main consequences [Sect. 5.2.1]? Can the Bianchi cases with scalar or vector (super)matter be improved or extended [Sects. 5.2.2 and 5.2.3]?

Problems 5.1 Time Gauge and SUSY Transformations General SUSY transformations will not preserve this gauge. Find out how a combination of SUSY transformations with appropriate Lorentz transformations can solve this. 5.2 Variation of the N = 1 SUGRA Action, Bianchi Models, and Boundary Terms Investigate how the action in homogeneous Bianchi A models changes under SUSY transformations. 5.3 Invariance of the N = 1 SUGRA Action, Bianchi Models and Boundary Terms Discuss how the action in homogeneous Bianchi A models is invariant under SUSY transformations if a correction boundary term is added.

108


5.4 Quantization of Bianchi Models and Boundary Terms How can the quantization of Bianchi class A models in SQC be changed if the boundary terms mentioned above are included?

References 1. D’Eath, P.D.: The canonical quantization of supergravity. Phys. Rev. D 29, 2199 (1984) 87 2. D’Eath, P.D.: Supersymmetric Quantum Cosmology, 252pp. Cambridge University Press, Cambridge (1996) 87, 88, 91, 93 3. Macias, A.: The ideas behind the different approaches to quantum cosmology. Gen. Rel. Grav. 31, 653–671 (1999) 87 4. Moniz, P.V.: A supersymmetric vista for quantum cosmology. Gen. Rel. Grav. 38, 577–592 (2006) 87 5. Casalbuoni, R.: On the quantization of systems with anticommuting variables. Nuovo Cim. A 33, 115 (1976) 87 6. Casalbuoni, R.: The classical mechanics for Bose–Fermi systems. Nuovo Cim. A 33, 389 (1976) 87 7. Moniz, P.V.: Supersymmetric quantum cosmology – shaken not stirred. Int. J. Mod. Phys. A 11, 4321–4382 (1996) 87 8. Bailin, D., Love, A.: Supersymmetric Gauge Field Theory and String Theory. Graduate Student Series in Physics, 322pp. IOP, Bristol (1994) 88 9. Muller-Kirsten, H.J.W., Wiedemann, A.: Supersymmetry: An introduction with conceptual and calculational details, pp. 1–586. World Scientific (1987). Print-86–0955 (Kaiserslautern) (1986) 88 10. Nilles, H.P.: Supersymmetry, supergravity and particle physics. Phys. Rep. 110, 1 (1984) 88 11. Sohnius, M.F.: Introducing supersymmetry. Phys. Rep. 128, 39–204 (1985) 88 12. Donets, E.E., Pashnev, A., Rosales, J.J., Tsulaia, M.: Partial supersymmetry breaking in multidimensional N = 4 SUSY QM. hep-th/0001194 (1999) 88 13. Donets, E.E., Pashnev, A., Juan Rosales, J., Tsulaia, M.M.: N = 4 supersymmetric multidimensional quantum mechanics, partial SUSY breaking and superconformal quantum mechanics. Phys. Rev. D 61, 043512 (2000) 88 14. Donets, E.E., Tentyukov, M.N., Tsulaia, M.M.: Towards N = 2 SUSY homogeneous quantum cosmology: Einstein–Yang–Mills systems. Phys. Rev. D 59, 023515 (1999) 88 15. Obregon, O., Rosales, J.J., Socorro, J., Tkach, V.I.: The wave function of the universe and spontaneous breaking of supersymmetry. hep-th/9812156 (1998) 88 16. Alty, L.J., D’Eath, P.D., Dowker, H.F.: Quantum wormhole states and local supersymmetry. Phys. Rev. D 46, 4402–4412 (1992) 89 17. D’Eath, P.D., Hughes, D.I.: Supersymmetric minisuperspace. Phys. Lett. B 214, 498–502 (1988) 89 18. D’Eath, P.D., Hughes, D.I.: Minisuperspace with local supersymmetry. Nucl. Phys. B 378, 381–409 (1992) 89 19. Ferrara, S., Gliozzi, F., Scherk, J., van Nieuwenhuizen, P.: Matter couplings in supergravity theory. Nucl. Phys. B 117, 333 (1976) 91 20. Ferrara, S., Scherk, J., Zumino, B.: Algebraic properties of extended supergravity theories. Nucl. Phys. B 121, 393 (1977) 91 21. Ferrara, S., Scherk, J., Zumino, B.: Supergravity and local extended supersymmetry. Phys. Lett. B 66, 35 (1977) 91 22. Freedman, D.Z.: SO(3) invariant extended supergravity. Phys. Rev. Lett. 38, 105 (1977) 91 23. Freedman, D.Z., Das, A.: Gauge internal symmetry in extended supergravity. Nucl. Phys. B 120, 221 (1977) 91 24. Freedman, D.Z., Schwarz, J.H.: Unification of supergravity and Yang–Mills theory. Phys. Rev. D 15, 1007 (1977) 91

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25. Freedman, D.Z., Schwarz, J.H.: N = 4 supergravity theory with local SU(2) SU(2) invariance. Nucl. Phys. B 137, 333 (1978) 91 26. Cheng, A.D.Y., D’Eath, P.D., Moniz, P.R.L.V.: Quantization of a Friedmann–Robertson– Walker model in N = 1 supergravity with gauged supermatter. Class. Quant. Grav. 12, 1343–1354 (1995) 91 27. Moniz, P.V., Mourao, J.M.: Homogeneous and isotropic closed cosmologies with a gauge sector. Class. Quant. Grav. 8, 1815–1832 (1991) 91 28. Cheng, A.D.Y., D’Eath, P.D., Moniz, P.R.L.V.: Quantization of a Friedmann–Robertson– Walker model in N = 1 supergravity with gauged supermatter. gr-qc/9503009 (1995) 95 29. Cheng, A.D.Y., D’Eath, P.D., Moniz, P.R.L.V.: Quantization of the Bianchi type IX model in supergravity with a cosmological constant. Phys. Rev. D 49, 5246–5251 (1994) 96 30. D’Eath, P.D.: Quantization of the supersymmetric Bianchi I model with a cosmological constant. Phys. Lett. B 320, 12–15 (1994) 96 31. Csordas, A., Graham, R.: Nontrivial fermion states in supersymmetric minisuperspace. grqc/9503054 (1994) 96 32. Csordas, A., Graham, R.: Supersymmetric minisuperspace with nonvanishing fermion number. Phys. Rev. Lett. 74, 4129–4132 (1995) 96 33. Csordas, A., Graham, R.: Hartle–Hawking state in supersymmetric minisuperspace. Phys. Lett. B 373, 51–55 (1996) 96 34. Graham, R., Csordas, A.: Quantum states on supersymmetric minisuperspace with a cosmological constant. Phys. Rev. D 52, 5653–5658 (1995) 96, 106 35. Cheng, A.D.Y., D’Eath, P.D.: Diagonal quantum Bianchi type IX models in N = 1 supergravity. Class. Quant. Grav. 13, 3151–3162 (1996) 96 36. Moniz, P.V.: Back to basics? . . . or how can supersymmetry be used in a simple quantum cosmological model. gr-qc/9505002 (1994) 96, 98, 100 37. Moniz, P.V.: Quantization of the Bianchi type IX model in N = 1 supergravity in the presence of supermatter. Int. J. Mod. Phys. A 11, 1763–1796 (1996) 96, 98, 100 38. Moniz, P.V.: FRW minisuperspace with local N = 4 supersymmetry and self-interacting scalar field. Annalen Phys. 12, 174–198 (2003) 100 39. Van Nieuwenhuizen, P.: Supergravity. Phys. Rep. 68, 189–398 (1981) 101, 105 40. Cheng, A.D.Y., Moniz, P.V.: Quantum Bianchi models in N = 2 supergravity with global O(2) internal symmetry. In: 6th Moscow Quantum Gravity, Moscow, Russia, 12–19 June 1995 101, 105, 106 41. Cheng, A.D.Y., Moniz, P.V.: Canonical quantization of Bianchi class A models in N = 2 supergravity. Mod. Phys. Lett. A 11, 227–246 (1996) 101, 105, 106 42. Moniz, P.V.. Why two is more attractive than one . . . . or Bianchi class A models and Reissner– Nordstroem black holes in quantum N = 2 supergravity. Nucl. Phys. Proc. Suppl. 57, 307–311 (1997) 101 43. Pimentel, L.O.: Anisotropic cosmological models in N = 2, D = 5 supergravity. Class. Quant. Grav. 9, 377–381 (1992) 101 44. Pimentel, L.O., Socorro, J.: Bianchi V models in N = 2, D = 5 supergravity. In: 7th Marcel Grossmann Meeting on General Relativity (MG 7), Stanford, CA, 24–30 July 1994 101 45. Pimentel, L.O., Socorro, J.: Bianchi VI(0) models in N = 2, D = 5 supergravity. Gen. Rel. Grav. 25, 1159–1164 (1993) 101 46. Pimentel, L.O., Socorro, J.: Bianchi V models in N = 2, D = 5 supergravity. Int. J. Theor. Phys. 34, 701–706 (1995) 101 47. Van Nieuwenhuizen, P.: Supergravity. Phys. Rep. 68, 189–398 (1981) 48. Graham, R., Paternoga, R.: Physical states of Bianchi type IX quantum cosmologies described by the Chern–Simons functional. Phys. Rev. D 54, 2589–2604 (1996) 107 49. Paternoga, R., Graham, R.: The Chern–Simons state for the non-diagonal Bianchi IX model. Phys. Rev. D 58, 083501 (1998) 107 50. Mena Marugan, G.A.: Is the exponential of the Chern–Simons action a normalizable physical state? Class. Quant. Grav. 12, 435–442 (1995) 107

Chapter 6

Connections and Loops Within SQC

This is an area of SQC where there exists a wide range of possibilities for exploration. Some progress has been made, in which cosmological models were investigated either through a reduction of SUGRA formulated in terms of connections (or loops) [1–19] or through a (hidden) N = 2 SUSY framework based upon Bianchi models described ab initio with connections [20, 21]. Further motivation will surely arise from recent developments in loop quantum cosmology [22–43], inheriting some of the main features and principles from loop quantum gravity [44–66]. In this chapter, we explain why the outlook is so promising in this area.

6.1 A Brief Summary of the Connection Representation The conventional quantization of gravity, i.e., from the ADM geometrodynamics formulation (see Chap. 2 of Vol. I) is a programme involving (and this is not entirely obvious) two specific problems: • We may recall that the canonical variables h i j and πi j are replaced by operators acting on the state Ψ [h i j ], which is determined from the action of the (quantum) constraints, that is, the Hamiltonian and momentum constraints leading to, e.g., the Wheeler–DeWitt equation. However, it is extremely difficult to proceed from this equation (by which we mean that it is difficult to completely solve it). In particular, within the full superspace context as indicated in Sect. 2.5 of Vol. I, it is a non-polynomial expression. The apparently naive (but as it turns out, rather interesting) approach is just to simplify it. But can this be done? The answer is affirmative, but at a price. In doing so, we also have to discuss the other problem. • The canonical variables are of a (classical) tensorial nature, with non-trivial transformation properties, inducing no coordinate-independent meaning for the various components. So how can we proceed to a quantum description?

6.1.1 Two Spinor Representation The first stage of the process mentioned above requires the introduction of what are now known as Ashtekar canonical variables [44]. We shall employ a representation Moniz, P.V.: Connections and Loops Within SQC. Lect. Notes Phys. 804, 111–126 (2010) c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-11570-7_6

112

6 Connections and Loops Within SQC

using a 2-spinor notation, given the framework we have been following in this book, since Chap. 4 of Vol. I. To be more specific, we will not present a detailed overview of loop quantum gravity (LQG) or cosmology (LQC), since rather complete references are already available [30]. We will point out some of the key features with relevance to SUGRA, following [1–4, 8] for the main part. At the purely gravitational level, the essential (canonical) variables are the densitized spatial triad, E AB i , i.e., the spatial components of the tetrad defined by 1

E ABk ≡ (2h)1/2 ie A A k n B A = √ εi jk e A A i e B A j , 2

(6.1)

satisfying E AB i E AB j = hh i j as expected (see Exercise 6.1), and a connection A j AB , which is actually the spatial part of the known 4D connection. These two variables will constitute a canonical (conjugate) pair and the starting basis for the Ashtekar canonical treatment of (quantum) gravity. Through this change of variables, the algebraic form of the constraints will become polynomial, bringing a considerable simplification. As we shall see in Sect. 6.2, a similar structure and consequences arise when applying it to SUGRA. A few other interesting properties are as follows: • The triad (6.1) has (initially) an ‘additional’ transformation property, by taking j the index i as merely labeling and allowing SO(3) rotations such that Oi E AB i will lead to invariant metrics. This means including a Gauss-like constraint in this broader gauge theory of gravity. • The other rather interesting point is that the extrinsic curvature (see Chap. 2 of Vol. I and Appendix A) is canonically conjugate to E ABi . But noticing that the spatial components of the spin connection (see again Chap. 2 of Vol. I and Appendix A), being functionals of the tetrad, have vanishing brackets with E ABi , we can then create a new connection, called the Ashtekar connection, by Ai AB ∼ ωi AB ⊕ γ K i AB ,

(6.2)

where γ is the Barbero–Immirzi parameter (which is positive) [51, 54]: – The expression (6.2) is of a qualitative nature. To be more precise, we have Ai AB = ± ω(3s) i AB + iϑi AB ,

(6.3)

where ω(3s) i AB is the torsion-free (spatial) metric connection. Invoking the i E AB = K i , i.e., the extrinsic curvaconstraint J AB = 0, it follows that1 ϑ AB j j ture (second fundamental form).

1 The constraint J AB enforces SL(2,C) covariance (see Appendix A) and generalizes the Gauss constraint of vacuum general relativity.

6.1

A Brief Summary of the Connection Representation

113

– But, being careful, whereas Ei AB is real, the SL(2,C) spatial connection A j AB is complex. Their canonical relationship is not that of a strictly conjugate pair. Either we take a complex phase space or we impose conditions to restrict to the real section, in the form

† E AB i = E AB i ,

ω(3s)i AB + iϑ AB i

†

= ω(3s)i AB − iϑ AB i ,

these constituting the reality conditions2 (from which a Hilbert space with a suitable inner product can be constructed).3 • As indicated in [4, 55], a test of ‘reality’ in unprimed spinors can be performed using, e.g., the Hermitian property

employing the metric Hermitian.

M AB √

†

≡ 2n AA n BB M

A B

= M AB ,

†

2n A A . Hence, E AB i = E AB i , given that e AB i is

It is also of interest (since it is part of the corresponding formulation of SUGRA) to discuss the way the action is re-expressed. The Einstein–Hilbert Lagrangian becomes (noting that R[λμν]ρ = 0) √

√

−gg λν g μρ Rλμνρ

√ i αβ λν μρ = −gg g Rλμνρ + ελμ Rαβνρ 2 √ λν μρ ˘ ≡ −gg g Rλμνρ √

μ ˜ AB , ≡ −geλA A e B A Rλμ

L=

−g R =

(6.4) (6.5)

where R˘ λμνρ is the self-dual part of the curvature and the last line is the spinor version of it. The action (6.5) will henceforth be considered as a functional of Ei AB and A j AB .

2 It is possible to proceed differently. We have to resort to a description in Euclidean terms, where all the Ashtekar variables become real. But non-polynomial expressions resurface, although there are claims that a satisfactory quantization is possible [3]. 3 The (scientifically subversive) reader may nevertheless question the use of ‘reality conditions’, and rightly so. There are several works on this possibility (either in general relativity or N = 1 SUGRA). The innovative setting known as the Barbero–Sawaguchi canonical transformation uses real Ashtekar variables, but at a price. One loses the polynomiality of the Hamiltonian constraint and has to deal with a more complicated form of Dirac brackets (although these can nevertheless be made simpler by switching to new variables) [15].

114


6.1.2 Hamiltonian Formulation The action (6.5) can also be rewritten in the spinor form i μνρσ A A

˜ A B ρσ , eμ e B A ν R ε 2

L=

(6.6)

˜ A B ρσ is the corresponding two-form (curvature) or field strength associated where R with A ABi . The corresponding Hamiltonian and constraints are then4 √ L = i 2 E ABi A˙ ABi − H ,

(6.7)

H = e A A 0 H A A + A AB0 J AB = e AB0 H

AB

+ A AB0 J

AB

(6.8)

,

(6.9)

1

˜ B A jk , H A A = √ εi jk e B A i R 2 B A ˜ ij H AB = Ei E j R ,

(6.11)

J AB = Di E ABi ,

(6.12)

(6.10)

where e AB0 = ih −1/2 e A A 0 n B A or alternatively, in a now familiar formulation,

H A A = −n A A H⊥ + e A A i Hi , ˜ ij , H⊥ = Tr Ei E j R ˜ ij . Hi = Tr E j R

(6.13) (6.14) (6.15)

6.1.3 Quantization Once the constraints above have been retrieved, the next step is to consider a quanABi and A tization scheme. Concerning the use j AB , the of the canonical variables E states will be functionals Ψ A ABi and E ABi −→

δ . δA ABi

(6.16)

Then, J AB = 0 implies invariance under (local) Lorentz transformations and 4

Here Di is the covariant derivative with connection Ai A B and curvature ˜ i j A B ≡ 2∂[i A j]A B + 2A[i| A C A j]C B . R

6.1

A Brief Summary of the Connection Representation

˜ ABi j H⊥ Ψ A ABi = R

˜ ABi j Hi Ψ A ABi = R

δ2 B δAiAC δAiC

115

Ψ =0,

δ Ψ =0. δA AB j

(6.17) (6.18)

Second-order differential equations are still required, but a solution to the constraints is eventually found by solving (6.17) and (6.18): • A holomorphic wave function exp(iS) ≡ exp(iSCS ) = exp(−ICS ) ,

(6.19)

was obtained for general relativity without fermionic matter using Ashtekar variables [67, 56, 14] for a closed FRW and a Bianchi IX model. From this a semiclassical estimate can be made for the Hartle–Hawking state [68, 69]. Of course, the reader will be wondering what information can be recovered from (6.19) concerning structure formation and the evolution of the universe [3]. Regarding this question, more research will be needed, but note that a semiclassical state of the form (6.19) can be obtained (with a careful analysis regarding the feature of a negative cosmological constant) in the theory of N = 1 SUGRA [8]. • Another rather bold and interesting line of exploration has also been proposed [47, 62–64]. This is loop quantization. Although not pursued further here and seldom explored in SUGRA (but see [6, 12]), we note here that it involves the use of holonomies 8 (6.20) hγ (A) exp A AB τi e˙i dt , γ

for all curves γ in space and ‘reproducing’ a connection, as well as fluxes fΣ (E)

Σ

d2 y n i E AB τi ,

(6.21)

for surfaces Σ in space. A holonomy–flux algebra can be constructed, leading to a loop quantization representation. For example, holonomies are multiplication operators, creating a complete set of graph-dependent states

Ψ Aia = Ψ˜ hγ 1 (A), . . . , hγ n (A) ,

(6.22)

from the basic state, i.e., Ψ Aia = 1, and with fluxes as derivative operators. The implications are rather promising: – Only (the above) holonomies are represented. Connections are not represented directly (as they would be in a Wheeler–DeWitt equation). – Fluxes have discrete spectra.

116


– (Wilson) Loops constitute associated phase coordinates for general relativity. – Corresponding states may form (representation) graphs with edges that are also created.

6.2 N = 1 SUGRA But what about N = 1 SUGRA? This is where the Ashtekar–Jacobson formulation [8] comes onto the scene. The seminal work by [8] is mandatory for the serious researcher,5 bearing in mind that the action (6.5) will be treated as a functional of

μ independent variables e A A μ and the (complex) connection6 A AB , or more precisely,

μ a functional of E ABi and A j AB plus the gravitino ψiA , ψ iA , but where A AB will now be a functional of the tetrad and gravitinos. The action (6.5) is extended to L = ε μνρσ

i A A

˜ A B ρσ − e A A μ ψ A ν Dρ ψ Aσ e μ e B A ν R 2

,

(6.23)

˜ A B ρσ is the corresponding (extended) two-form (curvature). where R

6.2.1 Hamiltonian Formulation The N = 1 SUGRA Hamiltonian becomes √ L = i 2 E ABi A˙ ABi + π Ai ψ˙ Ai − H ,

(6.24)

H = e A A 0 H A A + A AB0 J AB + ψ A0 S A + S A ψ A 0 A

HAA

H AB

= e AB0 H AB + A AB0 J AB + ψ A0 S A + S ψ A 0 , 1

˜ B A jk − iψ A i D j ψ Ak , = √ εi jk e B A i R 2 B A B ˜ ij = Ei E j R + 21/2 h −1 εi jk Em Ek π m Di ψ jA ,

S˜ A = i2−1/2 εi jk e A A i D j ψ Ak

≡ S A + i2−1/2 εi jk Dk e A A i ψ A j ,

5

(6.25) (6.26) (6.27) (6.28)

(6.29)

For higher N SUGRA, see [5, 9, 10, 13, 16, 18, 19].

The action is nevertheless not complex, due to the fact that R[λμν]ρ = 0. In addition, the imaginary part of the Lagrangian becomes a total derivative when the A ABi equation is satisfied.

6

6.2

N =1 SUGRA

117

S A = D j π Ai ,

(6.30)

J AB = Di E ABi − π (Ai ψ A) j ,

(6.31)

where π Ai ≡ i2−1/2 i jk e A A j ψ A k is the other fermionic variable, constituting a pair with ψiA from which a canonical framework can be obtained.7 The point to note here is that the constraints (6.25), (6.26), (6.27), (6.28), (6.29), (6.30), and (6.31) have a ‘polynomial’ form8 with these canonical variables.

6.2.2 Quantization Regarding quantization, what do these properties allow us to extract, even if at a preliminary or heuristic level? States will be of the functional form Ψ A ABi , ψ Cj , adopting the same representation as in (6.16) and hence inheriting the same implications from the Lorentz constraints, now extended as in (6.31). The new elements for quantization are the SUSY constraints: S A Ψ = 0 = Di

δ Ψ ⇒ δ(SUSY) ψ Ai = Di ε A , δψ Ai

S (1)A Ψ = 0 = D[i ψC j]

δ2 Ψ , δA ABi δA jC B

(6.35) (6.36)

together with an algebra ensuring that H(1)AB is automatically satisfied. In [6, 12], the authors investigated several aspects of the canonical quantization of N = 1 SUGRA (in terms of the Ashtekar–Jacobson variables) extended to a loop representation. The solutions seem to be of a similar nature to general relativistic gravity, but with the difference that only closed loops are included (although physical states are given by knot invariants [64]). This was claimed to be a direct consequence of SUSY.

7 This means that we will employ Poisson variables instead of having to resort to the Dirac brackets, since we will have no second-class constraints of the type discussed in Chap. 4 of Vol. I.

Or almost polynomial, as the h −1 term indicates in some expressions. Nonetheless, suitable redefinitions of the Lagrange multipliers can bring the full set of constraints into the desired polynomial form. In summary, we have to use

8

(1) ψ A0 ≡ h −1/2 n A A ψ A 0 , A

S A −→ S (1)A ≡ E j Ek D[ j ψk] ,

(6.33)

H(1)AB ≡ H AB + 2h −1 EC A i π Bi S (1)C ,

to get a full polynomial set of constraints in N = 1 SUGRA.

(6.32)

(6.34)

118


6.2.3 FRW with Cosmological Constant One interesting application includes a cosmological constant Λ [70] and investigates the consequences in the case of a typical cosmological model, e.g., the FRW cosmology. Hamiltonian Formulation The S A , S (1)A , and H(1)AB constraints acquire new terms associated with the presence of9 Λ ≡ −6Υ 2 , namely, A √ , S (new)A ≡ S A + 2 2Υ Ek ψk A √ (1)A S(new) ≡ S (1)A − 2 2Υ Ek π k , 1 (1)AB (1)A † S (new)A (x), S(new) (y) = H(new) (x)δ 3 (x, y) , 2

(6.37) (6.38) (6.39)

maintaining the polynomial form. Restricting to an FRW minisuperspace, we use 1 ds 2 = −N 2 dt 2 + e2α E ABi E AB dx i dx j , j 8

(6.40)

and the generic decomposition of the canonical variables given by A ABi ≡ iωh 1/2 E ABi , √ i 2h 1/2 A

ζ e A i n B A , E ABi ≡ − 3V i Θ ABC B

h 1/2 E ABi θ B − √ ψ Ai ≡ √ e A i n C A , 2 V 24V 7 32 1/2 i 4i

π Ai ≡ h E AB η B + √ h 1/2 χ ABC e B A i n C A , 3V V

(6.41) (6.42) (6.43) (6.44)

but where ω, ζ , η A , θ A , Θ ABC , and χ ABC , are purely time-dependent functions (and likewise for the Lagrange multipliers). We then take the case of a closed space with S 3 volume V = π 2 /4, ζ = 12Ve2α , and finally impose Θ ABC = χ ABC = 0 to obtain local SUSY in the FRW model, i.e., to satisfy the SUSY constraints (see Exercise 6.2).

We recall that, in N = 1 SUGRA, only a zero or negative cosmological constant is allowed [70]. Note the different use signs and constants with regard to [4, 8, 13, 14].

9

6.2

N =1 SUGRA

119

Quantization At a quantum mechanical level, from the Poisson brackets [ω, ζ ] = 1 , θ A , η B = −δ A B ,

(6.45) (6.46)

and the representation ζ −→ −i

∂ , ∂ω

η A −→ −

∂ , ∂θ A

(6.47)

we have to consider the constraints i 1/2 A B h θ η + θ BηA , (6.48) 6V 7 6 1/2 A S =i (6.49) h 2ωη A − $ γ θ A ζ , V hσ 2 γ

S (1)A = (6.50) 2 (i − ω) θ A + η A , √ 3$ 12V 2 6V

2hζ γ 2 γ A 1

2 A A + γ θ θ + η H⊥ = σ − $ ζ iω − ω + η 2 − ω) θ , (i A A 12 4 3λ

3V 2 (6.51)

2 γ Hi = − hη A 2 (i − ω) θ B + η B EiAB . (6.52) 3V 3$ J AB =

Applied to Ψ (ω, θ A ), this determines the quantum state Ψ (ω, θ A ) = exp(iS) ≡ exp(iSCS ) = exp(−ICS ) ,

(6.53)

which is of the Chern–Simons form10 (see Sect. 6.1.3) [3, 67, 56, 10, 68, 69]. These results for an FRW background with cosmological constant include a set of particular analytical solutions that can be obtained from the condition that the gravity sector can be solved analytically by itself and that the gravitino exists in that background. As expected, hyperbolic universes H 4 are retrieved. Given that generic solution of the full set of equations by analytical means proves too difficult, a numerical study has been carried out to retrieve a broader perspective [14]. The presence of the gravitino determines other types of (spacetime geometric) solutions, 10

In general terms, SCS ∼

1 Υ2

2 εi jk Ai AB ∂ j Ak AB + Ai AB A j B C AkC A + 2Υ ψiA D j ψk A . 3

120


namely compact universes where pure gravity leads to a hyperbolic non-compact universe or the presence of initial or final (time) singularities.

6.2.4 Bianchi Class A Models In terms of the Jacobson formulation using Ashtekar variables, the equivalent presentation to Sect. 5.2 of Vol. I goes as follows (see [1] for more details). Consider a triad of basis vectors X ai satisfying [X a , X b ]i = Cab c X ci , where Cab c are the structure constants characterizing the Bianchi model. The basis dual to X ai is obtained from X ai χib = δab . We then have c υb] , Cab c = εabd M dc + 2δ[a

where M a is symmetric. Models with υa = 0 are called type A. Ashtekar–Jacobson Formulation The Ashtekar–Jacobson variables for N = 1 SUGRA are then ˇ aABχa , Ai A B ≡ A i

(6.54)

Ei AB ≡ (det χ )eˇa AB X ai ,

(6.55)

ψˇ aA χia

ψiA

≡

π

≡ (det χ )πˇ

iA

,

(6.56) aA

X ai

,

(6.57)

where det χ denotes the determinant of χia , which is introduced in order to dedensitize the initial variables. The Poisson brackets reduce to . i eˇa AB , Aˇ bM N = √ δba δ(M A δ N ) B , 2 . i πˇ a A , ψˇ bM = √ δba δ M A . 2

-

(6.58) (6.59)

Dropping the check on the symbols from here on, the supersymmetry constraints become S A = (Cba b ε AB + Aa AB )π Ba = 0 ,

1 S †A = − Cab c δCD + AaC D δbc ψcD e[a|AB| eb] B C = 0 , 2

(6.60) (6.61)

6.2

N =1 SUGRA

121

together with the constraints B A B A H AB = − ea eb Cab c Ac + e[a eb] Aa Ab − π a eb Cab c ψc e AB + 2 π [a eb] Aa ψb e AB − π a Cab c ψc eb AB + 2 π [a Aa ψb eb]AB = 0 ,

(6.62)

J AB = C b ba ea AB + 2Aa C(A e|a|B) C − π a(A ψaB) = 0 .

(6.63)

We have . i S A , S †B = √ H AB 2 2

(6.64)

when Cba b = 0, as it is for type A Bianchi models, where the supersymmetry constraints reduce further to A A 1 S †A ≡ − εabd M dc ea eb ψc + e[a eb] Aa ψb , 2 S A ≡ Aa AB π Ba .

(6.65) (6.66)

Quantization As described in [4], quantum states may be represented by wave functions Ψ ≡ Ψ [e, ψ] that depend on the triad and on the gravitino field. Momenta may be represented by 1 δΨ , Aa AB Ψ −→ √ 2 δea AB

(6.67)

1 δΨ π a A Ψ −→ √ . 2 δψa A

(6.68)

Quantum mechanically, the issue of factor ordering arises in the second term of (6.65). There are three possibilities: (i) eeA, (ii) eAe, or (iii) Aee. The following form is suggested in [1]:

S

†A

A

A 1 1 dc a b [a b] δ Ψ (e, ψ) = − εabd M ψb e e ψc + √ e e 2 δea 2 1 + √ mea AC ψaC Ψ 2 =0,

(6.69)

122


where the constant m characterizes the factor ordering, and m = 0, 1, 2 for the orderings (i), (ii), (iii), respectively. The constraint (6.66) takes the form S A Ψ [e, ψ] =

1 δ2 Ψ =0, 2 δea AB δψa B

(6.70)

In addition, a physical state must also satisfy J AB Ψ = 0 ,

(6.71)

that is, it must be invariant under SL(2,C) rotations, i.e., the wave function must be an SL(2,C) scalar. We can then write Ψ [e, ψ] = Ψ(0) [e] + Ψ(2) [e, ψ] + Ψ(4) [e, ψ] + Ψ(6) [e, ψ] ,

(6.72)

where the subscript indicates the number of gravitino fields.11 The physical states for a quantum Bianchi type A model, restricted to the restrictive formulation in [71] and without including the improvements advanced in [72, 73], have a bosonic part Ψ(0) = k(0) h α/2 exp(2iF) ,

(6.73)

Ψ(6) = k(6) h 1/3 [βψ2 ][ρψ2 ]2 ,

(6.74)

and fermionic filled sector

where k(0) and k(6) are constants, h i j is the homogeneous metric, and h := det h i j . The constant m is determined by the factor ordering chosen. F is a bosonic functional, namely the homogeneous specialization of the generating function of the canonical transformation from the ADM variables to the Jacobson variables. [βψ2 ], [ρψ2 ] are Lorentz invariants built with the gravitino field. In particular, i F = − h −1/2 h ab M ab , 2

(6.75)

and the irreducible spin 1/2 and spin 3/2 parts of the gravitino field are ψ ABC ≡ ψaA ea BC = ψ A(BC) ,

(6.76)

ψ ABC ≡ ρ ABC + ε A(B β C) ,

(6.77)

with

11

The reader will recall that only even powers appear because of the SL(2,C) invariance. There is no mixing of fermionic number (in this case), so the quantum constraints (6.69), (6.70), and (6.71) can be solved order by order.

Problems

123

where ρ ABC = ρ (ABC) represents the spin 3/2 part, and β A = (2/3)ψ B AB the spin 1/2 part. The only Lorentz invariant combinations are: second order

[ρψ2 ] = ρ ABC ρ ABC ,

fourth order

[ρψ2 ]2 ,

sixth order

[ρψ2 ]2 [βψ2 ] .

[βψ2 ] = β A β A ,

[ρ 2 ][βψ2 ] ,

More precisely [1], 2 A ψa ψb A h ab − (2h)−1/2 ψa A ψb B εabc ec AB , 3 √ 2 A [βψ2 ] ≡ ψa ψb A h ab + 2h −1/2 ψa A ψb B εabc ec AB . 9 [ρψ2 ] ≡

(6.78) (6.79)

Then the most general expression for wave functions that depend on the gravitino field has the form Ψ(2) [e, ψ] = F1 (h)[ρψ2 ] + F2 (h)[βψ2 ] ,

(6.80)

Ψ(4) [e, ψ] = G1 (h)[ρψ2 ]2 + G2 (h)[ρψ2 ][βψ2 ] ,

(6.81)

Ψ(6) [e, ψ] = H(h)[ρψ2 ]2 [βψ2 ] ,

(6.82)

where the functions F, G, and H depend on ei AB only in the combination h i j = Tr(ei e j ). It turns out that, in general, the only physical states of second and fourth order are the trivial ones, i.e., Ψ(2) = 0 = Ψ(4) .

(6.83)

However, it is important to note that there are exceptions, given by special cases like Bianchi I, and with a specific factor ordering [1]. This should be contrasted with Sect. 5.2.3 of Vol. I, and certainly needs to be reassessed in view of [3]. It is shown in [2] that the quantum state for minisuperspaces of N = 1 supergravity with Λ = 0 has no non-trivial physical states for class A Bianchi models (recall [74]). However, this result was obtained before the improvements in [72, 73] and may subsequently acquire a different content.

Problems 6.1 Spinor Symmetry Show that E ABi is symmetric in the spinor indices. 6.2 FRW with Cosmological Constant Write down the (dimensionally) reduced Lagrangian and Hamiltonian for the FRW case described in Sect. 6.2.3.

124


6.3 N = 2 Hidden SUSY and Ashtekar’s Variables Discuss how (hidden) N = 2 supersymmetrization of Bianchi cosmologies can emerge straight from bosonic general relativity, using Ashtekar’s new variables [21].

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Chapter 7

SQC Matrix Representation

In this chapter we will report on another direction of investigation for SQC, which has received some attention. Instead of using a differential operator representation for the fermionic momenta (see Chap. 5 of either volume), a matrix representation is employed. The ground-breaking importance of the matrix representation approach should not be ignored. Since it was introduced to the SQC community in the late 1980s, this line of research has indeed provided many routes for study. In addition, there are still some unresolved issues to explore, and these deserve due consideration. In this chapter, we thus present this research programme [1–10]. Before proceeding, let us raise a question, even though the answer will remain incomplete, as it is still an open problem: Is there an easy way to translate the formalism employed in Chap. 5 to that used in the present chapter? That is, assuming they constitute chapters with different descriptions of the same landscape, how can the information in one be described in terms of the information in the other? In particular, how does the matrix representation approach differ from the one in Chap. 6 as far as the construction of the wave function is concerned? In the latter, the wave function is a superfield functional (see Chap. 3 of Vol. I), that is, it is expressed as a function of the fermionic and bosonic variables with values in the SuperRiem(Σ) Grassmannian manifold (see Chap. 4 of Vol. I), and by construction, it is an invariant scalar, obviously satisfying the Lorentz constraints. In the former, however, the wave function will instead take values in a linear representation space of the fermionic variables, i.e., it is represented as a vector on which the fermionic matrices can act. We can be more specific as follows. Defining fermionic variables εi (satisfying the canonical anticommutation relations such that π i are the canonical conjugate momenta of εi ), the wave function can, in general, be written in terms of the fermionic variables εi in the form, e.g., (1)

Ψ = ' (0) + εi 'i

+ εi1 εi2 'i(2) + · · · + εi1 εi2 εi3 εi4 εi5 εi6 'i(6) , 1 i2 1 i2 i3 i4 i5 i6

(7.1)

where the coefficients (n) in ε(n) induce a vector structure and i is an index that relates to the number of spatial components for the fermions, i.e., gravitinos (see Sect. 7.3.3). Moniz, P.V.: SQC Matrix Representation. Lect. Notes Phys. 804, 127–161 (2010) c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-11570-7_7

128

7 SQC Matrix Representation

In the approach in Chap. 5, only even combinations of the εi are admissible, being Lorentz invariant.1 But in (7.1), each one of the powers of ε can be represented as matrices, resulting in another matrix that acts on its corresponding ' (n) vector, thereby generating vectors that will eventually form Ψ (these vectors having been calculated using the Lorentz constraint equations). Furthermore, following the discussion in Chaps. 4 and 5 of either volume, we will show how the matrix representation has focused on an interesting issue in comparison with a differential operator representation for the fermionic momenta. In particular, in the matrix representation approach some rather apposite criticisms have been raised concerning the use of the Lorentz constraint in Chap. 5. In fact, in most works in SQC, the Lorentz constraints have not been solved explicitly.2 The wave function is expanded in blocks, each constituted by the fermionic (and bosonic) variables and each being a Lorentz scalar invariant. By assembling such blocks, it is proposed to construct the most general Lorentz invariant wave function. Any quantity in which those indices (or, instead, spinorial indices of the Infeld–van der Waerden symbols) are all contracted (or even combinations of them) will be Lorentz invariant. In this manner, Lorentz invariant wave functions could have been written down. However, it may not be such a trivial matter to write down all those invariants, and Ψ may even be Lorentz invariant overall while some of the blocks are not. We can consider a more generic Lorentz invariant Ψ , where Lorentz transformations map block(s) into block(s) without each one being Lorentz invariant, although Ψ as a whole would be. For this reason, and in spite of the self-consistency of the fermionic differential operator representation and its success, several authors [1, 6, 9] have suggested that some care must be exercised in assuming any kind of Lorentz invariance: • there are six constraints J ab , • they should all be represented as operators, and • each one of them should act on the wave function. This means imposing and solving explicitly all the equations resulting from the action of all the Lorentz constraints on the wave function. It is not necessary to search for all the Lorentz invariants, in terms of which the wave function should be expanded. This methodology could be brought to bear in Chap. 5 for the fermionic differential operator representation, but here we follow the alternative approach with the fermionic variables not being taken as continuously varying variables (and therefore, not represented after quantization by the derivatives of their conjugate variables), but instead represented by matrices.

1

We are referring to those products of the fermionic components where the indices are all contracted, possibly with the assistance of other variables, e.g., the tetrad (see Sect. 5.2.3 of Vol. I). 2 Recall that the Lorentz constraints act (classically as generators) on the flat (tangent space) indices present in the, e.g., vierbein and the Rarita–Schwinger variables (denoted by the latin letters a, b, . . . throughout this book).

7.1

Motivation and Historical Background

129

7.1 Motivation and Historical Background From the preceding paragraphs, we have some motivation for the matrix representation. But let us aim to be more precise and at the same time broaden our outlook. In a bosonic quantum cosmological setting, an equation formally similar to the Klein– Gordon equation is employed, namely the Wheeler–DeWitt equation (see Chap. 4 of Vol. I), with the well known problems, e.g., of interpreting solutions as state functions leading to probability densities. In the early 1970s and even earlier, as pointed out in [11], this motivated the search for a square root structure, i.e., a set of first order differential equations (linear in the momenta), in analogy with the relation between the Klein–Gordon and Dirac equations of relativistic quantum mechanics. This attempt was successful and the framework applied to the Bianchi universes. In particular, the resulting (Dirac-like) system of equations was explicitly solved for the Bianchi type I model, and a wave function with two components emerged (see Exercise 7.1) from a linear 2-component (spinor-like) equation for the Bianchi I universe, without any use of elements from SUSY or SUGRA.3 However, this procedure lacked a general rule, so the Dirac-like system of equations had to be established case by case for each Bianchi model under consideration. Moreover, at the time there was no clear interpretation for the components of the wave function. ψ1 , and Pauli matrices came into play, The solution was a vector of the form ψ2 assigning an apparent spin context. However, to describe this structure as spinorial would require a specific set of transformation properties in the minisuperspace. It was only after SUSY and SUGRA had been introduced that an interpretation could be envisaged,4 largely because N = 1 SUGRA provides a natural square root of gravity in a Dirac-like manner. In this context, the components of Ψ can then be interpreted in terms of states in SUGRA, as made up of eigenstates of the components of the gravitino field, associated through the Pauli matrices (something that would have been difficult to get approval for in the early 1970s and prior to the full development of SUGRA). To illustrate this discussion, consider for the Dirac equation the (classical) constraints (see Appendix B of Vol. I and [9]), which can be written as S ≡ ζ μ Pμ + ζ 5 m ,

(7.2)

H ≡ P +m ,

(7.3)

2

2

where ζ μ and ζ 5 are anticommuting real variables. The following non-vanishing Dirac brackets can then be retrieved: 9 μ ν: (7.4) ζ , ζ D = iημν , 3 4

Compare results in Exercise 7.1 with what we retrieve here in Sect. 7.2.1.

See Sects. 3.4.3 and 4.1.2 of Vol. I and Sect. 7.2.1 of Vol. II. The reader may recall in particular that assigning a spin structure to a theory (of a point particle, string, or membrane, or general relativity) amounts to introducing SUSY properties [12–19].

130


-

.

9

:

ζ 5, ζ 5

x μ , Pν

D D

=i,

(7.5)

= δνμ ,

(7.6)

together with {S, S}D = iH .

(7.7)

The argument now is that, if we represent the fermionic variables ζ μ and ζ 5 in the form of matrices, viz.,

1/2 h¯ γ5 γ μ , ζ ≡i 2

1/2 h¯ ζ5 ≡ γ5 , 2 μ

(7.8) (7.9)

with the standard choice Pμ = −ih¯ ∂μ

(7.10)

for the bosonic variables, the quantum mechanical Dirac equation is obtained. Hence, to quantize the classical equation, we can promote the five anticommuting real variables ζ μ , ζ 5 , and the bosonic variables x μ , P μ to operators (in accordance with their Dirac brackets) so that the fermionic variables are represented as products of gamma matrices and the bosonic Pμ as differential operators. In addition, (7.4, 7.5, 7.6, and 7.7) reveal an underlying SUSY setting (see Sects. 3.2 and 3.3 of Vol. I). The above actually summarizes the quantization of the classical supersymmetric spin 1/2 particle [20]. The Dirac equation represents a quantization of this classical formulation. In this chapter, we will quantize in accordance with the Dirac brackets, on which the algebra of constraints is based, analogously to what was done for the spin 1/2 particle introduced here, adapting the approach with a quantization in which the fermionic degrees of freedom are realized in terms of matrices, so that they realize the algebra of the fermionic field components. Note also that we will be using a four-component spinor formalism once again.

7.2 Supersymmetric Bianchi Models Considering what was said in the last section, the reader should have a fair, albeit incomplete, taste of the new methodology to be employed in the context of the matrix representation. It should lead us to discover more about the canonical quantization of N = 1 SUGRA, and hence of SQC. This is the aim in the remainder of this chapter. From a pedagogical standpoint, it will be useful in the framework of the matrix representation approach to SQC to begin by studying the Bianchi I model [3, 4]

7.2

Supersymmetric Bianchi Models

131

(see Sect. 7.2.1), as it is undoubtedly the simplest case, being exactly soluble. Then we compare the expressions and results with the Bianchi IX model [4] in Sect. 7.2.2, where couplings appear. This contains some particularly interesting physical cases, such as the Taub model (Sect. 7.2.3) [4, 10] and FRW geometry (Sect. 7.2.4) [4]. For the Bianchi I case, the state vector will comprise four 2-component vectors. In the interacting self -coupling case of the Bianchi IX model, the general solution to the square root equations is quite difficult to retrieve, but solutions can be found for the (particular) Taub microsuperspace configuration. It will be useful to understand and further investigate the reason why the resulting Dirac-like equations contain a coupling term in contrast to the squared equation for bosonic quantum cosmology. Similarly, for the FRW configuration (where a cosmological constant term can be added) the iterated (i.e., squared) equation will have two extra terms, in comparison with the corresponding Wheeler–DeWitt equation in the bosonic description (see Chap. 2 of Vol. I). The starting point will be the 4D N = 1 SUGRA action written here in a slightly different way [see (3.124), (4.1), or (4.49) of Vol. I]: L=

i 1√ −g R − εμνρσ ψ μ γ5 γν Dρ ψσ . 2 2

(7.11)

The ‘deviation’ due to the definition of the gamma matrices and corresponding representations (see Appendix A for details, conventions, and notation) and the basic field variables will be the tetrad (vierbein) ea μ and the gravitino ψ[a]μ . With a view to application of the matrix representation for fermions in homogeneous cosmologies, most research has been along the following lines: • In the above formulation of SUGRA, the condition of space homogeneity is imposed. The general Bianchi metric can be parametrized as in Sect. 2.3.1 of Vol. I or Sect. 6.1 of Vol. I. Specifically, in terms of a Misner–Ryan parametrization and a corresponding basis ωi , the orthonormal vierbein basis ea μ is given for the Bianchi IX case as (0)

e(1) j

(0)

(i)

e0 = eΩ e−βi j N j , = e−Ω eβ11 − sin x 3 δ 1j + sin x 1 cos x 3 δ 2j ,

e0 = N ,

ei

=0,

(7.12) (7.13)

(2) e j = e−Ω eβ22 cos x 3 δ 1j + sin x 1 sin x 3 δ 2j ,

(7.14)

(3) e j = e−Ω eβ33 cos x 1 δ 2j + δ 3j ,

(7.15)

where x 1 , x 2 , x 3 are angle variables. • The Rarita–Schwinger (gravitino) fields are also redefined so that the vector index becomes a local index (we use a 4-spinor notation in this chapter), i.e., ψa [a] = ea μ ψμ [a] ,

(7.16)

132


where [a] is the spin 1/2 index. We also assume the gravitinos to satisfy the Majorana condition, i.e., all the γ matrices are purely imaginary, the components of the gravitino vector spinor are real, and C = −iγ 0 (see Sect. A.3). To integrate over the spatial directions of a Bianchi model, and eventually fixing the gauge to Ni = 0, i.e., imposing a coordinate condition (see Sect. 2.3.2 and Note 2.8 of Vol. I), we define here the following new variables: ψ0 ≡ e3Ω/2 ψˇ 0 + eΩ/2 N i eβii ψˇ i ,

ψi ≡ eΩ/2 eβii ψˇ i ,

i = 1, 2, 3 , (7.17)

i.e., we use densitized local components when we assume as a homogeneity ansatz that the gravitino field does not depend on the spatial coordinates, in order to obtain an action written in canonical form. • We can further simplify this SUGRA formulation by requiring that ψμ[a] can be expanded in an anticommuting basis of order 2 [21] (see also [3, 4, 9, 10]): ψμ[a] ≡ ψμ[a]1 ε1 + ψμ[a]2 ε2 ,

(7.18)

where the ψμ[a]i are ordinary functions and the εi are anticommuting constants: εi ε j = −ε j εi . The quantized degrees of freedom are the ψμ[a]i , which are written as ψ[a]μ for simplicity.

Note 7.1 Whereas in Chap. 5 we used a Dirac superspace methodology (see Sect. 2.5 and Note 2.7 of Vol. I), in this chapter, we adopt the ADM-type formulation as in Sect. 2.3 of Vol. I.

Note 7.2 Use of (7.18) in the Lagrangian is actually a severe restriction. It also involves an assumption, namely that a vector–spinor (e.g., the gravitino) can be expressed in the form ψ=

(1)

ψμi εi ωμ +

(3)

ψμi jk εi ε j εk ωμ + · · · ,

(7.19)

where the εi anticommute. Equation (7.18) is the simplest possible expansion. The variables are expanded in even elements, effectively in ‘powers’ of the torsion, leading to general relativity at lowest order, with the gravitino as a linear field. It provides an algorithm for solving the SUGRA equations, leading to a sequence of coupled equations, with the usual vacuum Einstein equations, and linear field expressions in a curved setting [21].

7.2


133

7.2.1 Bianchi I Cosmology In this section we consider the simplest case, namely, the diagonal Bianchi I model [3, 4]. This means we have ωi = dx i . Classical Constraints and Algebra In this case, employing the expressions (7.11) and (7.17), we subsequently obtain the corresponding reduced Lagrangian for the diagonal Bianchi I universe written in a canonical form: N e3Ω (−p2Ω + p2+ + p2− ) 12 √ √ − 3p+ + 3p− ψˇ 1 γ 1 γ 0 γ 3 ψˇ 3 − 3p+ − 3p− ψˇ 2 γ 2 γ 0 γ 3 ψˇ 3

L = pΩ Ω˙ + p+ β˙+ + p− β˙− +

N e3Ω 24 √ √ + 3p+ + 3p− ψˇ 3 γ 3 γ 0 γ 1 ψˇ 1 + 3p+ − 3p− ψˇ 3 γ 3 γ 0 γ 2 ψˇ 2

+i

√ √ −2 3p− ψˇ 1 γ 1 γ 0 γ 2 ψˇ 2 + 2 3p− ψˇ 2 γ 2 γ 0 γ 1 ψˇ 1 √ 1 +iN k e3Ω pΩ ψˇ k γ i ψˇ i + iN 1 e3Ω p+ + 3p− ψˇ 1 γ i ψˇ i 12 √ 1 + iN 2 e3Ω p+ − 3p− ψˇ 2 γ i ψˇ i 12 1 3 3Ω i ˇ ˇ + iN e (2p+ ) ψ 3 γ ψi 12 3 1 i 3Ω (γ i ψˇ i ) − p+ (γ 1 ψˇ 1 + γ 2 ψˇ 2 − 2γ 3 ψˇ 3 ) pΩ + ψˇ 0 e 12 2 i=1

√ 3 1 ˇ 2 ˇ − p− (γ ψ1 − γ ψ2 ) 2 3 i 1 3Ω − (ψˇ i γ i ) − p+ (ψˇ 1 γ 1 + ψˇ 2 γ 2 − 2ψˇ 3 γ 3 ) e pΩ 12 2 i=1

√

3 − p− (ψˇ 1 γ 1 − ψˇ 2 γ 2 ) 2

ψˇ 0 ,

(7.20)

where the dot over a symbol denotes a derivative with respect to Ω time (see Sects. 2.3.1 and 6.1 of Vol. I).

134


Note 7.3 The momenta conjugate to the fermion ψˇ i are (linear combinations of) the fermions themselves, and we may substitute for them here accordingly. Of course, corresponding Dirac brackets are used throughout. Note that, in the Lagrangian above, all terms of fermionic fourth order have disappeared and will thus not contribute to the supersymmetric constraints. This is in fact due to the assumption (7.18). The non-dynamical variables N , Ni , and ψˇ 0 in (7.20) multiply the constraints H0 , Hi , and S. The SUSY constraints S, S will therefore have four spinor components.5 For example, S1 =

1 3Ω e pΩ (ψˇ 11 + ψˇ 24 − ψˇ 32 ) 12 √ 3 1 − p+ (ψˇ 11 + ψˇ 24 + 2ψˇ 32 ) − p− (ψˇ 11 − ψˇ 24 ) . 2 2

(7.21)

The reader may be wondering at this stage whether restricting generality as in (7.18), possibly implying deviations from Pilati’s [13] formulation discussed in Sect. 4.1 of Vol. I, we will still produce from these constraints a closed algebra similar to the one introduced in [14, 15]. In fact, if the Dirac bracket algebra [3, 4, 9, 10] 9

:

S[a] (x), S[b] (x )

D

μ

= γ[a][b] Hμ δ(x, x )

(7.22)

is converted to anticommutation relations, it is possible to obtain a homogeneous form with the following square-root property (note that this homogeneous reduction of the algebra is not automatically guaranteed): :

9 1 {S1 , S1 }D = − S1 , S 4 D = − e3Ω γ 14 H⊥ + 18 iψˇ j γ j γ 0 S1 , (7.23) 1 32 : 9 1 3Ω {S2 , S2 }D = S1 , S 3 D = γ 23 H⊥ + 18 iψˇ j γ j γ 0 S2 , e 2 32

(7.24)

:

9 1 {S3 , S3 }D = − S1 , S 2 D = − e3Ω γ 32 H⊥ + 18 iψˇ j γ j γ 0 S3 , (7.25) 3 32 : 9 1 3Ω {S4 , S4 }D = S1 , S 1 D = γ 41 H⊥ + 18 iψˇ j γ j γ 0 S4 , e 4 32

5

The index 1 in S1 is the spinor index associated with ψˇ [a] 0 , [a] = 1.

(7.26)

7.2


135

: 9 with S[a] , S[b] D = 0 when [a] = [b]. It should be noted that the homogenization procedure has led (in contrast with the algebra displayed in Chaps. 4 and 5 of Vol. I) to extra terms proportional to S. These are therefore automatically weakly zero (see Appendix B of Vol. I). The Hamiltonian constraint is H⊥ =

1 3Ω 2 e −pΩ + p2+ + p2− = 0 , 12

(7.27)

corresponding to the Wheeler–DeWitt equation, and will therefore be a consequence of the algebra (7.23), (7.24), (7.25), and (7.26). Nevertheless, the reader would be right to ask where the Lorentz constraints have gone? These play a fundamental role, as described in Chap. 4 of Vol. I. Although a thorough discussion of this issue will be postponed to Sects. 7.2.2 and 7.3.3, where it becomes more relevant, let us just note that it will relate to the terms in ψˇ 0 and ψˇ 0 in ˇ η (7.20) which have not yet been combined by means of the relations ηγ ψˇ = −ψγ μ ν ρ μ ν ρ ˇ ˇ and ψγ γ γ η = ηγ γ γ ψ. Quantization In order to quantize this model, we will convert pΩ , p+ , p− , and ψˇ to operators that act on the state wave function for the diagonal Bianchi I universe. Quantization is carried out in the following canonical manner. We assign for the bosonic momenta the derivative of the corresponding conjugate variable {pΩ , p+ , p− } −→

i

∂ ∂ ∂ , −i , −i ∂Ω ∂β+ ∂β−

.

(7.28)

From the Dirac brackets, the momenta conjugate to the ψˇ i[a] are the ψˇ i[a] themselves, and a matrix realisation6 can be found (see the example at the beginning of this chapter and [20]) for the ψˇ i[a] , subject to -

ψˇ i[a] , ψˇ j[b]

. D

=

1 i γ j γi [a][b] , 8

(7.29)

consistently with (7.23), (7.24), (7.25), 9 and (7.26). : To be more precise, we have to note that, due to the fact that we have S[a] , S[b] D = 0 in (7.23), (7.24), (7.25), and (7.26) when [a] = [b], we can take each of the S[a] to act on orthogonal subspaces. The quantum constraint SΨ = 0

(7.30)

6 Generically, they will satisfy a Clifford algebra via a matrix representation of the algebras, taking ψˇ i[a] to contain all Grassmann dependence (and constituting a classical limit of Dirac matrices).

136


is then written as ⎡

⎤⎛ ⎞ S1 0 0 0 Ψ1 ⎢ 0 S 0 0 ⎥⎜Ψ ⎟ 2 ⎢ ⎥⎜ 2⎟ ⎢ ⎥⎜ ⎟ = 0 , ⎣ 0 0 S3 0 ⎦ ⎝ Ψ3 ⎠ Ψ4 0 0 0 S4

(7.31)

where each of the S[a] is a matrix operator of the smallest possible rank. Each S[a] can be rewritten in a more compact manner as a linear combination of the ψˇ i[a] , e.g., S[a] =

1 3Ω pΩ M[a]1 + p+ M[a]2 + p− M[a]3 , e 12

(7.32)

where, e.g., M11 ≡ −ψˇ 11 − ψˇ 24 + ψˇ 32 , 1 ψˇ 11 + ψˇ 24 + 2ψˇ 32 , M12 ≡ 2 √ 3 ψˇ 11 − ψˇ 24 , M13 ≡ 2

(7.33) (7.34) (7.35)

or, quantum mechanically, S[a] ∼

∂ ∂ 1 3Ω ∂ − iM[a]3 e M[a]1 − iM[a]2 i , 12 ∂Ω ∂β+ ∂β−

(7.36)

whence 9 : S[a] , S[a] D ∼ e6Ω −p2Ω + p2+ + p2− .

(7.37)

Therefore, it implies the following algebra for the matrices M[a]i : 9

:

3 i, 8 : : 9 9 3 M[a]2 , M[a]2 D = M[a]3 , M[a]3 D = − i , 8 : 9 M[a]i , M[a] j D = 0 , i = j . M[a]1 , M[a]1

D

=

(7.38) (7.39) (7.40)

For all [a], the minimum possible rank for the matrices is 2 and the algebra can simply be represented by means of Pauli matrices: 7 M[a]1 =

3i σ3 , 8

7

M[a]2

3i =i σ2 , 8

7 M[a]3 = i

3i σ1 . 8

(7.41)

7.2


137

In order to further strengthen the relation M[a]i and ψ[a]i between 0 1 which are eigenvecand as matrices [3, 4], take, e.g, the vectors 1 0 √ tors of the Hermitian matrix iM11 with eigenvalues ∓1, in other words, a trilinear combination of eigenstates of ψˇ 11 , ψˇ 24 , and ψˇ 32 as in (7.35). One can transform between eigenstates of the twelve M[a]i and twelve ψˇ [a]i with non-singular matrices and write the equations in (7.30) and (7.31) in the new basis, where each eight-vector is of the form Note 7.4

⎛ ⎞ 1 ⎜0⎟ ⎜ ⎟ ⎜ ⎟ , ⎜ .. ⎟ ⎝.⎠ 0

⎛ ⎞ 0 ⎜1⎟ ⎜ ⎟ ⎜ ⎟ , ⎜ .. ⎟ ⎝.⎠

... ,

(7.42)

0

and an eigenstate of some spinorial component ψˇ i[a] . Of course, S[a] are written in a less comfortable and manageable form.

Hence, remarkably, the operator equation (7.30) for the 2D vectors Ψ[a] that compose Ψ in (7.31) reduce to, e.g.,

i

∂ ∂ ∂ + iσ2 − iσ1 ∂Ω ∂β+ ∂β−

Ψ[a] = 0 .

(7.43)

For each Ψ[a] , this is (B.182) as displayed in Exercise 7.1, when deriving a square root for the Wheeler–DeWitt equation without any SUSY or SUGRA background and methodology. For comparison and discussion in the next sections, note that with no coupling potential in the Wheeler–DeWitt equation, the free Bianchi I case has no coupling terms in the Dirac-like equations. For each Ψ[a] , the solution is Ψ[a] ∼

exp i (p+ β+ + p− β− − EΩ)

, exp i (p+ β+ + p− β− − EΩ)

(7.44)

with ± (p+ β+ + p− β− )1/2 = E and p+ , p− , and Ω as constants. The two solutions correspond to expanding (E > 0) and contracting (E < 0) universes. However, it is now natural to ask about the interpretation of the various components of the eight-vector Ψ . Let us consider the equation S1 Ψ1 = 0 ,

(7.45)

138


since the conclusions from the others will be the same. What we find for S1 is

∂ ∂ 1 3Ω ∂ − iM13 e M11 − iM12 i . (7.46) 12 ∂Ω ∂β+ ∂β− The point is that any state with a given probability of finding the universe with ψˇ having a specific value can be formed as a linear combination of eigenstates of the M[a]i matrices. Hence, the components Ψ[a] can be interpreted as probabilities of ˇ finding the universe at a given time with given values of Ω, β± , and ψ.

7.2.2 Bianchi IX Cosmology We now go beyond the useful but limited grounds of the Bianchi I cosmology, to consider the Bianchi IX model [4]. The basis corresponding to this concrete model is (7.12), (7.13), (7.14), and (7.15). Using this framework, we will proceed in a similar way to the interesting physical cosmologies of the Taub and closed FRW universes in Sects. 7.2.3 and 7.2.4. Classical Constraints and Algebra As for the current case study, employing the expressions (7.16) and (7.17) introduced in Sect. 7.2, we retrieve the reduced action for the Bianchi IX universe in a canonical manner: N e3Ω - 2 −pΩ + p2+ + p2− L = pΩ Ω˙ + p+ β˙+ + p− β˙− + 12 √ √ 4 −4Ω 1 −8β+ 2 4β+ −2β+ −e + e cosh 2 3β− e cosh 4 3β− −1 − e 3 3 3 √ √ N e3Ω 3p+ + 3p− ψˇ 1 γ 1 γ 0 γ 3 ψˇ 3 + 2 3p− ψˇ 1 γ 1 γ 0 γ 2 ψˇ 2 −i 12 √ + 3p+ − 3p− ψˇ 2 γ 2 γ 0 γ 3 ψˇ 3 √ N Ωe ψˇ 3 γ 3 γ 0 γ 5 γ 1 ψˇ 1 e−4β+ + 2e2β+ sinh(2 3β− ) 2 √ +ψˇ 2 γ 2 γ 0 γ 5 γ 3 ψˇ 3 e−4β+ − 2e2β+ sinh(2 3β− ) . √ −ψˇ 2 γ 2 γ 0 γ 5 γ 1 ψˇ 1 e−4β+ − 2e2β+ cosh(2 3β− ) 3 i 3Ω (γ i ψˇ i ) − 1/2p+ (γ 1 ψˇ 1 + γ 2 ψˇ 2 − 2γ 3 ψˇ 3 ) + ψˇ 0 e pΩ 12 −i

i=1

7.2


139

√ − 3/2p− (γ 1 ψˇ 1 − γ 2 ψˇ 2 ) . √ √ +N eΩ e2β+ +2 3β− γ 5 γ 1 ψˇ 1 + e2β+ −2 3β− γ 5 γ 2 ψˇ 2 + e−4β+ γ 5 γ 3 ψˇ 3 3 i 3Ω e − pΩ ψˇ i γ i − 1/2p+ ψˇ 1 γ 1 + ψˇ 2 γ 2 − 2ψˇ 3 γ 3 12 i=1

− 3/2p− ψˇ 1 γ 1 − ψˇ 2 γ 2 √

. √ √ +N eΩ e2β+ +2 3β− γ 5 ψˇ 1 γ 1 + e2β+ −2 3β− γ 5 ψˇ 2 γ 2 + e−4β+ γ 5 ψˇ 3 γ 3 ψˇ 0 i ψˇ 1 γ 1 γ 0 γ ψ˙ˇ 2 + γ ψ˙ˇ 3 + ψˇ 2 γ 2 γ 0 γ 1 ψ˙ˇ 1 + γ 3 ψ˙ˇ 3 2 + ψˇ 3 γ 3 γ 0 γ 1 ψ˙ˇ 1 + γ 2 ψ˙ˇ 2

−

√ +iN k e3Ω pΩ ψˇ k γ i ψˇ i + iN 1 cosh 3β+ − 3β− ψˇ 2 γ 0 ψˇ 3 √ √ +iN 2 cosh 3β+ + 3β− ψˇ 3 γ 0 ψˇ 1 + iN 3 cosh 2 3β− ψˇ 1 γ 0 ψˇ 2 . (7.47) Here a dot over a symbol denotes the derivative with respect to Ω time and, as above for the Bianchi I case, all fermionic fourth order terms have disappeared and do not therefore contribute to the supersymmetric constraints. The SUSY constraints S are then 1 3Ω 1 pΩ (−ψˇ 11 − ψˇ 24 + ψˇ 32 ) + p+ (ψˇ 11 + ψˇ 24 + 2ψˇ 32 ) S1 = − e 12 2 √ 3 p− (ψˇ 11 − ψˇ 24 ) + 2 √ √ +3e−2Ω e2β+ +2 3β− ψˇ 12 + e2β+ −2 3β− ψˇ 23 + e−4β+ ψˇ 31 , (7.48) 1 3Ω 1 S2 = − e pΩ (ψˇ 12 + ψˇ 23 + ψˇ 31 ) + p+ (−ψˇ 12 − ψˇ 23 + 2ψˇ 31 ) 12 2 √ 3 p− (−ψˇ 12 + ψˇ 23 ) + 2 √ √ +3e−2Ω e2β+ +2 3β− ψˇ 11 + e2β+ −2 3β− ψˇ 24 + e−4β+ ψˇ 32 , (7.49)

140


1 3Ω 1 e pΩ (−ψˇ 13 + ψˇ 22 + ψˇ 34 ) + p+ (ψˇ 13 + ψˇ 22 + 2ψˇ 34 ) 12 2 √ 3 + p− (ψˇ 13 + ψˇ 22 ) 2

S3 = −

√ √ +3e−2Ω −e2β+ +2 3β− ψˇ 14 + e2β+ −2 3β− ψˇ 21 − e−4β+ ψˇ 33 , (7.50)

1 3Ω 1 pΩ − (ψˇ 14 − ψˇ 21 + ψˇ 33 ) + p+ (−ψˇ 14 + ψˇ 21 + 2ψˇ 33 ) S4 = − e 12 2 √ 3 ˇ ˇ − p− (ψ14 + ψ21 ) 2 √ √ +3e−2Ω e2β+ +2 3β− ψˇ 13 + e2β+ −2 3β− ψˇ 22 + e−4β+ ψˇ 34 .

(7.51)

The constraints S[a] and H⊥ satisfy an algebra similar to (7.23), (7.24), (7.25), and (7.26), with 9

S[a] , S[b]

: D

=0,

(7.52)

when [a] = [b]. As an example, : 9 {S1 , S1 }D = − S1 , S 4 D =

1 3Ω γ 14 H⊥ + 18 iψˇ j γ j γ 0 S1 e 1 32 √ + 54ie−2Ω S1 e2β+ +2 3β− γ 2 γ 0 γ 3 ψˇ 1 √

+e2β+ −2

3β−

γ 3 γ 0 γ 1 ψˇ 2 + e−4β+ γ 1 γ 0 γ 2 ψˇ 3

(7.53)

. 1

.

However, the homogeneization procedure of (7.47) with (7.16) and (7.17) leads to even more terms proportional to:S that will also close weakly (see Sect. 7.2.1). 9 Once again, since S[a] , S[b] D = 0 when [a] = [b], each of the S[a] acts on an orthogonal subspace. In view of the subsequent quantization, the bosonic momentum corresponds to the derivative of the corresponding conjugate variable (as in the Bianchi I case) and the momenta conjugate to the ψˇ i are the ψˇ i themselves. They will satisfy a Clifford algebra via a matrix representation of the algebras, as in Sect. 7.2.1.

7.2


141

The constraints can thus be put into a compact matrix form: 1 3Ω e {(pΩ M11 + p+ M12 + p− M13 ) 12 √ √ +e−2Ω e2β+ +2 3β− M21 − M22 − 3M23

S1 = −

+e2β+ −2

√

3β−

M21 − M22 + .

+e−4β+ (M21 + 2M22 )

√

3M23

,

(7.54)

1 3Ω e {(pΩ M21 + p+ M22 + p− M23 ) 12 √ √ +e−2Ω e2β+ +2 3β− −M11 + M12 + 3M13

S2 = −

√ 3β−

+e2β+ −2

−M11 − M12 −

. +e−4β+ (−M11 − 2M12 )] .

√ 3M13 (7.55)

Note that the linear combinations of ψˇ in the kinetic term of S1 occur in the interaction coupling terms of S2 and vice versa. A similar pattern is present for S3 , S4 . Note 7.5 Before proceeding, let us reiterate the following interesting feature of the matrix representation approach. It is known from a calculation with a second order formulation that the classical Lagrangian for the diagonal Bianchi I and IX models does not lead to the SUGRA equations (i, j) for the metric components h i j (i = j), i.e., there are non-diagonal equations that the Lagrangian cannot produce [3, 4, 9]. However, these non-diagonal equations reduce to a set of algebraic expressions in the gravitinos that constitute a set of constraints on the final solutions. In fact, as indicated in [9], it can be further demonstrated by means of the Lorentz constraints J ab (and the remaining equations) that all terms in these i = j equations reduce to a set of quadratic expressions in the gravitino components through those Lorentz constraints. Similarly to the Bianchi I case [3], the configuration here also suffers from the drawback that the equations of motion obtained by varying the Lagrangian will require additional features to be proven to be fully equivalent to those retrieved from the Einstein and Rarita–Schwinger equations

142


for N = 1 4D SUGRA (gauge choices like N = 1, Ni = 0, ψˇ 0 = 0 and γ i ψˇ i = 0). Only in Sect. 7.3.3 will the situation be fully clarified. In particular, from the N i we get the (i0) equations for the Einstein and Rarita–Schwinger system. Given these and the others for Ω, β± , N , ψˇ 0 , and ψˇ i , we recover the (0i) equations. By means of the Lorentz constraints J ab (and the remaining equations), all terms in these i = j equations reduce to a set of quadratic expressions in the gravitino components. At the quantum level these are equivalent to the Lorentz constraints themselves.

Quantization In contrast to the Bianchi I cosmology, we now have six matrices M[a]i and the minimum possible rank is eight, so each Ψ[a] will now be an eight-dimensional vector. The quantum equations can then be cast in the form 0 = iΓ 1

√ √ ∂ ∂ ∂ − iΓ 3 + e−2Ω e2β+ +2 3β− Γ 4 − Γ 5 − 3Γ 6 − iΓ 2 ∂Ω ∂β+ ∂β−

+e−2Ω e2β+ −2

√

3β−

Γ4 −Γ5 −

√

3Γ 6 + e−2Ω e−4β+ Γ 4 + 2Γ 5 ΨI , (7.56)

0 = iΓ 4

√ √ ∂ ∂ ∂ − iΓ 6 + e−2Ω e2β+ +2 3β− −Γ 1 + Γ 2 + 3Γ 3 − iΓ 5 ∂Ω ∂β+ ∂β− √

+e−2Ω e2β+ −2

3β−

√ −Γ 1 +Γ 2 − 3Γ 3 +e−2Ω e−4β+ −Γ 1 − 2Γ 2 ΨII , (7.57)

√ √ ∂ ∂ ∂ − iΓ 3 + e−2Ω e2β+ +2 3β− −Γ 4 + Γ 5 + 3Γ 6 0 = iΓ 1 − iΓ 2 ∂Ω ∂β+ ∂β− √

+e−2Ω e2β+ −2

3β−

√ −Γ 4 +Γ 5 − 3Γ 6 +e−2Ω e−4β+ −Γ 4 −2Γ 5 ΨIII , (7.58)

7.2


143

√ √ ∂ ∂ ∂ 0 = iΓ 4 − iΓ 6 + e−2Ω e2β+ +2 3β− Γ 1 − Γ 2 − 3Γ 3 − iΓ 5 ∂Ω ∂β+ ∂β− √ √ +e−2Ω e2β+ −2 3β− Γ 1 − Γ 2 + 3Γ 3 + e−2Ω e−4β+ Γ 1 + 2Γ 2 ΨIV . (7.59)

Note 7.6 Note that if there is a coupling potential in the Wheeler–DeWitt equation, this leads in the general Bianchi IX case to corresponding coupling terms in the Dirac-like equations [4]. This will be seen in the following sections for the Bianchi IX (Taub) and FRW models, but not in the Taub microsuperspace. Concerning the solutions of the equations (7.56), (7.57), (7.58), and (7.59), the state vector can be written ⎞ ⎛ ΨI ⎜Ψ ⎟ ⎜ II ⎟ (7.60) ⎟ , ⎜ ⎝ ΨIII ⎠ ΨIV but with each of ΨI , ΨII , ΨIII , and ΨIV an eight-component vector. Changing the sign of the potential in each of the equations, the state vector would be related to the previous one by ⎞ ΨIII ⎜Ψ ⎟ ⎜ IV ⎟ ⎟ , ⎜ ⎝ ΨI ⎠ ⎛

(7.61)

ΨII transforming into one another as positive and negative energy solutions. The system of equations (7.56), (7.57), (7.58), and (7.59) is nevertheless rather complicated, and no solution has been found. Only by restricting it can some solutions be established, as for the Taub case in Sect. 7.2.3, or when the Lorentz constraint is brought into the analysis (see Sect. 7.3, and especially Sect. 7.3.3).

7.2.3 Taub Minisuperspace The Taub model is the particular configuration of the Bianchi IX case where we take β− = p− = 0, remembering that the parameters β± determine the degree of anisotropy of the model [11].

144


Classical Constraints With the redefinition β+ ≡ x, the four S[a] components of the Taub model are written in the form 1 1 S1 = − e3Ω pΩ (−ψˇ 11 − ψˇ 24 + ψˇ 32 ) + px (ψˇ 11 + ψˇ 24 + 2ψˇ 32 ) 12 2 +3e−2Ω e2x ψˇ 12 + ψˇ 23 + e−4x ψˇ 31 , (7.62) 1 3Ω 1 ˇ ˇ ˇ ˇ ˇ ˇ pΩ (ψ12 + ψ23 + ψ31 ) + px (−ψ12 − ψ23 + 2ψ31 ) S2 = − e 12 2 −2Ω 2x ˇ −4x +3e e ψ11 + ψˇ 24 − e ψˇ 32 , (7.63) 1 3Ω 1 e pΩ (−ψˇ 13 + ψˇ 22 + ψˇ 34 ) + px (ψˇ 13 + ψˇ 22 + 2ψˇ 34 ) 12 2 +3e−2Ω e2x −ψˇ 14 + ψˇ 21 − e−4x ψˇ 33 , (7.64)

S3 = −

1 3Ω 1 e pΩ − (ψˇ 14 − ψˇ 21 + ψˇ 33 ) + px (−ψˇ 14 + ψˇ 21 + 2ψˇ 33 ) 12 2 +3e−2Ω e2x − ψˇ 12 + ψˇ 22 + e−4x ψˇ 34 , (7.65)

S4 = −

where the Hamiltonian is 1 H⊥ = −p2Ω + p2x + e−4Ω e−8x − 4e−2x , 3

(7.66)

9 : and S[a] , S[b] D = 0 when [a] = [b], using each of the S[a] to act on orthogonal subspaces.7 Hence in (7.62), (7.63), (7.64), and (7.65), the linear combinations of ψˇ i[a] can be made to correspond to four matrices, represented here as γ matrices α

γ ≡

0 σLα σRα 0

,

σLα ≡ (1, −σ ) ,

σRα ≡ (1, σ ) ,

(7.67)

where σ are the Pauli matrices. We then write

∂ ∂ i 0 = iγ 0 − iγ 1 − 2e−2Ω e2x γ 3 − iγ 2 + 2e−2Ω e−4x γ 3 + γ 2 ΨI , ∂Ω ∂x 2 (7.68) Each quantum constraint is written S[a] Ψ[a] = 0, where each S[a] represents a separate square root for the Hamiltonian constraint H⊥ , and the S[a] are matrices of the smallest possible rank that produces the appropriate algebra for S[a] and the ψˇ i[a] . 7

7.2


145

∂ ∂ 1 0 = −γ 2 − iγ 3 − 2e−2Ω e2x γ 0 − γ 1 − 2e−2Ω e−4x γ 1 + γ 0 ΨII , ∂Ω ∂x 2 (7.69)

∂ ∂ i 0 = iγ 0 − iγ 1 − 2e−2Ω e2x −γ 3 + iγ 2 + 2e−2Ω e−4x −γ 3 − γ 2 ΨIII , ∂Ω ∂x 2 (7.70)

∂ ∂ 1 −γ 2 − iγ 3 − 2e−2Ω e2x −γ 0 + γ 1 − 2e−2Ω e−4x −γ 1 − γ 0 ΨIV = 0. ∂Ω ∂x 2 (7.71) We have a set of coupled matrix-valued terms acting on the state vector for the Taub model, which is divided into 4-component vectors. As expected from Sect. 7.2.2, equation (7.68) produces (with a similar result for the others) ∂Ψ14 ∂Ψ13 + − 2ie−2Ω e2x (Ψ13 − Ψ14 ) = 0 , ∂Ω ∂x

(7.72)

∂Ψ13 ∂Ψ14 + − 2ie−2Ω e2x (Ψ13 − Ψ14 ) = 0 , ∂Ω ∂x

(7.73)

∂Ψ12 ∂Ψ11 − − 2ie−2Ω e2x (−Ψ11 + Ψ12 ) = 0 , ∂Ω ∂x

(7.74)

∂Ψ11 ∂Ψ12 − − 2ie−2Ω e2x (−Ψ11 + Ψ12 ) = 0 , ∂Ω ∂x

(7.75)

where the second index in, e.g., Ψ1[a] denotes the corresponding component of the four-vector Ψ1 . A solution is given by ⎛

⎞ c 2 (1 + i)e−a1 2 ⎜ ⎟ ⎜ ⎟ c 2 ⎜ − (1 − i)e−a1 ⎟ ⎜ ⎟ 2

⎟ ⎜ ΨI = ⎜ ⎟ , i 1 ⎜ da1 a22 − a12 ⎟ ⎜ ⎟ 2 3 ⎜

⎟ ⎝ ⎠ i 1 −da1 a22 + a12 2 3

(7.76)

with c, d constants and a1 = e−Ω ex and a2 = e−Ω e−2x the two radii of the Taub model. It is curious to note that there is an exact solution in pure bosonic quantum cosmology [22]: Ψ ∼ e−a1 /3 e−a2 /6 . 2

2

(7.77)

146


Taub Microsuperspace Let us now consider the microsuperspace sector for the Taub model, which corresponds to the domain of very large x (at large Ω). In this case the Hamiltonian reduces and we have H⊥ = −p2Ω + p2x −→ −

∂ 2Ψ ∂ 2Ψ + =0, 2 ∂Ω ∂x2

(7.78)

with solutions of the form Ψ = f(Ω − x) + g(Ω + x) ,

(7.79)

where f and g are arbitrary functions of Ω −x and Ω +x, respectively. The important point to note is that, in (7.68), (7.69), (7.70), and (7.71), the last terms vanish for large x (large Ω), while the first potential term remains. However, in this limit, the Dirac-like equations do not correspond to the square root of (7.78). The Dirac-like equations for the microsuperspace sector contain only one potential term, and its iterated square implies a potential term that does not arise in the bosonic quantum cosmology scenario. Moreover, for the generic Taub model, the two potential terms in the Dirac-like equations (7.68), (7.69), (7.70), and (7.71) will reproduce the potential in (7.66). Notice also that the solution (7.76) will decay in the microsuperspace limit as 2 e−a1 for the two first components and as zero for the others. However, the solution from bosonic quantum cosmology in this limit takes the form

1 2 Ψ ∼ e−a1 /3 1 − a22 . 6

(7.80)

Note 7.7 The Dirac-like equations (7.56), (7.57), (7.58), and (7.59) and (7.68), (7.69), (7.70), and (7.71) above are nevertheless similar to the Dirac oscillator equations [4, 10], where the momentum pi is replaced by pi − ˘ i and β˘ is a matrix, transforming into one another by changing the imωβr sign of the potential, and thereby relating positive–negative energy solutions. The Taub equations above do indeed have the form pΩ + ζ1 px + ζ2 e−2Ω e2x ζ3 e−2Ω e−4x ψ = 0 ,

(7.81)

where the ζi matrices satisfy ζ12 = ζ32 = constant and ζ22 = 0, so that the square of this equation reproduces the potential e−4Ω e−8x − 4e−2x in (7.66).

7.3

Bianchi Models and Lorentz Invariance

147

7.2.4 FRW Minisuperspace The closed FRW model corresponds, in the Bianchi IX context, to choosing β± = p± = 0 (spatial isotropy) [4]. Defining e−Ω ≡ a for the radius or scale factor of the FRW universe, the four S[a] components are written S1 = iγ 1 pa − 3iγ 2 a + 2Λγ 0 a 2 ,

(7.82)

S2 = iγ 2 pa + 3iγ 1 a − 2iΛγ 3 a 2 ,

(7.83)

S3 = iγ 3 pa + 3γ 0 a + 2iΛγ 2 a 2 ,

(7.84)

S4 = iγ 0 pa − 3iγ 3 a − 2iΛγ 1 a 2 ,

(7.85)

where the factor Λ is retrieved by adding the term −Λψ μ σ μν ψν to the Lagrangian to represent the effect of a cosmological constant inducing a gravitino mass. The pertinent feature here is that the iterated squared equations from (7.82)– (7.85) produce pa2 + 9a 2 + 4Λ2 a 4 + 3iγ 1 γ 2 − 4Λγ 1 γ 0 a ΨI = 0 .

(7.86)

The first three terms correspond to what one finds in the standard description of quantum cosmology. However, the two extra terms have no counterpart there.

7.3 Bianchi Models and Lorentz Invariance For the attentive reader, the matrix representation formulation in Sect. 7.2 will have raised an important question: Where did we deal with the Lorentz constraint? In Chaps. 4 and 5 of Vol. I, we discussed the importance of writing the Hamiltonian for N = 1 SUGRA in the form 1 H = N H⊥ + N i Hi + ω0AB J AB + ψ 0 S , 2

(7.87)

where H⊥ , Hi , and J AB are the usual Hamiltonian, diffeomorphism, and Lorentz constraints, respectively, and S the supersymmetric fermionic constraint. In the matrix formulation, it seems that the Lorentz constraint is only present in a rather indirect manner (see Sect. 7.2.3), through algebraic equations that relate to the difference between the set constituted by the Einstein and Rarita–Schwinger equations and those equations of motion extracted directly from the reduced (homogeneous) Lagrangian. But, in any case, they were not applied directly to Ψ . In this section we will describe particular contributions [23, 6, 9] to clarifying (in some generality) what exactly might be the implications of the Lorentz constraint on

148


the wave function of the universe.8 A rather radical conclusion seems to be that no physical states will exist in the minisuperspace sector of the theory (see Sect. 7.3.1), apart from the exception provided when the Lorentz generator trivializes and rest frame solutions are acceptable, being valid for an arbitrary Lorentz generator (see Sect. 7.3.2). However, as the matrix approach has provided an alternative to the fermionic differential operator representation, it also provides a source for selfanalysis. In Sect. 7.3.3, a different use of the explicit form of the Lorentz constraints leads to somewhat different conclusions. Overall, this is good news, because it means that there is still much to explore and study in SQC.

7.3.1 Implications of the Lorentz Constraints We recall that, in the matrix representation, the gravitinos are taken as 4-component spinors in a real Majorana representation (see Sect. 7.2). We will be using densitized local components ψˇ a = e ea i ψi with e = (3) e = det(ei a ). We now proceed to a Bianchi class A background [23, 6], taking the basic field variables, the tetrad vierbein ea μ , and the gravitino field ψˇ μ to be spatially homogeneous. More precisely, in the description and choice of basis presented in Sect. 7.2, we will have ψˇ μ = ψˇ μ (t), and the full set of dynamical variables is thus Ω(t), βi j (t), and ψμ[a] (t) (compare with the Misner representation explained in Sects. 2.3.1 and 6.1 of Vol. I). In addition, the following transformation is used: ⎧ √ ⎪ ⎨x = Ω − β+ − √3β− , u k ≡ y = Ω − β+ + 3β− , ⎪ ⎩ z = Ω + 2β+ .

(7.88)

This will modify the appearance of the SUSY constraints. With these coordinates and for the ωi basis characteristic of the particular Bianchi model, it follows that e0 0 = N (t), ei 0 = Ni , and e j i = exp (u ( j) )δ ij , with ωi jk =

1 exp (u i )Ci jk , 2

e ≡ exp [−(x + y + z)],

ψˇ a = exp [−(x + y + z)]ψa .

Note that, in exp (u (i) )Ci jk , there is no sum over i, and Ci jk = εi jk in the Bianchi IX case. As a consequence, using the real Majorana representation in the matrix representation, the supersymmetric constraints for the class A models are [23, 6]

8 Chapter 5 of Vol. I focused on constructing inequivalent Lorentz invariants, i.e., providing scalar wave functions as fermionic and bosonic power series expansions that automatically fulfill the Lorentz condition.

7.3


149

1 S = e−1 iγ 0 γ 1 γ 2 ψˇ 2 + γ 3 ψˇ 3 ∂x − exp (x)(m 22 + m 33 ) γ 3 ψˇ 2 − γ 2 ψˇ 3 2 1 +γ 2 γ 1 ψˇ 1 + γ 3 ψˇ 3 ∂ y + exp (y)(m 11 + m 33 ) γ 3 ψˇ 1 − γ 1 ψˇ 3 2 1 +γ 3 γ 1 ψˇ 1 +γ 2 ψˇ 2 ∂z + exp (z)(m 11 +m 22 ) γ 2 ψˇ 1 +γ 1 ψˇ 2 . 2 (7.89) The Lorentz constraint Jab is written explicitly (see also Sect. 7.3.3) Jab ≡ pa i ebi − pb i eai − π i σab ψi 1 T = 2 p[a i eb]i + ψ[a ψb] , 2

(7.90)

with πa =

i 0αδβ ψ δ γ5 γβ ε 2

the momentum conjugate to the gravitino field, taking ψ ≡ −iψ T γ 0 . The bosonic part of this expression vanishes when it is written in the basis of the coordinates (x 1 , x 2 , x 3 ) of SO(3) (in the Bianchi IX case) and then integrated over them, the fermionic part remaining. Up to an irrelevant factor, the Lorentz constraints written in the variables ψˇ defined in (7.17) turn out to be given by J ab ∼

ςi (ab) ψˇ iT γ αβ ψˇ j ,

(7.91)

i = j

where ςi (ab) =

1 if i = a, b , −1 otherwise .

(7.92)

More precisely, with ψˇ i ≡ −γ i ψˇ i (no sum), we find that the J ab constraints have the form [9] 1 01 J = ψˇ 2 γ 2 ψˇ 1 + ψˇ 3 γ 3 ψˇ 1 , 2

(7.93)

1 02 J = ψˇ 1 γ 1 ψˇ 2 + ψˇ 3 γ 3 ψˇ 2 , 2

(7.94)

1 03 J = ψˇ 1 γ 1 ψˇ 3 + ψˇ 2 γ 2 ψˇ 3 , 2

(7.95)

150


1 12 J = ψˇ 1 γ 0 ψˇ 2 , 2

(7.96)

1 13 J = ψˇ 1 γ 0 ψˇ 3 , 2

(7.97)

1 23 J = ψˇ 2 γ 0 ψˇ 3 . 2

(7.98)

The physical states Ψ thus satisfy SΨ = 0 ,

Ha Ψ = 0 ,

Jab Ψ = 0 ,

(7.99)

The choice of matrix representation seems to induce the matrix realization of the ψˇ i[a] to obey : i ψˇ i[a] , ψˇ j[b] = − (γi γ j )[a][b] , 8

9

(7.100)

where [a] and [b] are spinor indices (see Sect. 7.2), and the wave function of the universe is taken to be [see (7.60) and (7.61)] ⎛

⎞ ΨI ⎜ ΨII ⎟ ⎟ Ψ ≡⎜ ⎝ΨIII ⎠ . ΨIV

(7.101)

From this point, some significant differences arise with respect to the descriptions in Chaps. 4 and 5 of Vol. I, and even with respect to Sect. 7.3.3 of this volume. In [23, 6], physical states were required to satisfy ⎛

⎞ ⎞⎛ 0 0 0 0 ΨI ⎜0 ⎟ ⎜ 0 J12 J13 ⎟ ⎟ ⎜ ΨII ⎟ = 0 . Jab Ψ = 0 ⇒ ⎜ ⎝0 − J12 0 J23 ⎠ ⎝ΨIII ⎠ 0 − J13 − J23 0 ΨIV

Note 7.8

(7.102)

We may make the following remarks here:

• From (7.102), the operator Jab is defined on the tangent space (using a local orthonormal Lorentz coordinatization, even though a vector representation can be transferred to a spinor formulation as explained in Appendix A) and acts on (irreducible) orthogonal (sub-)Hilbert spaces of quantum states.

7.3


151

• It should also be mentioned that the formulas (7.101) and (7.102) were introduced and used in [23, 6] before any discussion of the ‘implications’ of (7.103), (7.104), and (7.105). • Equation (7.102) leads to J12 ΨIII = −J13 ΨIV ,

J12 ΨII = J23 ΨIV ,

J13 ΨII = −J23 ΨIII , (7.103)

whence, using Ja ≡

1 ε0abc J bc ⇒ J0 = 0 , 2

(7.104)

we can write instead J3 ΨIII = J2 ΨIV ,

J3 ΨII = J1 ΨIV ,

J2 ΨII = J1 ΨIII .

(7.105)

• At this stage the reader should note the highly relevant fact that there is no condition in (7.103) involving ΨI . This will play an important role in what follows. • Moreover, (7.103) suggests that the components Jab of the Lorentz generator (7.102) and also the components of the wave function of the universe (7.101) could be represented as 4 × 4 and 4 × 1 non-singular matrices, respectively. It then follows (see Exercise 7.2) that there is a transformation J1 = J3 (J2 )−1 J1 (J3 )−1 J2 , J2 = J1 (J3 )−1 J2 (J1 )−1 J3 ,

(7.106)

J3 = J1 (J2 )−1 J3 (J1 )−1 J2 . • In addition, we observe the following: – When using ei0 = Ni = 0 in Jab = 0, which determines ω00i ∼ ∂Ni = 0 from g0i = 0, we have no role for Hi and J0i . – In Jab , the space of states used to formulate (7.101) is decomposed into orthogonal blocks induced from the algebra 9

:

S(x), S(x )

D

= γ a Ha δ(x, x ) ,

(7.107)

from which (7.101) follows, producing a partition of the Hilbert space of states. Hence, we would expect the application of each Jab to any orthogonal subspace (see Sect. 7.3.3), as determined by (7.52), (7.60), (7.100) [see also (7.22), (7.29), (7.30), and (7.31)]. This subspace structure is established through the SUSY algebra in (7.52) and (7.100).

152


Applying (7.102) [23, 6] seems in contradiction with the previous argument, and too severe a restriction to impose on Ψ . • With (7.88), {S, S}D would probably still imply a structure similar to (7.22), (7.23), (7.24), (7.25), (7.26), and (7.31), but the above assumes that (7.101) is valid generically. In particular, this assumption was further strengthened in the Gowdy (midisuperspace) T 3 model, in which solutions were found where none were possible when restricted to a minisuperspace background [24, 25].

For the remainder of this section and the following, we use the requirement suggested in [23, 6]. We then have two possibilities for interpreting the conditions (7.106): • First, consider them as an invariance between the different components of the Lorentz generator. A matrix representation that corresponds suitably to this feature is possible with J 1 = −γ 1 γ 0 ,

J 2 = γ 1γ 3 ,

J 3 = −γ 3 γ 0 ,

(7.108)

where

J i, J

j

=

1 i jk ε Jk , 2

and the components of the gravitino field are then ψˇ 1 = −iγ 3 ,

ψˇ 2 = −iγ 1 ,

ψˇ 3 = −iγ 0 .

(7.109)

• Now, for the second possibility [23, 6], we consider the case where the Lorentz constraint trivializes. For example, consider J12 (J13 )−1 = ±1 ,

(7.110)

J12 = ∓ (J13 )−1 .

(7.111)

determining

Then similar results follow for the other components, and this means that all those components are proportional, i.e., J3 = ±J2 = ±J1 . Consequently, this implies that

(7.112)

7.3


ψˇ 1 = −ψˇ 2 = ψˇ 3 ,

153

(7.113)

and the components ψˇ i[a] are thus pure real numbers. What is more relevant is that the above does lead to ΨII = ΨIII = ΨIV .

(7.114)

Since the Lorentz operator trivializes, only two components are independent in the wave function of the universe. Hence, it remains to apply the supersymmetry constraints on ΨI and, e.g., ΨII . The supersymmetry constraints reduce to

1 1 0 = −i ψˇ 11 + ψˇ 13 ∂x + ex ΨI + i ψˇ 11 + ψˇ 12 ∂ y + e y ΨI 2 2

1 +i ψˇ + ψˇ ∂z + ez ΨI (7.115) 2 and

1 x 1 y ˇ ˇ ∂x + e ΨII + i ψ − ψ ∂ y − e ΨII 2 2

1 +i ψˇ + ψˇ ∂z + ez ΨIV . (7.116) 2

0 = −i ψˇ 12 − ψˇ 14

7.3.2 Physical States In this section we extract the physical consequences of the Lorentz conditions established in Sect. 7.3.1, at the level of explicit solutions for Ψ . Rest Frame Solution Let us first discuss the fact that the Lorentz conditions do not constrain the first component ΨI of the wave function of the universe [23, 6]. Hence, assume the case (rest frame type solution) ⎛ ⎞ ΨI ⎜0⎟ ⎟ Ψ =⎜ ⎝0⎠ . 0

(7.117)

Note that the action of the Lorentz constraint is arbitrary. The supersymmetry constraint (7.89) can be written as

154


0=

a b exp (x) + iΓ 3 ∂2 + Γ 4 exp (y) 2 2 . c + iΓ 5 ∂3 + Γ 6 exp (z) ΨI , 2 iΓ 1 ∂1 − Γ 2

(7.118)

where the Γ i are the 4 × 4 matrices involving the gravitino components which appear in (7.89), a = m 22 + m 33 , b = m 11 + m 33 , and c = m 11 + m 22 , referring to the particular type of Bianchi model in class A. For a matrix realization of the set of six independent Γ i matrices satisfying {Γ A , Γ B }D = 0, with A = B = 1, . . . , 6, the following Majorana representation is used: Γ 1 = −γ 0 γ 1 , Γ 2 = γ 0 , Γ 3 = −γ 0 γ 3 , Γ 4 = γ 0 γ 5 , Γ 5 = γ 0 γ 2 , Γ 6 = γ 1 . (7.119) This means that ΨI is a 4-component vector. The solution of (7.118) is ⎛

Ψ01 eae

x /2

e−be

y /2

e−ce

z /2

+ e−ce

z /2

E i (z)l1

⎞

⎟ ⎜ x y z z ⎜ Ψ02 e−ae /2 ebe /2 ece /2 − ece /2 E i (z)l2 ⎟ ⎟ , ⎜ ΨI = ⎜ ⎟ aex /2 ebe y /2 ecez /2 + ecez /2 E (z)l i 3 ⎠ ⎝ Ψ03 e x y z z Ψ04 e−ae /2 e−be /2 e−ce /2 − e−ce /2 E i (z)l4

(7.120)

with l j separation constants, Ψ0 j integration constants ( j = 1, . . . , 4), and the exponential integral function E i is given by E i (x) =

ee

α /2

dα =

−x

eu du . u

∞

States from a (Non-)Trivial Lorentz Constraint Embracing the trivialization option (7.112), (7.114) determined from the Lorentz constraint action, we have only two components to consider. The action to investigate is that of the supersymmetry constraints on ΨI and another, say, ΨII . The solution of the equations (7.115), (7.116) is then ΨI = Ψ0I eae

−x /2

e−be

ae−x /2

ΨII = Ψ0II e

y /2

be y /2

e

ece

z /2

−cez /2

e

− ece

z /2

E i (z)n z ,

cez /2

+e

E i (z)m z ,

(7.121) (7.122)

where n z , m z are separation constants and Ψ0I and Ψ0II are integration constants. The conclusion extracted from (7.89) is that, for a representation induced by imposing the full non-trivial Lorentz constraint, the wave function of the universe vanishes for all Bianchi class A models: SΨ = 0 ⇒ Ψ = 0 .

(7.123)

7.3


155

This means that there are no physical states consistent with the nontrivial Lorentz constraint. In more detail, Jab Ψ = 0 for all Bianchi class A models would lead, according to [23, 6], to no physically occurring states in the supersymmetric approach to quantum consmology other than the rest frame type (concerning only the first component of Ψ , which is not constrained by the Lorentz condition). These results have the merit of broadening the discussion of minisuperspace studies in SQC to the perspective of Chap. 4 for the canonical quantization of N = 1 SUGRA. However, the next section, where the Lorentz constraint is applied without the particular reductive assumption (7.102) on its structure, will add to this investigation.

7.3.3 Lorentz Constraint Components and Bianchi IX Diagonal Models Here we discuss another perspective on the Lorentz constraints, in the context of a diagonal Bianchi IX configuration [9]. The following points should be emphasized: • No assumptions are made regarding the automatic fulfillment of the Lorentz constraints via any fermionic (or bosonic) power series expansion of the wave function. • No other assumption is imposed on what Lorentz invariance should mean. • All the Lorentz constraints are solved explicitly. The context here will be that of the diagonal Bianchi IX model, integrating over the SO(3) spatial directions, fixing the gauge to Ni = 0, and specifically defining the new variables ψˇ 0 ≡ e−3Ω/2 ψ0 , ψˇ i ≡ e−Ω/2 e−βii γi ψi ,

(7.124) i = 1, 2, 3 .

(7.125)

Assuming homogeneity, i.e., the fields do not depend on the spatial coordinates, the following action is obtained in canonical form [compare with (7.47) in Sect. 7.2.4]: N e3Ω - 2 −pΩ + p2+ + p2− − 12 √ √ 4 2 4β+ −2β+ + e cosh 2 3β− cosh 4 3β− − 1 − e 3 3

L = pΩ Ω˙ + p+ β˙+ + p− β˙− + −4Ω

−e

1 −8β+ e 3

√ √ N e3Ω 3p+ + 3p− ψˇ 1 γ 0 ψˇ 3 + 2 3p− ψˇ 1 γ 0 ψˇ 2 12 √ + 3p+ − 3p− ψˇ 2 γ 0 ψˇ 3

−i

156


−i

√ N Ωe ψˇ 3 γ 0 γ 5 ψˇ 1 e−4β+ + 2e2β+ sinh(2 3β− ) 2

√ +ψˇ 2 γ 0 γ 5 ψˇ 3 e−4β+ − 2e2β+ sinh(2 3β− ) . √ −ψˇ 2 γ 0 γ 5 ψˇ 1 e−4β+ − 2e2β+ cosh(2 3β− ) √ 3 i 3 1 3Ω ˇ ˇ ˇ ˇ ˇ ˇ ˇ + ψ0 e (ψi ) − p+ (ψ1 + ψ2 − 2ψ3 ) − p− (ψ1 − ψ2 ) pΩ 12 2 2 i=1

. √ √ +eΩ e2β+ +2 3β− γ 5 ψˇ 1 + e2β+ −2 3β− γ 5 ψˇ 2 + e−4β+ γ 5 ψˇ 3 √ 3 i 3 1 3Ω ˇ ˇ ˇ ˇ ˇ ˇ − (ψ i ) − p+ (ψ 1 + ψ 2 − 2ψ 3 ) − e p− (ψ 1 − ψ 2 ) pΩ 12 2 2 i=1

. √ √ +eΩ e2β+ +2 3β− γ 5 ψˇ 1 + e2β+ −2 3β− γ 5 ψˇ 2 + e−4β+ γ 5 ψˇ 3 ψˇ 0 −

i . ψˇ 1 γ 0 ψ˙ˇ 2 + ψ˙ˇ 3 + ψˇ 2 γ 0 ψ˙ˇ 1 + ψ˙ˇ 3 + ψˇ 3 γ 0 ψ˙ˇ 1 + ψ˙ˇ 2 2

(7.126)

Classical Constraints and Algebra Let us recall the Lorentz constraints with the form (7.94), (7.95), (7.96), (7.97), and (7.98). Exercise 7.3 shows how the Lorentz constraints relate to missing nondiagonal equations of the Einstein–Rarita–Schwinger set. By means of the Lorentz constraints J ab , all terms in the Einstein–Rarita–Schwinger equations with i = j are reduced to a set of quadratic expressions in the gravitino components, equivalent to the Lorentz constraints themselves at the quantum level. This is a consequence (shown here for the diagonal Bianchi IX model) of not retrieving the full SUGRA equations (i, j) for the metric components gi j (i = j). We now find that the supersymmetric constraints can be written [9] √ 3 3 1 ˇ ˇ ˇ ˇ ˇ ˇ (ψi ) − p+ (ψ1 + ψ2 − 2ψ3 ) − p− (ψ1 − ψ2 ) pΩ 2 2

S=e

3Ω

i=1

+eΩ e2β+ +2

√ 3β−

γ 5 ψˇ 1 + e2β+ −2

√ 3β−

γ 5 ψˇ 2 + e−4β+ γ 5 ψˇ 3

. (7.127)

It is relevant at this point to introduce some algebraic features. In particular, the Dirac brackets for the ψˇ components will satisfy

7.3


-

.

157

i δ[a][b] , D 2 . i ψˇ i[a] , ψˇ j[b] = − δ[a][b] , D 2 ψˇ i[a] , ψˇ i[b]

=

(7.128) i = j ,

(7.129)

which can be diagonalized by the linear transformation 1 χ1 = √ (1 + i) ψˇ 1 + ψˇ 2 + ψˇ 3 , 3 1 χ2 = (1 + i) ψˇ 1 − ψˇ 2 , 2 1 χ3 = √ (1 + i) ψˇ 1 + ψˇ 2 − 2ψˇ 3 , 2 3

(7.130) (7.131) (7.132)

where the factor (1 + i) is chosen so that the χ are real. Hence, {χ1A , χ1B }D = − {χ2A , χ2B }D = − {χ3A , χ3B }D = δ AB ,

(7.133)

and therefore 1 S = (pΩ χ1 − p− χ2 − p+ χ3 ) + γ 5 (α1 χ1 + α2 χ2 + α3 χ3 ) , 3

(7.134)

where the potentials α1 ≡ −(E 1 2 + E 2 2 + E 3 2 ) , √ α2 ≡ 3(E 1 2 − E 2 2 ) ,

(7.136)

α3 ≡ E 1 2 + E 2 2 − 2E 3 2 ,

(7.137)

(7.135)

can also be written αi ≡

1 ∂i u , 2

(7.138)

with u ≡ E12 + E22 + E32

(7.139)

and √

E 1 ≡ e−Ω+β+ +

3β−

,

(7.140)

E 2 ≡ e−Ω+β+ −

3β−

,

(7.141)

√

E 3 ≡ e−Ω−2β+ .

(7.142)

158


Quantization For the bosonic momenta, we assign the derivative of the corresponding conjugate variable as above. The physical fermionic variables are ψˇ i[a] (or the equivalent χi[a] ), which satisfy a Clifford-type algebra. Hence, the canonical quantization of these 12 fermionic variables can be achieved by a matrix representation. If we want to represent a Clifford algebra with 12 generators, we need matrices of dimension at least 212/2 = 64. In essence, the wave function Ψ will satisfy the set of matrix equations given by J ab Ψ = 0 ,

S[a] Ψ = 0 .

(7.143)

With the 64 × 64 matrices (see the last section), the wave function will be a vector with 64 components which are functions of Ω, β+ , and β− . The Hermitian representation of the Dirac matrices can be found in [9]. Now the task is to solve the 384 coupled algebraic equations corresponding to the Lorentz constraints [noting that the solution of the Lorentz generators in (7.143) is the same for many different possible orderings]: J ab Ψ = 0 .

(7.144)

As described in [9], the result is that Ψ has only two nonvanishing components, both of them arbitrary functions of Ω, β+ , and β− , which are situated at positions 39 and 52 of Ψ (labelled Ψ1 and Ψ2 , respectively): ⎛

⎞ 0 ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ Ψ1 ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ . ⎟ ⎟ Ψ =⎜ ⎜ .. ⎟ . ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ Ψ2 ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎝ . ⎠ 0

(7.145)

Inserting this solution into the system of 256 differential equations S[a] Ψ = 0 [9], the 64 equations S1 Ψ = 0 reduce to 6 equations:

7.3


159

3

∂ Ψ1 + α1 Ψ1 = 0 , ∂Ω

(7.146)

3

∂ Ψ1 + α2 Ψ1 = 0 , ∂β−

(7.147)

3

∂ Ψ1 + α3 Ψ1 = 0 , ∂β+

(7.148)

3

∂ Ψ2 − α1 Ψ2 = 0 , ∂Ω

(7.149)

3

∂ Ψ2 − α2 Ψ2 = 0 , ∂β−

(7.150)

3

∂ Ψ2 − α3 Ψ2 = 0 . ∂β+

(7.151)

Similarly to what we have been observing throughout this chapter, if we change the sign of the potentials u in the first three equations (7.146), (7.147), and (7.148), we get the second set of three equations (7.149), (7.150), and (7.151), just as happens with the positively and negatively charged Dirac states. The solution is Ψ1 = e−φ/6 ,

(7.152)

Ψ2 = eφ/6 ,

(7.153)

with u as defined in (7.139). Let us elaborate further on an issue introduced in this chapter, just before Sect. 7.2. In the approach of Chap. 5, the coefficients of the fermionic power of the wave function are invariant singlets. The 64-vector wave function could be displayed in terms of this type of expansion. To do so, define new fermionic variables as follows: 1 εi ≡ √ (χ2i−1 − iχ2i ) , 2 1 π i ≡ √ (χ2i−1 + iχ2i ) , 2

(7.154) (7.155)

where i = 1, . . . , 6. Then these variables satisfy the canonical anticommutation relations with π i the canonically conjugate momenta of εi , and the wave function can, in general, be written as an expansion in the fermionic variables ηi as follows: (1)

Ψ = Ψ (0) + εi Ψi

(2)

(6)

+ εi1 εi2 Ψi1 i2 + · · · + εi1 εi2 εi3 εi4 εi5 εi6 Ψi1 i2 i3 i4 i5 i6 ,

(7.156)

where the coefficients Ψ (n) are now 64-vectors. In the approach of Chap. 5, since only even combinations of the εi are Lorentz invariant (those products of the

160


fermionic components, possibly with tetrad terms, where the indices are contracted), only these are considered. In the expansion (7.156), each of the powers of εi , when these are represented as matrices, also gives a matrix that multiplies its corresponding 64-vector Ψ (n) , thereby generating six 64-vectors. Summary and Review. To end this chapter devoted to the matrix representation of fermionic variables, we include a few revision questions: 1. How does the matrix representation meet the historical background of SQC and the ADM methodology [Sect. 7.1]? 2. How is the case of Bianchi I cosmology dealt with [Sect. 7.2.1]? 3. What complexity does spatial curvature bring in [Sect. 7.2.2]? 4. Why is the Taub case so interesting to discuss in this context [Sect. 7.2.3]? 5. What is the ‘issue’ with the Lorentz constraints? What different analyses exist [Sects. 7.3.1, 7.3.2, and 7.3.3]?

Problems 7.1 An Avant Garde (Non-SUSY) Square Root Formulation Obtain a Dirac-like system of equations for the (diagonal) Bianchi I model (involving a two-spinor component wave function) and solve it. Note that this is basically Exercise 11.3 in [11]. The reader should find this somewhat surprising! 7.2 From the Lorentz Constraints to the Lorentz Conditions Use (7.103) to determine J1 = J3 (J2 )−1 J1 (J3 )−1 J2 , J2 = J1 (J3 )−1 J2 (J1 )−1 J3 , J3 = J1 (J2 )

−1

J3 (J1 )

−1

(7.157)

J2 ,

as conditions implied by the Lorentz constraints. 7.3 From the Lorentz Constraints to the Non-Diagonal ‘Missing’ Equations Using the Lagrangian, the corresponding equations of motion, and the Lorentz constraints displayed in Sect. 7.3.3, establish that the non-diagonal equations (i, j) for the metric components h i j (i = j) in the SUGRA equations can indeed be found through this manipulation.

References 1. Macias, A.: The ideas behind the different approaches to quantum cosmology. Gen. Rel. Grav. 31, 653–671 (1999) 127, 128 2. Macias, A., Mielke, E.W., Socorro, J.: Supersymmetric quantum cosmology: The Lorentz constraint.In: 8th Marcel Grossmann Meeting on Recent Developments in Theoretical and

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8.

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

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Experimental General Relativity, Gravitation and Relativistic Field Theories (MG 8), Jerusalem, Israel, 22–27 June 1997 127 Macias, A., Obregon, O., Ryan, M.P.: Quantum cosmology: The supersymmetric square root. Class. Quant. Grav. 4, 1477–1486 (1987) 127, 130, 132, 133, 134, 137, 141 Macias, A., Obregon, O., Socorro, J.: Supersymmetric quantum cosmology. Int. J. Mod. Phys. A 8, 4291–4317 (1993) 127, 130, 131, 132, 133, 134, 137, 138, 141, 143, 146, 147 Macias, A., Ryan, M.P.: Quantum cosmology for the Bianchi type IX models. In: 7th Marcel Grossmann Meeting on General Relativity (MG 7), Stanford, CA, 24–30 July 1994 127 127 Macias, A., Mielke, E.W., Socorro, J.: Supersymmetric quantum cosmology: The physical states. Phys. Rev. D 57, 1027–1033 (1998) 127, 128, 147, 148, 150, 151, 152, 153, 155 Obregon, O.: Conserved current and the possibility of a positive definite probability density in quantum cosmology. In: 8th Jorge Andre Swieca Summer School: Particles and Fields, Rio de Janeiro, Brazil, 5–18 February 1995 127 Obregon, O., Ramirez, C.: SUSY quantum cosmology with matrix representation for fermions. In: 5th Mexican Workshop of Particles and Fields, Puebla, Mexico, 30 October–3 November 1995 127 Obregon, O., Ramirez, C.: Dirac-like formulation of quantum supersymmetric cosmology. Phys. Rev. D 57, 1015–1026 (1998) 127, 128, 129, 132, 134, 141, 147, 149, 155, 156, 158 Socorro, J., Obregon, O., Macias, A.: Supersymmetric microsuperspace quantization for the Taub model. Phys. Rev. D 45, 2026–2032 (1992) 127, 131, 132, 134, 146 Ryan, M.P., Shepley, L.C.: Homogeneous Relativistic Cosmologies. Princeton Series in Physics, 320pp. Princeton University Press, Princeton, NJ (1975) 129, 143, 160 Casalbuoni, R.: The classical mechanics for Bose–Fermi systems. Nuovo Cim. A 33, 389 (1976) 129 Pilati, M.: The canonical formulation of supergravity. Nucl. Phys. B 132, 138 (1978) 129, 134 Tabensky, R., Teitelboim, C.: The square root of general relativity. Phys. Lett. B 69, 453 (1977) 129, 134 Teitelboim, C.: Supergravity and square roots of constraints. Phys. Rev. Lett. 38, 1106–1110 (1977) 129, 134 Teitelboim, C.: Surface integrals as symmetry generators in supergravity theory. Phys. Lett. B 69, 240–244 (1977) 129 Teitelboim, C.: Spin, supersymmetry, and square roots of constraints. In: Proceedings of Current Trends in Theory of Fields, Tallahassee, FL, 6–7 April 1978 129 Teitelboim, C.: The Hamiltonian structure of space-time (1978) 129 Teitelboim, C.: How commutators of constraints reflect the space-time structure. Ann. Phys. 79, 542–557 (1973) 129 Galvao, C.A.P., Teitelboim, C.: Classical supersymmetric particles. J. Math. Phys. 21, 1863 (1980) 130, 135 Finkelstein, R.J., Kim, J.: Solutions of the equations of supergravity. J. Math. Phys. 22, 2228 (1981) 132 Moncrief, V., Ryan, M.P.: Amplitude real phase exact solutions for quantum mixmaster universes. Phys. Rev. D 44, 2375–2379 (1991) 145 Macias, A., Mielke, E.W., Socorro, J.: Supersymmetric quantum cosmology for Bianchi class A models. Int. J. Mod. Phys. D 7, 701–712 (1998) 147, 148, 150, 151, 152, 153, 155 Macias, A., Quevedo, H., Sanchez, A.: Gowdy T 3 cosmological models in N = 1 supergravity. gr-qc/0505013 (2005) 152 Macias, A., Camacho, A., Kunz, J., Laemmerzahl, C.: Midisuperspace supersymmetric quantum cosmology. Phys. Rev. D 77, 064009 (2008) 152

Chapter 8

N =2 (Local) Conformal Supersymmetry

In this chapter we will present yet another way of charting SQC. It employs an N = 2 local conformal (time) supersymmetry, which is a subgroup of the 4D spacetime supersymmetry and has been presented within the framework of a superfield description (see Chap. 3 of Vol. I). We will describe in detail the FRW and Bianchi class A cosmological configurations. The corresponding SUSY action is displayed, extended to a set of complex (homogeneous) scalar supermultiplets that can be related to the dilaton–axion (and chiral) components of SUGRA theory. In this framework, where this local conformal time supersymmetry determines the structure of the scalar field potential, supersymmetry breaking (see Sects. 3.2 and 5.1 of this volume) and the recovery of a suitable inflationary scenario are discussed in [1–16]. Before proceeding, the reader should note that there will be some similarities, and also some differences, with respect to the methodology of Chap. 6, Vol. I. This is part of the interest in the approach here. Another motivation was criticism [4] that the Grassmannian quantities used in Chap. 6 of Vol. I should be introduced in a ‘clearer’ (SUSY) manner.

8.1 Motivation and Superfield Description The argument for this approach runs as follows. The action of the cosmological models obtained from the 4D Einstein–Hilbert action (by spatial reduction) preserves the invariance under the local time reparametrization (see, e.g., [17]). In fact, the FRW action

1 d a 2 a˙ a a˙ 2 + kN a + dt (8.1) S= − 2N 2 dt 2N is invariant under (time reparametrization) t → t + f (t) ,

(8.2)

provided that a(t) and N (t) transform according to Moniz, P.V.: N = 2 (Local) Conformal Supersymmetry. Lect. Notes Phys. 804, 163–188 (2010) c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-11570-7_8

8 N =2 (Local) Conformal Supersymmetry

164

δa = f a˙ ,

δN = ( f N )· .

(8.3)

We thus extend this group of local time reparametrizations (of the cosmological models) to the N = 2 local conformal time supersymmetry [1, 4, 7, 11]. This means defining ‘odd’ (Grassmanian-like) time parameters η and η (where η is the complex conjugate of η). These are the superpartners of the usual time parameter t. As a consequence, any (new) functions, which were previously just functions of time t, now become superfunctions depending on (t; η, η). These are called superfields (see Sect. 3.3 of Vol. I). Following this superfield procedure,1 a cosmological action can be retrieved, which is invariant under the N = 2 local conformal time supersymmetry: • The Lagrangian involves superfields on the superspace (t, η, η). • It describes the evolution of the bosonic and additional Grassmann degrees of freedom (η, η).

8.1.1 Empty Matter Sector We begin by implementing this superfield formulation for the FRW action using the above methodology. Bianchi class A models are addressed in the next section. FRW Universe The N = 2 local conformal time SUSY (t, η, η) transformations (whose notation we adopt throughout this chapter) are [4, 8–11] (see also footnote 7 in Sect. 3.3.1 and Note A.6): i i δt = f (t) + ηε + η ε , 2 2 1 ˙ 1 f + ib η + δη = ε + 2 2 1 1 ˙ f − ib η − δη = ε + 2 2

(8.4) i εηη , 2 i εηη , 2

(8.5) (8.6)

or instead, in a more condensed manner, 1 1 δt ≡ L(t, η, η) + ηDη L(t, η, η) − ηDη L(t, η, η) , 2 2 i i δη ≡ Dη L(t, η, η) , δη = − Dη L(t, η, η) , 2 2

(8.7) (8.8)

1 This methodology for the generalization of a local time reparametrization is well known in the formulation of spinning particles and superparticles [18].

8.1

Motivation and Superfield Description

165

with the superfunction L(t, η, η) defined by L(t, η, η) ≡ f (t) + iηε (t) + iηε (t) + b(t)ηη ,

(8.9)

where2 Dη ≡

∂ ∂ + iη , ∂η ∂t

Dη ≡ −

∂ ∂ − iη , ∂η ∂t

are the corresponding (super)covariant derivatives of this global conformal supersymmetry, f (t) is a local time reparametrization parameter, and ε (t) ≡ N −1/2 ε(t) is the Grassmann complex parameter of the local conformal SUSY transformations. Note 8.1 b(t) is the parameter of local U(1) rotations on the complex coordinate η. It is due to this property that the action will be invariant under N = 2 (N = 1 in the complex calculation) local SUSY transformations, with a U(1) internal subgroup [4].

Note 8.2 Note that in an SQM perspective, the contents of Chap. 6, Vol. I, follow a process which, from a bosonic setting, extend to a global N = 2 SUSY, which is then made local with (only) time t reparametrization. In the present context, it would be different, as we shall see, employing the construction above from the start, with subsequent use of SUSY constraints S, S, which are gauged to depend on t, and with {S, S}D ∼ H. The superfield generalization of the action thus has the form (with k2 = 1) √ k 2 N−1 ADη ADη A + A dηdηdt , − 2 2

S=

(8.10)

where we note the following points: • N(t, η, η) is a real 1D superfield which has the form N(t, η, η) ≡ N (t) + iηψ 0 (t) + iηψ0 (t) + ηηϑ (t) ,

2

(8.11)

The reader should note that a whole new set of derivative operators will be required here [5].


166

with ψ0 (t) ≡ N 1/2 ψ0 (t), ψ 0 (t) ≡ N 1/2 ψ 0 (t), and ϑ (t) ≡ N ϑ − ψ 0 ψ0 . This superfield transforms according to3 i i δN = (LN)· + Dη LDη N + Dη LDη N . 2 2

(8.12)

N (t) is an einbein (i.e., the lapse function!), ψ0 (t) and ψ 0 (t) are the (corresponding and equivalent) timelike components of the Rarita–Schwinger fields ψμ[a] and ψ [a] μ , and ϑ(t) is a U(1) gauge field. • In addition, the superfield A(t, η, η) has the form [1, 4] A(t, η, η) ≡ a(t) + iηψ (t) + iηψ (t) + ηηB (t) ,

(8.13)

where N 1/2 ψ (t) ≡ √ ψ(t) , a

N 1/2 ψ (t) ≡ √ ψ(t) , a

and 1 B (t) ≡ N B − (ψ 0 ψ − ψ0 ψ) . 2 The transformation rule for the scalar superfield A(t, η, η) is i i δA = LA˙ + Dη LDη A + Dη LDη A . 2 2

(8.14)

The component B(t) is an auxiliary degree of freedom, while ψ(t) and ψ(t) are the fermionic superpartners of the scale factor a(t), being dynamical degrees of freedom obtained from the spatial part of the Rarita–Schwinger field. These superfield transformations are thus the generalization of the transformations for N (t) and a(t). • Finally, let us include the transformation law for the components N (t), ψ(t), ψ(t), and ϑ(t) of the superfield: i δN = ( f N )· + (εψ 0 + εψ0 ) , 2 ˇ + i bψ ˆ 0, δψ0 = ( f ψ0 )· + Dε 2

3

(8.15) (8.16)

In fact, the superfield N transforms as a 1D vector field under the local conformal time supersymmetric transformations.

8.1


167

ˇ − i bψ ˆ 0, δψ 0 = ( f ψ 0 )· + Dε 2 ˙ δϑ = ( f ϑ)· + bˆ ,

(8.17) (8.18)

where i Dˇ ε ≡ ε˙ + εϑ , 2

i Dˇ ε ≡ ε˙ − εϑ , 2

are the U(1) covariant derivatives and 1 (εψ 0 − εψ0 ) . bˆ = b − 2N In addition, i δa = f a˙ + (εψ + εψ) , 2

ε Da δψ = f ψ˙ + − iB − 2 N

ε Da + iB + δψ = f ψ˙ + 2 N

(8.19) iˆ bψ , 2

(8.20)

iˆ bψ , 2

(8.21)

1 δB = f B˙ + (ε D˜ ψ − εD˜ ψ) , 2N

(8.22)

where i Da = a˙ − (ψ0 ψ + ψ 0 ψ) , 2

ˇ − i Da + iB ψ0 , ˜Dψ = Dψ 2 N

ˇ − i Da − iB ψ 0 , D˜ ψ 0 = Dψ 2 N

(8.23) (8.24) (8.25)

are the supercovariant derivatives, and i Dˇ ψ ≡ ψ˙ + ϑψ , 2

i Dˇ ψ ≡ ψ˙ − ϑψ , 2

are the U(1) covariant derivatives. In order to get the component action, we use the expansion in Taylor series of the superfields N(t, η, η), A(t, η, η), with respect to η and η (see Sect. 3.3 of Vol. I). After integrating over the Grassmann variables η, η (see Chap. 3), the action (8.10)


168

becomes a component action with the auxiliary field B(t), satisfying an algebraic equation B∼−

√ 1 ψψ + k . 2 3a

(8.26)

Substituting it into the action, we then obtain [4, 8–11] S=

√ i ˙ N k 1 kN a i ˙ ˙ 2 1 a(a) ψψ + ψψϑ (8.27) + + ψ ψ + ψψ + − 2 N 2 2 2 2a 2 √ √ k√ i a a˙ √ 1 + a(ψ 0 ψ − ψ0 ψ) + a(ψ 0 ψ + ψ0 ψ) − ψ 0 ψ0 ψψ . 2 2N 4N

The conjugate momenta are πa = −

a i a˙ − √ (ψ0 ψ + ψ 0 ψ) , N 2 a

i π ψ ≡ πψ + ψ = 0 , 2 i π ψ ≡ πψ + ψ = 0 . 2

(8.28) (8.29) (8.30)

As often indicated in this book, these fermionic expressions constitute second class constraints and hence require one to use Dirac brackets (see Chaps. 4 and 5, and also Appendix B of Vol. I). The second-class constraints can be eliminated by the Dirac procedure, elimination of the momenta conjugate to the fermionic variables leaving us with the non-zero Dirac bracket relations. We also get first class constraints from the variation of the action (8.10) with respect to N , ψ0 , ψ 0 , and ϑ (which are Lagrange multipliers): H=0,

S=0,

S=0,

F =0,

(8.31)

where, in the gauge ψ0 = 0, N = 1 [admissible due to the invariance of the action under SUSY transformations (8.7) and (8.8)]: √ k ka πa2 ψψ , − + H≡− 2a 2 2a

√ √ πa S ≡ √ −i k a ψ , a

√ √ πa S = √ +i k a ψ , a

(8.33)

F = ψψ .

(8.35)

(8.32)

(8.34)

8.1


169

These constraints form a closed algebra under the Dirac brakets and follow from the invariance of the action under the local SUSY transformations used here. The total classical canonical Hamiltonian is i 1 i H = N H + ψ 0 S + ψ0 S + ϑF , 2 2 2

(8.36)

where H is the Hamiltonian of the system, S and S are SUSY constraints,4 and F is the U(1) (rotation) operator corresponding to a generator and the fermion number as well! Let us turn now to another rather interesting feature. Note 8.3 If we replace a(t) in the Hamiltonian constraint (8.32) by x(t) ≡ ˙ 2 /2 to x˙ 2 /2, and the 2a 3/2 (t)/3, then the kinetic term changes from a(a) Hamiltonian becomes H=−

1 πx2 − [W (x)]2 + W

ψψ , 2 2

(8.37)

where primes denote differentiation with respect to x and √ 34/3 kx 4/3 . W(x) ≡ 27/3

(8.38)

The signs of the first and second term in (8.37) arise as a consequence of the pure gravity field, while the third term is a consequence of SUSY and is positive-definite. But the relevant point with the superpotential W(x) is that we retrieve the known structure of N = 2 SQC [with the action (8.10) being a local generalization of those SUSY models], as illustrated in Chap. 6 of Vol. I. Constraints with the form (8.32), (8.33), (8.34), and (8.35) are present in Chap. 6 of Vol. I and related to hidden SUSY in cosmological models, but the algebra of these generators corresponds therein to global SUSY, although with local time t reparametrization. [We are constructing a ‘wider’ N = 2 SUSY with two fermionic charges, which are then gauged to get local SUSY by means of a U(1) symmetry (compare with N = 2 SUGRA).]

Bianchi Class A Models The superfield formulation above can be extended to the case of Bianchi class A models [11]. The action is written as

4 S will correspond to the single complex supersymmetric charge of the N = 2 SUSY quantum mechanics.


170

S=

1 2

1 G X Y (q Z )q˙ X q˙ Y + N U (q Z ) dt , N

(8.39)

being invariant under the reparametrization t −→ t + f (t) with [see (8.2) and (8.3)] δq X (t) = f (t)q˙ X ,

δN (t) = ( f N )· ,

(8.40)

(0)

and q X ≡ {α, β+ , β− }, G X Y ≡ diag(−1, 1, 1) fixed only up to an arbitrary confor˜ (0) ˜ mal factor, usually written as exp[2Ω(q)], i.e., G X Y = e2Ω(q) G X Y . Subsequently, X the fields q (t) and N (t) become superfields Q(t, η, η), N(t, η, η) with a corresponding supermetric G(Q), a function of the superfield Q(t, η, η). Here the usual superpotential W[q] ≡ W[a, φ] will be W[Q Z ]. The superfield generalization of the action (8.10) is then S=

1 2

N−1 G X Y (Q Z )Dη QY Dη QY + W(Q Z ) dηdηdt ,

(8.41)

where, for the FRW case, we have to replace G X Y by G00 = −A(t, η, η), with A related to the scale factor a(t),√and the potential (which is the well known superpotential!) W(Q Z ) translates to kA2 (t, η, η) [see (8.10)]. The superfield Q Z (t, η, η) has the form Q Z (t, η, η) ≡ q Z (t) + iηψ X (t) + iηψ X (t) + ηηB X (t) ,

(8.42)

where ψ X (t) ≡ N 1/2 ψ X (t) ,

ψ X (t) ≡ N 1/2 ψ(t) ,

and 1 B X (t) ≡ N B X − (ψ 0 ψ X − ψ0 ψ X ) . 2 In addition, and as expected, we will have U (q) =

1 X Y ∂W(q) ∂W(q) . G 2 ∂q X ∂q Y

(8.43)

By integrating over the Grassmann variables η, η, the action (8.41) acquires the auxiliary field B(t) with an equation that is algebraic (see Sect. 3.3): B X = % ZXY ψ Z ψ Y − G X Y Substituting this in, we then obtain

∂W . ∂q Y

(8.44)

8.1


S=

171

∂W ∂W 1 1 G X Y D˘ q X D˘ q Y − N G X Y X Y − iG X Y ψ X D˘ ψ Y 2N 2 ∂q ∂q 1 + N R X Y ZU ψ X ψ Y ψ Z ψ U − N ∇ X ∇Y Wψ X ψ Y 2 1 1 X X X X + ψ0 ψ ∂ W − ψ 0 ψ ∂ W dt , 2 2

(8.45)

where the minisuperspace derivatives are (in the present framework) i D˘ q X ≡ q˙ X − (ψ 0 ψ X + ψ0 ψ X ) , 2

(8.46)

i D˘ ψ X ≡ Dˇ ψ X + %YXZ q˙ Z λY = ψ˙ X − ϑψ + %YXZ q˙ Z ψ Y , 2

(8.47)

and Dˇ ψ X is a U(1) covariant derivative, R X Y ZU is the curvature tensor in N = 2 SUSY minisuperspace with metric G X Y , and %YXZ are the Christoffel symbols in the definition of the covariant derivatives ∇ X for the SUSY minisuperspace (see Chap. 6 of Vol. I and Sect. 3.3 herein). The canonical conjugate momenta to q X are given by πX =

GX Y Y D˘ q − iω X b c ψ b ψ c , N

(8.48)

where the minisuperspace spin connections ωY b c are functions of the q X defined by ωY b c = −eb X ec X ;Y = −ωY c b ,

(8.49)

and ec X ;Y denotes the Riemann-like covariant derivative of the minisuperspace vielbein fields. Furthermore, the following constraints are obtained H≡

1 1 1 X p p X − R X Y ZU ψ X ψ Y ψ Z ψ U + ∂ X W∂ X W + ∇ X ∇Y Wψ X ψ Y , (8.50) 2 2 2

S ≡ ( p X − i∂ X W) ψ X ,

(8.51)

S ≡ ( p X + i∂ X W) ψ X ,

(8.52)

F ≡ ψ X ψX ,

(8.53)

with p X ≡ π X + iωY b c ψ b ψ c .

(8.54)


172

8.1.2 Complex Scalar Fields In this section, we will describe a generic interacting action for this N = 2 conformal supersymmetric FRW model with a set of spatially homogeneous complex scalar matter superfields written here # [3, 10, 12]. (See the description in Sect. 3.2.3 of Vol. I for the Wess–Zumino model.) The framework presented here will be useful when discussing spontaneous symmetry breaking in the context of local N = 2 SUSY (see also [5, 8, 13]). There are then several steps to follow: 1. Construct a general superfield action including two superfunctions, i.e., the kinetic term T(# I , # I ) and the superpotential W(#). 2. Perform a Weyl rescaling of the superfields. 3. Include a recombination of the function T(# I , # I ) and W(#) in the Kähler superfunction G(# I , # I ). 4. Eliminate the auxiliary fields (with a possible analysis of spontaneous SUSY breaking). 5. Finally, retrieve the classical and quantum Hamiltonian and supercharges (SUSY constraints). Let us follow [14] with spatially homogeneous and complex5 scalar matter superfields # I (t, η, η) and # I (t, η, η) = (# I )∗ , consisting of a set of spatially homogeneous matter fields φ I (t) and φ I (t) (I = 1, 2, . . . , n), four fermionic degrees of freedom χ I (t), χ I (t), λ I (t), and λ I (t), and also two bosonic auxiliary fields F I (t) and F I (t). The components of the matter superfields # I (t, η, η) and # I (t, η, η) are # I (t, η, η) ≡ φ I (t) + iηχ I (t) + iηλ I (t) + ηηF I (t)

(8.55)

# I (t, η, η) ≡ φ I (t) + iηλ I (t) + iηχ I (t) + ηηF I (t) ,

(8.56)

and

where χ I (t) ≡ N 1/2 χ I (t),

(8.57)

λ I (t) ≡ N 1/2 λ I (t) ,

(8.58)

1 F I (t) ≡ N F I − (ψχ I − ψλ I ) , 2

(8.59)

1 F I (t) ≡ N F I − (ψλ I − ψχ I ) . 2

(8.60) i

The transformation rules for the superfields #i (t, η, η) and # (t, η, η) take the form 5

In [8], we have a setup for a real scalar field, two fermions, and one auxiliary field.

8.1


173

i i ˙ I + Dη LDη # I + Dη LDη # I , δ# I = L# 2 2 i ˙ I + D LD # I + i D LD # I , δ# I = L# η η η η 2 2

(8.61) (8.62)

which translate into the transformation law for the components of the matter superfields: i δφ I = a φ˙ I + (εχ I + ελ I ) , 2 i δφ I = a φ˙ I + (εχ I + ελ I ) , 2 I ˇ ε i D λ ˆ I+ δχ I = a χ˙ I + bχ − iF I , 2 2 N ˜φI i ε D I I I I ˆ + + iF , δχ = a χ˙ − bχ 2 2 N

iˆ I ε D˜ φ I I I I ˙ + iF δλ = a λ − bλ + , 2 2 N ε D˜ φ I iˆ I ˙ I I I − iF , δλ = a λ + bλ + 2 2 N 1 δF I = a F˙ I + 2N 1 δF I = a F˙ I + 2N

εD˜ λ I − εD˜ χ I εD˜ χ I − εD˜ φ I

(8.63) (8.64) (8.65)

(8.66)

(8.67)

(8.68)

,

(8.69)

,

(8.70)

where i D˜ φ I = φ˙ I − (ψλ I + ψχ I ) , 2 i D˜ φ I = φ˙ I − (ψλ I + ψχ I ) , 2 1 D˜ λ I = Dˇ λ I − (N −1 Dφ I + iF I )ψ , 2 1 D˜ λ I = Dˇ λ I − (N −1 D˜ φ I − iF I )ψ , 2 1 D˜ χ I = Dˇ χ I − (N −1 D˜ φ I − iF I )ψ 2 1 D˜ χ I = Dˇ χ I − (N −1 D˜ φ I + iF I )ψ , 2

(8.71) (8.72) (8.73) (8.74) (8.75) (8.76)


174

are the supercovariant derivatives, and i Dˇ λ I = λ˙ I + ϑλ I , 2 i Dˇ λ I = λ˙ I − ϑλ I , 2 i Dˇ χ I = χ˙ I − ϑχ I , 2 i Dˇ χ I = χ˙ I + ϑχ I , 2

(8.77) (8.78) (8.79) (8.80)

are the U(1) covariant derivatives. Therefore, in more detail, for step 1, we write the superfield action in the form √ 1 −1 2 Dη N A Dη A − Dη N−1 A2 Dη A T N−1 ADη ADη A − kA2 − 2 1 1 −1 3 ∂ 2 T + N−1 A3 T−1 Dη TDη T + N A Dη # I Dη # J + Dη # J Dη # I 4 2α ∂# I ∂# J 2A3 1 −1 3 −1 ∂T ∂T I J J I α − N A T Dη # Dη # + Dη # Dη # − 3 |W(#)| dηdηdt , 2α k ∂# I ∂# J (8.81)

S=

where k = 0, 1 stands for plane and spherical FRW, k2 = 8π G, and α is an arbitrary constant parameter (not fixed by the local conformal time SUSY). We can see from (8.81), that the interaction depends on two arbitrary superfunctions T(# I , # I ) and W(# I ). We then proceed to step 2 by implementing a Weyl conformal transformation [1–3, 5, 9, 10, 12, 14, 15]: αK N, N → exp 6

αK A → exp A, 6

αK T exp 3

=−

3 . k2

(8.82)

In addition (see step 3), we introduce the superfunction G(# I , # I ) as a special combination of the present superKähler potential K(# I , # I ) and of the spatially homogeneous superpotential W(#): G(#, #) ≡ K(#, #) + log |W(#)|2 .

(8.83)

The derivatives of the Kähler superfunction are denoted by (see Chap. 3) ∂G = G, I ≡ G I , ∂φ I

∂G ∂φ I

= G, I ≡ G I ,

(8.84)

8.1


175

and this Kähler supermetric is G I J = G J I , with G I J such that G I J G J K = δ KI . The superfield action (8.81) thus becomes S=

−

3√ 2 3 −1 N ADη ADη A + 2 kA2 − 3 A3 eα G/2 2 k k k

1 −1 3 I J J I + 2 N A G I J Dη # Dη # + Dη # Dη # dηdηdt . 2k

(8.85)

Note that this action is determined only by terms depending on one arbitrary Kähler superfunction, i.e., G. Moreover, the superfield action (8.85) is invariant under the N = 2 local conformal time supersymmetric transformations of (t, η, η), i.e., the transformations (8.7), (8.8), (8.12), (8.14), (8.61), and (8.62). We further redefine the fields by φ φ≡ √ , 3 a

χA χA ≡ √ 3 , a

λA λA ≡ √ 3 . a

After integrating over η and η, the (superfield) Lagrangian L contains terms with auxiliary fields B(t), F(t), and F(t), so L total = L˜ + L aux . The Lagrangian without auxiliary fields is √ √ ˜L = − 3 a(D˜ a)2 + 2i ψ D˜ ψ + k a (ψ 0 ψ − ψ0 ψ) 3 k N k2 √ a3 i 2N k ψψ + G Gφ I Gφ J + D˜ φ J (φG I J χ I + λG I J λ I ) + 3a 2k N k2 I J i i + D˜ φ I (λG I J λ J + ψG I J χ J ) − 2 G I J χ I D˜ χ J 2k k i N G λ I D˜ λ J − 2 3 G I J K L χ I χ J λ K λ L k2 I J k a 1 − √ 3 (ψ0 ψ − ψ 0 ψ)G I J (χ I χ J + λ J λ I ) 4k a N (λG I J K χ K − ψG I J K χ K )λ J λ I + 2ka 3 −

N N (ψG I J K λ K − ψG I J K λ K )χ I χ J + 3 G I J (χ I χ J + λ J λ I )ψψ 2ka 3 3a 4N α G/2 N 2N − ψψ − 2 3 (eα G/2 ),I J χ I λ J − 3 (eα G/2 ),I J λ I χ J e 3k k k 2N α G/2 2N α G/2 I I J J I − 3 (e ),I J (χ χ + λ λ ) − 2 ψ (e ) I λ + (eα G/2 ) I χ I k k +


176

√a 3 2N α G/2 I α G/2 I ψ (e ) χ + (e ) λ + 3 (eα G/2 ),I (ψ0 χ I − ψ 0 λ I ) I I k2 k √ 3 √ 3 a a + 3 (eα G/2 ),I (ψ0 λ I − ψ 0 χ I ) − 2 (ψ 0 ψ − ψ0 ψ)eα G/2 , (8.86) k k +

with the Lagrangian for the auxiliary fields being potential

kinetic + L aux L aux = L FRW aux + L aux

F,F ≡ LB aux + L aux ,

(8.87)

where N −1 L B aux

= −3a B + − 2

−

6 2 α G/2 a e k2

√ k 6 k 3 ψψ + a+ G (χ I χ J + λ J λ I ) 3a k 2ka I J ,

(8.88)

and 1 2a 3 1 G I J (ψχ I − ψλ I ) + 2 G I J K λ K χ I − 3 (eα G/2 ),J 2k k k 1 2a 3 1 +F I G J I (ψλ J − ψχ J ) + 2 G I J K χ K λ J − 3 (eα G/2 ),I 2k k k

F,F = FJ N −1 L aux

+

a3 G FI FJ . k2 I J

(8.89)

The equations for the auxiliary fields B, F I , and F I are algebraic, with solutions √ k a 1 k ψψ + G I J (χ I χ J + λ J λ I ) − 2 eα G/2 , + B∼− 2 2 k 18a 4ka k FL ∼ −

k 1 2 (ψλ L − ψχ L ) − 3 G L I G I J K χ K λ J + G L I (eα G/2 ),I , (8.90) 3 k 2a a

FL ∼ −

k 1 2 (ψχ L − ψλ L ) − 3 G L J G I J K λ K χ I + G L J (eα G/2 ),J , 3 k 2a a

where ( ),I and ( ),I are derivatives with respect to φ I and φ I . Step 4 involves substituting these into the component action, to retrieve the total supersymmetric Lagrangian

8.1


177

√ 3 a(D˜ a)2 2i N k N 3 ˜ L =− 2 ψψ − eα G/2 ψψ − N a U (a, φ, φ) + ψ Dψ + N 3 3a k k √ k√ a3 i I˜ J J I I ˜ ˜ + a ψ 0 ψ − ψ0 ψ + G φ + ψG χ + ψG λ D φ D D φ IJ IJ k 2k N k2 I J i i + D˜ φ I ψG I J λ J + ψG I J χ J − 2 G I J χ I D˘ χ J + λ I D˘ λ J 2k k N i − 2 3 RI J K L χ I χ J λK λL − √ ψ0 ψ − ψ 0 ψ G I J χ I χ J + λ J λ I k a 4k a 3 2 3√k 3N I J J I I J J I G + χ χ + λ λ + χ χ + λ λ G 16k2 a 3 I J 2k2 a I J 3N − 3 eα G/2 G I J χ I χ J + λ J λ I 2k 2 2N α G/2 ),I J χ I λ J − 3 N (eα G/2 ),I J λ I χ J − 3 (e k k N 2N α G/2 − 3 (e ),I J χ I χ J + λ J λ I − 2 ψ (eα G/2 ),I λ I + (eα G/2 ),I χ I k k √ N a3 α G/2 I α G/2 I + 2 ψ (e ),I χ + (e ),I λ − 2 ψ 0 ψ − ψ0 ψ eα G/2 k k √ √ 3 3 a a + 3 (eα G/2 ),I ψ0 χ I − ψ 0 λ I + 3 (eα G/2 ),I ψ0 λ I − ψ 0 χ I , (8.91) k k where D˜ a, D˜ φ I , D˜ χ J , D˜ λ J , and D˜ ψ are defined as before, with ˘ J = D˜ χ J + % J φ˙ K χ L , Dχ KL

(8.92)

˘ = D˜ λ Dλ

(8.93)

J

J

+ % KJ L φ˙ K λ L

,

and R I J K L is the curvature tensor of the Kähler manifold defined by the coordinates φ I , φ J with the metric G I J , and % KJ L = G J I % I K L are the Christoffel symbols in the definition of the covariant derivatives and their complex conjugates. In the Lagrangian (8.91), the potential term is √ 3k 6 k α G/2 + ϑeff (φ, φ), U (a, φ, φ) = − 2 2 + 3 e k a k a

(8.94)

where the effective potential of the scalar matter fields is ϑeff

4 3 αG eα G α G/2 I L α G/2 ≡ 4 (e ) I G (e )L − e = 4 α2G A G A − 3 . 4 k k

(8.95)

Finally, in step 5, in order to obtain the corresponding Hamiltonian, we write down the momenta conjugate to a(t), φ I (t), and φ I (t):


178

6 a D˜ a , 2 Nk a3 i π I (φ) = G J I D˜ φ J + λG J I χ J + ψG J I λ J 2 2k Nk i J − 2 G M J % IJL χ M χ L + % AL λM λL 2k i ≡ p I (φ) − 2 G M J % IJL χ M χ L + % IJL λ M λ L , 2k 3 a i J J ˜φJ + π I (φ) = G ψG λ + ψG χ D I J I J I J 2k N k2 i + 2 G M J % IML χ L χ J + % IML λ L λ J 2k i ≡ p I (φ) + 2 G M J % IML χ L χ J + % IML λ L λ J , 2k πa = −

(8.96)

(8.97)

(8.98)

where p I (φ) and p I (φ) are the (covariant) momenta. With respect to the canonical Poisson brackets, we have [a, πa ] = −1 , φ J , π I = −δ IJ , φ J , π I = −δ IJ . (8.99) The dynamical Grassmann variables ψ(t), χ (t), and λ(t) lead to second class constraints i (8.100) π ψ ≡ πψ + ψ = 0 , 3 i π ψ ≡ πψ + λ = 0 , (8.101) 3 i π I (χ ) ≡ π I (χ ) − 2 G J I χ J = 0 , (8.102) 2k i (8.103) π I (χ ) ≡ π I (χ ) − 2 G I J χ J = 0 , 2k i (8.104) π I (λ) ≡ π I (λ) − 2 G J I λ J = 0 , 2k i (8.105) π I (λ) ≡ π I (λ) − 2 G I J χ J = 0 , 2k where πψ =

dL , dψ˙

π I (χ ) =

dL , dχ˙ I

π I (λ) =

dL , dλ˙ I

are the momenta conjugate to the anticommuting variables ψ(t), χ (t), and λ(t), respectively. They can be eliminated (see Exercise 8.1), and from the non-zero Dirac brackets, obtained as

8.1


179

[a, πa ]D = [a, πa ] = −1 , φ I , πφJ = φ I , πφJ = −δ IJ ,

(8.106)

D

φ I , πφJ

D

= φ I , πφJ = −δ IJ ,

(8.107) (8.108)

χI,χJ = −ik2 G I J ,

(8.109)

λI , λ J = −ik2 G I J ,

(8.110)

D

D

ψ, ψ

D

=

3 i, 2

(8.111)

and in addition, i J M L J M L G χ χ + % λ φ % , I L I L M J 2k2 i p I (φ) = π I − 2 G M J % IML χ L χ J + % IML φ L φ J , 2k p I (φ) = π I +

(8.112)

for which the following non-zero Dirac brackets are found:

pI , χ K

pI , p J

p I , χK pI , λK p I , λK

D

D

D

D

D

=−

i K L K L R χ + λ λ χ , I J K L k2

=

1 G %J χL , 2 KJ IL

(8.113)

=

1 G I J % IJL χ L , 2

(8.114)

=

1 G % J λL , 2 KJ IL

(8.115)

=

1 G % J λL . 2 KJ IL

(8.116)

Hence, the canonical Hamiltonian is the sum of all the first class constraints: H = NH + i

ψ0 ϑ ψ0 S +i S + F , 2 2 2

where H is the classical Hamiltonian constraint of the system given by

(8.117)


180

√ k k2 2 k2 ik 3 H=− ψψ + 3 p I G I J p J − 3 (ψχ I + ψλ I ) p I πa + a U (a, φ, φ) − 12a 3a a 2a ik 1 1 (ψλ I + ψχ I ) p I + 2 3 R I J K L χ I χ J λ K λ J + eα G/2 ψψ k 2a 3 k a 2 3 1 I J J I χ χ + λ λ G + 3 ψψG I J χ I χ J + λ J λ I − I J 4a 16k2 a 3 √ 3 k 3 − 2 G I J χ I χ J + λ J λ I + 3 eα G/2 G I J χ I χ J + λ J λ I 2k a 2k 2 α G/2 2 α G/2 I J ),I J χ λ + 3 e λJ χ I + 3 (e ,I J k k 2 + 3 eα G/2 χ I χ J + λB λI ,I J k 1 α G/2 I α G/2 I λ + e χ + 2ψ e ,I ,I k 1 α G/2 I α G/2 I − 2 ψ (e ),I χ + e λ , ,I k −

(8.118)

and S and S are the classical SUSY generators for this model: 2i √ √ 2i * 3 α G/2 i k I J J I k a− 2 a e + √ GI J χ χ + λ λ ψ S = √ πa + k k 3 a 2k a 3 2i * 1 2i * 1 χK, λ K + √ p K − 3 a 3 eα G/2 + √ p K − 3 a 3 eα G/2 ,K ,K k k a3 a3 (8.119)

S=

k 2i √ √ 2i * i k a + 2 a 3 eα G/2 − √ G I J χ I χ J + λ J λ I ψ √ πa − k k 3 a 2k a 3 2i * 3 α G/2 1 2i * 3 α G/2 1 K λ + √ pK + 3 a e χK, + √ pK + 3 a e ,K ,K k k a3 a3 (8.120)

while F is the classical U(1) rotation generator G 2 F = − ψψ + I2J χ I χ J + λ J λ I . 3 k

(8.121)

8.2

Quantum FRW Minisuperspace

181

Note 8.4 It is of interest to note the following feature of the action (8.81). The part for the scalar complex supermatter can be related to the part obtained by spatial reduction from the Wess–Zumino model in four dimensions (see Chap. 3), with an arbitrary superpotential W(Z ). Furthermore, it gives two complex supercharges S1 and S2 . Since it is invariant under the change # I ←→ # I , the supercharges then allow invariance under the change Q 1 ←→ Q 2 , hence combining them into a single complex supercharge S = S1 + S 2 and S = S 1 + S2 . A closed superalgebra for H, F and S, S is then obtained from the Dirac brackets (8.109) and (8.114):

S, S

= −2iH ,

[S, H]D = S, H D = 0 ,

(8.122)

[F, S]D = iS ,

F, S D = −iS,

(8.124) (8.125)

[F, S]D = 0 ,

(8.126)

[S, S]D = 0 ,

S, S D = 0 .

(8.127)

D

(8.123)

(8.128)

8.2 Quantum FRW Minisuperspace Having introduced the Hamiltonian description, we can now consider quantization of the FRW model (simplest case) [1, 2, 5]. In the procedure often used here, the Dirac brackets (8.109) are replaced by the following brackets: [a, πa ] = −i , φ I , π J = −iδ JI , φ I , π J = −iδ JI , : 9 3 ψ, ψ = − , 2 . I 2 I χ , χ J = k δJ , . λ I , λ J = k2 δ JI , where ψ, χ J , and λ J are Hermitian conjugates to ψ, χ J , and λ J .

(8.129) (8.130) (8.131) (8.132) (8.133) (8.134)


182

The first class constraints (8.118), (8.119), (8.120), and (8.121) associated with the invariance of the action (8.91) now become operator constraints on the wave function Ψ (a, φ, ψ, χ, λ). Any physically allowed states must obey HΨ = 0 ,

SΨ = 0 ,

SΨ = 0 ,

FΨ = 0 ,

(8.135)

with [see (8.122), (8.123), (8.124), (8.125), (8.126), (8.127), and (8.128)] 9

: S, S = 2H ,

[S, H] = S, H = 0 ,

F, S = −S ,

[F, H] = 0 ,

[F, S] = −S , 2

S2 = S = 0 ,

(8.136)

where H is the Hamiltonian, S is a single complex supersymmetric operator, and F is the fermion number operator. Within this canonical framework, the ‘even’ (i.e., bosonic) canonical variables are replaced by operators a −→ a ,

πa = i

∂ , ∂a

φ I −→ φ I ,

πI = i

∂ . ∂φ I

(8.137)

Regarding the ‘odd’ (fermionic) variables ψ, ψ, χ I , χ I , λ I , and λ I , we have two options: • We can implement a Fock space representation with ψ, χ I , λ I as creation and ψ, χ I , λ I as annihilation operators on |0, the Fock space vacuum, so that ψ|0 = χ I |0 = λ I |0 = 0, and the general quantum states can be written as vectors depending on a, φ I , and φ I in the corresponding Fock space. • We can write ψ, ψ, χ I , χ I , λ I , and λ I in the form of the direct product of (1+2n) 2 × 2 matrices (see Chap. 7). For the simple case I = n = 1, the fermionic variables ψ, ψ, χ , χ , λ, and λ are ψ = −σ − ⊗ 1 ⊗ 1 ,

ψ = σ+ ⊗ 1 ⊗ 1 ,

(8.138)

χ = σ3 ⊗ σ− ⊗ 1 ,

χ = σ3 ⊗ σ+ ⊗ 1 ,

(8.139)

λ = σ3 ⊗ σ3 ⊗ σ+ ,

λ = σ3 ⊗ σ3 ⊗ σ− ,

(8.140)

where σ ± = (σ1 ± iσ2 )/2 and σ1 , σ2 , and σ3 are the Pauli matrices. For the more general case (n complex matter supermultiplets), we then have the following matrix realization

8.2


7 ψ= 7 ψ = †

183

3 (−) ⊗ 12 ⊗ . . . ⊗ 12n+1 , σ 2 1

(8.141)

3 (+) ⊗ 12 ⊗ . . . ⊗ 12n+1 , σ 2 1

(8.142)

(3)

(3)

(−)

(3)

(3)

(+)

λi = kσ1 ⊗ . . . ⊗ σ2i−1 ⊗ σ2i ⊗ 12i+1 ⊗ . . . ⊗ 12n+1 ,

(8.143)

λi = kσ1 ⊗ . . . ⊗ σ2i−1 ⊗ σ2i ⊗ 12i+1 ⊗ . . . ⊗ 12n+1 ,

(8.144)

(−) ⊗ 12i+2 ⊗ . . . ⊗ 12n+1 , χ i = kσ1(3) ⊗ . . . ⊗ σ2i(3) ⊗ σ2i+1

(8.145)

(3)

(3)

(+)

χ i = kσ1 ⊗ . . . ⊗ σ2i ⊗ σ2i+1 ⊗ 12i+2 ⊗ . . . ⊗ 12n+1 .

(8.146)

In the matrix realization, the operators ψ, χ I , and λ I on the wave function Ψ = Ψ (a, φ I , φ I , ψ, χ , λ) produce 22n+1 component columns Ψi (a, φ I , φ I ), where i = 1, . . . , 22n+1 . Note 8.5 Concerning the operator order ambiguity [1, 12, 14], the (super) algebra (8.136) does not provide a positive-definite expression for the Hamiltonian. In fact, in this case the operator expression k2 −1/2 πa a −1/2 πa ) (a 12 corresponding to the energy of the scale factor a brings a negative contribution into the Hamiltonian, while the energy of the scalar fields φ I is positive. From the classical Hamiltonian (8.118), the energy of the scale factor is negative [due to the fact that the anticommutator for ψ, ψ, which are superpartners of the scale factor a (while9other :particle-like fluctuations do not associate with the scale factor), with ψ, ψ = −3/2, is negative], in contrast to the anticommutation relations for χ I , χ J , and λ I , λ J which are positive. In addition, the potential Veff of the scalar fields is not positive-definite, in contrast with standard supersymmetric quantum mechanics. This is a characteristic peculiarity of extending SQM to a minisuperspace approach (for a SUGRA framework coupled to a Wess–Zumino type model).

8.2.1 Supersymmetry Breaking In this section we discuss local supersymmetry breaking in the context of N = 2 (conformal) SUSY. The construction of the spontaneous SUSY breaking mechanism is related to the existence of vacuum states in SUGRA (see Sect. 3.2.2).


184

Expressing the effective scalar field potential in the local conformal supersymmetry [see (8.91), (8.92), (8.93), (8.94), and (8.95)] for the k = 0 case, we can put generally ϑeff ≡ ϑF + ϑD , where ϑF ≡

eα G 2 α GI GI L GL − 3 , 4 k

(8.147)

and the contributions of the D-terms to the effective potential ϑeff (see Sect. 3.2) can in some cases be chosen so as to ignore the minimum of the ϑD term. The scalar field potential (8.147) depends on the Kähler function G(φ I , φ I ) = K(φ I , φ I ) + log |W(φ I )|2 [see (8.83)]. With a view to an application (see Sect. 8.2.2), let us recall certain elements from Chap. 3 together with some new features. To be more precise, in contrast with global SUSY, the F-term part of the effective scalar potential in (8.147) is not positive semi-definite in general. Unlike global supersymmetry, it permits spontaneous SUSY breaking with vanishing classical vacuum energy (see the superHiggs effect in Sect. 3.2.2). To display some of the features of spontaneous SUSY breaking, the potential (8.147) in the k = 0 case can be presented in terms of the auxiliary fields f I of the matter supermultiplets: ϑF =

1 3 I I f I f I − 4 eα G(φ ,φ ) , k2 k

(8.148)

where the f I can be expressed as [see (8.95)] fI =

α α G(φ I ,φ I )/2 J K GI φ , φ e . k

(8.149)

Local SUSY is spontaneously broken if the auxiliary fields (8.149) of the matter supermultiplets get non-vanishing vacuum expectation values. Accordingly, at its minimum, the potential (8.148) is ϑF (φ I , φ J ) = 0, but f I = 0, so SUSY is broken. Following [5, 8, 13], we can use the fields φ I ≡ (ϕ, φ I ), where ϕ stands here for the dilaton field, and φ I for the spatially homogeneous chiral fields, thereby extending the scope of the discussion. Hence, at the classical level, the conditions √ for spontaneous SUSY breaking with vanishing vacuum energy become (α ≡ 3) ∂ϑF (ϕ0 , ϕ 0I , φ0I , φ 0I ) = 0 , ∂φ I

ϑF (ϕ0 , ϕ 0 , φ0I , φ 0I ) = 0 ,

FI = 0 , (8.150)

where ϕ0 , φ0I are the absolute minima. Moreover, we may note the following: • The first condition implies the existence of a minimum. • The second condition implies a vanishing ‘cosmological constant’. • The non-vanishing f -term for the auxiliary field f I implies spontaneous SUSY breaking. As a simple application, writing the corresponding Kähler function now as, e.g.,

8.2


G(φ I ) =

185

3 2 k k2 I J φ φ + log W(φ I ) , 2 2

(8.151)

and substituting (8.151) into (8.147), the effective potential becomes ϑF (φ I ) =

eα G 2 φI φI G G G − 3 . α I φ φI k4

(8.152)

In order to make this description clearer, for the potential presented in (8.91), (8.92), (8.93), (8.94), and (8.95), let us write it in terms of the auxiliary fields as U (a, φ, φ) =

f I GI J f J 3B 2 − , k2 a2

(8.153)

with fI =

α α G(φφ)/2 I G (φ, φ) , e k

√ k a − 2 eα G(φ,φ)/2 . B= k k

(8.154)

(8.155)

SUSY is spontaneously broken if the auxiliary fields (8.154) of the matter supermultiplets get nonvanishing vacuum expectation values. The potential (8.153) then comprises two terms: • The first is the potential for the scalar fields in the case of global supersymmetry, which is unbroken when the energy is zero (k = 0 and ϑeff = U ) due to the fact that f I = 0. • However, this (super)potential is not positive semi-definite, in contrast with the case of standard supersymmetric quantum mechanics, here allowing SUSY breaking with vanishing (classical) vacuum energy.

8.2.2 Towards an Inflationary Scenario SUSY may play an important role in an inflationary setting. Many models have been proposed describing the inflationary phase transition in both globally and locally SUSY theories [19, 20]. These models have been analyzed with and without supergravity corrections. SUGRA corrections affect the flatness of the inflaton potential because SUSY is broken during inflation. In this context, SQC models with local (conformal) time supersymmetry have been considered [2], corresponding to specific SQM models with spontaneous breaking of supersymmetry when the vacuum energy is zero, in order to find some simple (WKB) solutions to the Wheeler–DeWitt equation.


186

The framework is essentially that of Sect. 8.2.1 and extends from the solution to Exercise 8.3 to the case where there are more scalar fields. Basically, the extended superfield action for a homogeneous scalar supermultiplet interacting with the scale factor in the supersymmetric N = 2 FRW model now has the form [2] √ k 2 1 A3 1 A Dη # I Dη # I − 2A3 W(# I ) dηdηdt , S= 6 − 2 Dη ADη A + 2 A + 2N 2k N 2k (8.156) where I = 1, 2, 3 and for real scalar matter superfields # I = # I † . Using the global minimum in the potential ϑ(φ0 , ) = 0, the eigenstates of the corresponding Hamiltonian have sixteen components in the matrix representation (see Chap. 7). 16 can have the right behaviour when R → ∞, with

1 2 − πˆ + c1 a Ψ16 = 0 , 12Mp2 a

k=0,

(8.157)

1 2 3 − πˆ + c2 a Ψ16 = 0 , 12Mp2 a

k=1.

(8.158)

and

The semiclassical solutions of these equations become [2] √ * Ψ16 ∼ exp −i2 2 d1 a 3/2 ,

Ψ16 ∼ exp

−12i * 5/2 d2 a 5

k=0,

(8.159)

k=1.

(8.160)

,

The behavior of the scale factor corresponding to the situations of a flat and a closed universe are, respectively, a ∼ t 2/3 ,

a ∼ t2 ,

k=0,

k=1.

(8.161)

(8.162)

Then a flat universe with a scale factor can be obtained, such as a dust-dominated universe [2]. The last case corresponds to a scenario with power law inflation.

References

187

Summary and Review. As the main technical content of the book reaches completion, the reader may review the essential features of the approach in this chapter: 1. How do the ingredients introduced here extend those of Chap. 3 in either volume and Chap. 6 of Vol. I [Sect. 8.1.1]? 2. Is (super)matter more suitably treated (see Chap. 3 of Vol. I) and more closely related to SUGRA [Sect. 8.1.2]? 3. How is (spontaneous) SUSY breaking discussed [Sect. 8.2.1]? 4. How does it absorb some elements of the matrix representation approach (see Chap. 7) [Sect. 8.2.2]?

Problems 8.1 Invariance of the Action Under N=2 Conformal SUSY Discuss the invariance of the actions (8.10), (8.41), and (8.85) under the N = 2 conformal supersymmetric transformations associated with (8.7), (8.8), (8.12), (8.14), (8.61), and (8.62). 8.2 Dirac Brackets in N=2 Conformal SUSY From the second class constraints (8.102) and (8.103), determine the Dirac brackets (8.109). 8.3 FRW Wave Function in N=2 Conformal SUSY Find a normalized wave function for the supersymmetric quantum FRW cosmological model. The component of the (scalar) matter superfield satisfies (#+ = #) with # = φ(t) + iηχ (t) + iηχ (t) + F (t)ηη [see (8.55) and (8.56)].

References 1. Aceves de la Cruz, F., Rosales, J.J., Tkach, V.I., Torres, J.A.: SUSY cosmological models. Grav. Cosmol. 8, 101–106 (2002) 163, 164, 166, 174, 181, 183 2. Guzman, W., Socorro, J., Tkach, V.I., Torres, J.: Inflation from SUSY quantum cosmology. Phys. Rev. D 69, 043506 (2004) 163, 174, 181, 185, 186 3. Obregon, O., Rosales, J.J., Socorro, J., Tkach, V.I.: The wave function of the universe and spontaneous breaking of supersymmetry. hep-th/9812156 (1998) 163, 172, 174 4. Obregon, O., Rosales, J.J., Tkach, V.I.: Superfield description of the FRW universe. Phys. Rev. D 53, 1750–1753 (1996) 163, 164, 165, 166, 168 5. Obregon, O., Socorro, J., Tkach, V.I., Rosales, J.J.: Supersymmetry breaking and a normalizable wavefunction for the FRW (k = 0) cosmological model. Class. Quant. Grav. 16, 2861–2870 (1999) 163, 165, 172, 174, 181, 184 6. Rosales, J.J., Tkach, V.I., Pashnev, A.I.: On the Schroedinger equation for the supersymmetric FRW model. Phys. Lett. A 286, 15–24 (2001) 163

188


7. Rosales, J.J., Tkach, V.I., Torres, J.: Extended supersymmetry for the Bianchi-type cosmological models. Mod. Phys. Lett. A 17, 2199–2207 (2002) 163, 164 8. Tkach, V.I., Obregon, O., Rosales, J.J.: FRW model and spontaneous breaking of supersymmetry. Class. Quant. Grav. 14, 339–350 (1997) 163, 164, 168, 172, 184 9. Tkach, V.I., Rosales, J.J.: Supersymmetric action for FRW model with complex matter field. gr-qc/9705062 (1997) 163, 164, 168, 174 10. Tkach, V.I., Rosales, J.J., Martinez, J.: Action for the FRW model and complex matter field with local supersymmetry. Class. Quant. Grav. 15, 3755–3762 (1998) 163, 164, 168, 172, 174 11. Tkach, V.I., Rosales, J.J., Obregon, O.: Supersymmetric action for Bianchi type models. Class. Quant. Grav. 13, 2349–2356 (1996) 163, 164, 168, 169 12. Tkach, V.I., Rosales, J.J., Socorro, J.: General scalar interaction in the supersymmetric FRW model. hep-th/9807058 (1998) 163, 172, 174, 183 13. Tkach, V.I., Rosales, J.J., Socorro, J.: Spontaneous breaking of supersymmetry in cosmological models and supergravity theories. Mod. Phys. Lett. A 14, 1209–1216 (1999) 163, 172, 184 14. Tkach, V.I., Rosales, J.J., Socorro, J.: Supersymmetric FRW model and the ground state of supergravity. Class. Quant. Grav. 16, 797–812 (1999) 163, 172, 174, 183 15. Tkach, V.I., Rosales, J.J., Torres, J.: On the relation of the gravitino mass and the GUT parameters. Mod. Phys. Lett. A 14, 169–176 (1999) 163, 174 16. Tkach, V.I., Socorro, J., Rosales, J.J., Nieto, J.A.: Dual symmetry and the vacuum energy. Phys. Rev. D 60, 067503 (1999) 163 17. Kiefer, C.: Quantum Gravity . International Series of Monographs on Physics, vol. 136, 2nd edn., pp. 1–308. Clarendon Press, Oxford (2007) 163 18. Galvao, C.A.P., Teitelboim, C.: Classical supersymmetric particles. J. Math. Phys. 21, 1863 (1980) 164 19. Liddle, A.R., Lyth, D.H.: Cosmological Inflation and Large-Scale Structure, 400pp. Cambridge University Press, Cambridge (2000) 185 20. Mukhanov, V.: Physical Foundations of Cosmology, 421pp. Cambridge University Press, Cambridge (2005) 185

Chapter 9

More Obstacles and Results: From QC to SQC

Throughout this volume, we have either presented new frameworks within which SQC can be investigated, or new results, some extending from the settings in Vol. I, that lead to interesting but as yet untackled problems. In this sense, these routes to SQC were also initiated by the objectives and subsequent results from purely bosonic quantum cosmology. One remaining issue is whether or not these different approaches might all be equivalent. As we near the end of this volume, the reader will hopefully agree that SQC constitutes a vaster, more adequate, and more elegant perspective for the very early universe. With this in mind, we have in this volume discussed further essential elements from either QC or SUSY/SUGRA, with a view to obtaining a more realistic SQC.

9.1 Quantum Cosmology Additional and perhaps more observational features within purely bosonic quantum cosmology (QC) were discussed in Chap. 2 (for more thorough presentations, see [1–3]). In more detail: 1. Assuming we achieve a satisfactory framework for both the dynamical and the initial conditions, what type of predictions can be made? Would they be testable and if so how? In particular, can we establish that the most probable (classical) emergent spacetime will have a satisfactory inflationary phase, with suitable density fluctuations and gravitational waves?1 At a more fundamental level, can we thereby determine the probability distribution for the values of the constants of nature, depending on the different choices of the vacuum state (possibly related to different compactifications in string theory)? 2. We must face the fact that the cosmological wave function will constitute a superposition of states associated with all possible values for those constants previously mentioned. But then the usual Copenhagen interpretation is no longer

1 This means the necessary initial conditions for inflation must be implied by the boundary condition for the wave function of the universe.

Moniz, P.V.: More Obstacles and Results: From QC to SQC. Lect. Notes Phys. 804, 191–196 (2010) c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-11570-7_9

192

9 More Obstacles and Results: From QC to SQC

applicable [4–9, 1–3, 10, 11]. The ‘observer’ is now an integral part of the full quantum system. A promising alternative is the so-called many worlds interpretation [12], which seems particularly well suited for use in quantum cosmology. In particular, it has been claimed recently [13, 14] that quantum cosmology could actually lead to a physical test, i.e., empirical distinctions between the Copenhagen and many worlds interpretations. However, it has also been claimed [9, 1] that a generalization of quantum mechanics will be necessary for quantum gravity (standard quantum mechanics being based on a fixed background spacetime geometry, to specify the notion of time that characterizes the whole formalism), where spacetime geometry is not fixed. In more technical terms, the quantum cosmological version of the many worlds interpretation says that a definite prediction can be made if the wave function has one clear peak. But it is not so easy to find a clear peak in the solutions of the Wheeler–DeWitt equation (even using Wigner functions or semi-classical functions) as many others may occur. Decoherence effects are paramount in yielding a single significant peak for the Wigner function [6, 10, 11]. But then, what should ‘significant’ mean, i.e., what height must it have, and relative to what? And in decoherence, one traces out ‘small’ (less relevant) degrees of freedom. Is this definition clear and fully objective? 3. So the question arises [6, 9, 1, 10, 11] as to how effective quantum cosmological decoherence really is. How does a reliable classical cosmological spacetime emerge from the quantum description? Did the initial state have a low entropy initial state (evolving towards greater complexity, with a clear time asymmetry)? The above are still open questions, in spite of the wide range of contributing authors. But more importantly, this provides the opportunity for developments in SQC to assist in such a quest. To this end, in Part IV of this volume, we discuss what SQC could contribute to this debate (see the final paragraph of Sect. 9.2 and Sect. 9.3).

9.2 Supergravity Specifically for this volume, and with a view towards an ‘observational’ perspective in SQC, we have focused here on a few standard issues of SUSY breaking. Much more can be said on this subject than was presented in Chap. 3. This is an area where a great deal remains to be investigated within SQC. For practical reasons, it proves worthwhile to consider simple settings in SQM, and this was why it was brought in at this stage. And as in many related areas of SQC, only a small fraction has been explored within QC, even as regards applications of Nicolai maps. Although it constitutes a far more demanding route, another issue remains, concerning SUGRA: the problem is to obtain quantum physical solutions of the general theory of supergravity (dealing with infinite degrees of freedom [15, 16]), without retreating to simple truncated (minisuperspace) models. Following the Dirac procedure [17] (in the canonical representation), such states have to satisfy the Hamiltonian and the diffeomorphism constraints, H⊥ and Hi ,

9.3

Supersymmetric Quantum Cosmology

193

respectively, associated with general coordinate transformations, together with the supersymmetry constraints, S A , corresponding to supersymmetry transformations, and also the Lorentz constraints, J AB , for the Lorentz transformations [18, 4, 19]. Initially, there were claims about the existence of states with finite fermionic number, which were opposed by significant objections. In fact, it was shown in [15] that quantum states could only have an infinite fermionic number. A particular solution was presented in [16] and shown to correspond to a wormhole (Hawking– Page) solution in a minisuperspace sector [20]. In view of these results, it was imperative to attempt to establish whether there was a connection between such frameworks and solutions for the observed universe; in other words, to determine a semiclassical setting from canonical quantum gravity that would be applicable in SQC, and possibly provide an ‘observational’ context. In [21], the corresponding Hamilton–Jacobi Schrödinger and SUSY quantum gravity corrections to the Schrödinger equation were found, proposing a useful tool , for rewriting most expressions, namely, the DeWitt supermetric of the superRiem ( ) space of all tetrads and gravitinos on a spatial hypersurface Σ. Interestingly, from a different line of exploration, it was found that a wave function depending only on the tetrad could not be allowed, and spacetime would emerge with Grassmannian-like structures and possible ‘observational’ ranges.

9.3 Supersymmetric Quantum Cosmology In this volume, we have given a broader description of SQC. We have found additional sets of solutions, some still corresponding to what is present in ‘bosonic’ quantum cosmology (which is both desirable and unsurprising). Supersymmetry may constitute a tool for selecting the primordial solution on more fundamental grounds. Moreover, the issues of symmetry breaking [22] and a semiclassical limit for the resulting quantum states [21, 23] have brought some realism within reach of SQC as regards the real classical universe (more in Chap. 10).

9.3.1 FRW Models As in Vol. I, FRW models have provided the simplest setting in which SQC can be probed [24–26] (see also [27, 22]), specifically concerning less straightforward (super)matter contents here. In Chap. 5, new non-trivial solutions led to a discussion of SUSY breaking within SQC in the context of N = 1 SUGRA, with a suitable ansatz for the metric and gravitino variables, including (super)matter via a scalar supermultiplet, in the presence of a generic superpotential, with an expansion in powers of a parameter λ 1 [22]. Interestingly and perhaps somewhat disturbingly, when both a scalar and a vector supermultiplet are included, the only allowed state was found to be the trivial one (i.e., Ψ = 0) [28].

194


But SUSY FRW models have also been investigated along other lines. The Ashtekar–Jacobson formulation was briefly discussed in Chap. 6, where a short comment was made on how the matrix representation for fermionic momenta [29] can add information. In Chap. 8, Bianchi and FRW models were brought into a wider SUSY framework [30–35].

9.3.2 Spatial Anisotropy Bianchi models, with anisotropic gravitational degrees of freedom and more gravitino modes (see [36–38, 4, 39, 40] and in particular [41]) allow one to probe deeply into these new SQC frameworks, and especially the matrix representation [42, 29, 43–46]. The cases of Bianchi models with scalar and vector (super)matter were discussed and some components of the solutions found, in particular in the context of N = 1 SUGRA and N = 4 SUSY minisuperspaces. However, a considerable amount of research has also been carried out using the matrix approach (see Chap. 6), with particular emphasis on comparison with the Taub cosmology (special case of the Bianchi IX model) with and without SUSY, and also issues relating to proper application of the Lorentz constraints and the possible consequences of it [47, 48]. And, of course, a significant set of results and features was obtained using the approach of Chap. 8, assigning a wider (‘hidden’) N = 2 SUSY to several models, with particular interest in those extracted from (super)string and related theories [49–51]. Last but not least, loop quantum gravity and cosmology allowed an alternative quantization picture of gravity to be extended into SUGRA. Some of the arguments and results are rather interesting: an FRW model has instead a Chern–Simons form and the gravitino induces different types of ‘initial’ singularity avoidance. Table 9.1 reviews Vol. I and summarizes the type of solutions found so far using the canonical quantization of N = 1 and N = 2 supergravity. In order to go beyond the scope of SQC, i.e., the restricted minisuperspace setting, we can either include fluctuations of the fields around a background configuration (see Chap. 10) or attempt to further discuss the full theory, employing some insight from SQC into the Bianchi models (see Chap. 4). Table 9.1 Solutions for three cosmological models using different supergravity theories. HH noboundary Hartle–Hawking solution, WH wormhole Hawking–Page solution, CS Chern–Simons solution k = +1 FRW Bianchi class A Full theory Pure N = 1 N = 1 with Λ N =2 N = 1 with scalar fields N = 1 with vector fields N = 1 with general matter

HH, WH HH — Not quite HH or WH HH, WH Ψ =0

HH, WH CS & −→ WH, HH CS Neither WH nor HH Unknown Unknown

WH Unknown CS Unknown Unknown Unknown

References

195

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Chapter 10

More Routes Beyond the Borders

Hopefully the contents of the preceding chapters will have convinced the reader that SQC is a fascinating topic, in which many problems remain to be tackled. We have already mentioned some of these as we went along, and will now add considerably to the list. But let us also illustrate one avenue of exploration in which SQC is by no means a closed book. In Sect. 10.1, we go beyond the confines of minisuperspace into a SUSY midisuperspace, where we find a method for glimpsing structure formation and definition of the vacuum, although much remains open for investigation. In fact, we will be able to provide some of the detail, but there is much room for improvement, extension, and exploration of new areas [1, 2]. In Sect. 10.2, we present a longer list of possible lines of investigation, although only describing each one briefly, where methods, results, and implications are yet to be revealed.

10.1 Beyond Minisuperspace The objective of this section is more than just the mathematical exercise of extending the framework in Chap. 5, and in particular in Sect. 5.1.4 of Vol. I, with additional expansion modes, following the contents of Sect. 2.2.1 in this volume. Indeed, it may allow us to establish whether the inclusion of supersymmetry in a quantum cosmological scenario can lead to a realistic prediction for the spectrum of density fluctuations (see Sect. 2.1 of Vol. I). We will present here a model that describes perturbations about a supersymmetric FRW minisuperspace with complex scalar fields. It was hoped it would push SQC towards an ‘observational context’. This was also an attempt to find new quantum states that would have a physical significance as regards the following: 1. A period of evolution from supersymmetric quantum gravitational physics towards a semiclassical stage (see Sect. 4.2). 2. The existence of a quantum state associated with structure formation. 3. The relation between conventional, i.e., purely bosonic, quantum cosmology and this picture, e.g., an effective mean theory after some coarse-graining over SQC (and possibly with decoherence [3]). Moniz, P.V.: More Routes Beyond the Borders. Lect. Notes Phys. 804, 197–206 (2010) c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-11570-7_10

198

10 More Routes Beyond the Borders

4. The consistent establishment of a path from supersymmetric quantum cosmology physics down to a classical level. We report here on the second item, while recent progress on the first item can be found in [4]. Regarding the approach as a whole, note that two perspectives should be considered. Either supersymmetry has been entirely broken while quantum gravity continues to prevail afterwards (this may seem strange, in view of recent studies that analyse realistic supersymmetry-driven inflationary scenarios within a classical gravitational background [5]), or else conventional quantum cosmology simply constitutes a coarse-grained description, extracted from SQC by some ‘averaging’ process (e.g., quantum mechanically driven SUSY breaking), whose physical justification is yet to be established. Let us present our case study. The action for our model is recovered from the general action of N = 1 SUGRA with scalar supermultiplets, as represented in equation (25.12) of [6]. Our by now well known background supersymmetric minisuperspace is constituted by the gravitational field, which is represented by the

a σ A A (in two-spinor notation), where we recall that (see Chap. 5 of tetrad eμA A = eμ a Vol. I)

eaμ =

N (t) 0 0 a(t)eai ˆ

.

(10.1)

Here aˆ and i run from 1 to 3, eai ˆ is a basis of left-invariant 1-forms on the unit

S 3 , and N (t), a(t), σaA A (A = 0, 1) denote the lapse function, scale factor, and Infeld–Van der Warden symbols, respectively (see Appendix A). In addition, we also have the gravitinos, which must have the form (see Sect. 5.1.1 of Vol. I)

ψ A i = e A A i ψ A (t) ,

ψ A i = e A A i ψ A (t) ,

(10.2)

where ψ A , ψ A are time-dependent spinor fields and ψ0A (t), ψ 0A (t) are Lagrange multipliers. A set of time-dependent complex scalar fields, φ, φ, and their fermionic superpartners, χ A (t), χ A (t) are also included. Finally, we choose a flat Kähler manifold for the scalar fields. The novel elements enter at this stage. As far as the perturbations to the background minisuperspace are concerned, we take here the scalar fields as extended to (see Sect. 2.2.1) n (xi ) , ϕ(xi , t) ≡ φ(t) + Σnlm f nlm (t)Q lm

(10.3)

together with its complex conjugate, where the coefficients f nlm , f lm n are functions n are standard scalar spherical harmonics on S 3 , x of the time coordinate t, and Q lm i are coordinates on the three-sphere, and m = −l, . . . , l, while l = 0, . . . , n − 1 and n ∈ N [7]. The fermionic superpartners are expanded to give X A (xi , t) ≡ χ A (t) + a −3/2 Σmpq βm

pq

nq mq smp (t)ρ A (xi ) + t mp (t)τ A (xi ) , (10.4)

10.1

Beyond Minisuperspace

199 mq

mq

mq

mq

together with its Hermitian conjugate, where ρ A , ρ A , τ A , τ A are spinor hyperspherical harmonics on S 3 , while p, q = 1, . . . , (m+1)(m+2) with m ∈ N. In addition, the time-dependent coefficients tmp , smp , and their Hermitian conjugates are 2 = 2 1n . odd elements of a Grassmann algebra, where the matrix β npq satisfies βnpq Now inserting (10.2), (10.3), and (10.4) into the general action for N = 1 supergravity with scalar supermatter and using the properties of the harmonics mentioned in [8–10, 7, 2], we obtain (after integration) a reduced action which includes an infinite sum of time-dependent harmonic and Fermi oscillators. The next step is to construct the relevant constraint equations for our model. To write down the supersymmetry constraints, we first need to obtain the Hamiltonian of the theory, which has the form

H = N H + ψ0A S A + S A ψ 0A + M AB J AB + M A B J A B ,

(10.5)

where M AB , M A B are additional Lagrange multipliers, H⊥ represents the Hamiltonian constraint, and S A , S A , and J AB , J A B denote the supersymmetry and Lorentz constraints, respectively. After some suitable redefinitions of the ψ A and χ A variables (see Sects. 5.1.1 and 5.1.4 of Vol. I), the quantum supersymmetry

constraints of the model can be constructed from the coefficients in ψ0A , ψ 0A in the Hamiltonian. They take the form (0)

(perturbation)

SA ≡ SA + SA

,

(10.6)

with S (0) A = −iχ A

√ ∂ ∂ i aψ A ∂ − √ − 3a 2 ψ A − φχ B χ B ∂φ 8 ∂χ A 2 3 ∂a

∂ i 3 ψ B ψB ∂ B ∂ ψ − φχ A ψ B + χ + √ √ A 4 ∂ψ B ∂χ B 4 3 8 3 ∂ψ A

(10.7)

and (perturbation)

SA

∂ ∂ ∂ ∂ ψA i = √ Σm smp − tmp − tm + φΣm sm χA ∂smp ∂tmp 2 ∂sm ∂tm 3 −iχ A Σn

∂ + 2ia 2 Σn f lm n (n + 1)χ A , ∂ f nlm (0)

(10.8) (perturbation)

. Note that together with their Hermitian conjugates, S A = S A + S A (0) (0) S A , S A will denote the supersymmetry constraints on the unperturbed background, (perturbation)

(perturbation)

while S A , SA correspond to the perturbed sector and have the form needed to produce the corresponding bosonic Hamiltonian constraint. Hereafter, the labels n, l, m and m, p will be denoted simply by n and m, respectively.

200


At this point, we introduce a natural ansatz for the wave function of the universe, which has the form (see Note 5.3 in Chap. 5 of Vol. I) Ψ = A + Bψ C ψC + iCψ C χC + Dχ D χ D + Eψ C ψC χ D χ D = A(0) (a, φ, φ)Πn A(n) (a, φ, φ; f n f n )Πm A(m) (a, φ, φ, sm , tm ) +B(0) (a, φ, φ)Πn B(n) (a, φ, φ; f n f n )Πm B(m) (a, φ, φ, sm , tm )ψ C ψC +C(0) (a, φ, φ)Πn C(n) (a, φ, φ; f n f n )Πm C(m) (a, φ, φ, sm , tm )ψ C χC +D(0) (a, φ, φ)Πn D(n) (a, φ, φ; f n f n )Πm D(m) (a, φ, φ, sm , tm )χ C χC + +E(0) (a, φ, φ)Πn E(n) (a, φ, φ; f n f n )Πm E(m) (a, φ, φ, sm , tm )ψ C ψC χ D χ D , (10.9) where each wave functional A(n) , A(m) , . . ., E(n) , E(m) depends only on the individual perturbation modes f n or sm , tm .

Note 10.1 Several comments are in order at this point. First, the expression (10.9) satisfies the Lorentz constraints associated with the unperturbed field variables ψ A , ψ A , χ A , and χ A : J AB = ψ(A ψ B) − χ(A χ B) = 0 . Second, the perturbation modes of the scalar fields and the fermionic partners do not couple to each other in our approximation, and this is also translated in the ansatz (10.9). In addition, we also follow the approach described in [8–10], where the coefficients sm , tm , s m , t m are taken to be invariant under local Lorentz transformations to lowest order in perturbation. Overall, this approach is fully satisfactory. In fact, we will see in the following how we can extract a consistent set of solutions from (10.7), (10.8), and (10.9).

Now, let us substitute (10.9) into the supersymmetry constraint (10.7), (10.8) and their Hermitian conjugates. It is important to notice that the terms independent of the perturbation modes have to vanish separately from the ones involving the perturbation modes. Furthermore, the various terms with perturbation modes must also vanish independently, since they do not couple with one another (see [9], where this procedure was similarly employed). After dividing S A Ψ = 0 and S A Ψ = 0 by Ψ as in (10.9), we then obtain a set of first-order differential equations. Among them are:

10.1

Beyond Minisuperspace

201

∂ ∂ ∂A ∂A 1 A + Σn − tm + 2a 2 Σn (n + 1) f n A , − φΣm sm ∂φ 2 ∂sm ∂tm ∂ fn

∂ ∂ ∂E ∂E 1 E + Σn 0= − tm − 2a 2 Σn (n − 1) f n E , + φΣm sm 2 ∂sm ∂tm ∂φ ∂ fn

√ ∂ ∂ 2 a ∂A A, − tm 0= √ + 2 3a 2 A − √ φΣm sm ∂sm ∂tm 3 ∂a 3

√ 2 a ∂E ∂ ∂ 2 0= √ E. − tm − 2 3a E + √ φΣm sm ∂sm ∂tm 3 ∂a 3

0=

(10.10)

(10.11)

(10.12) (10.13)

Concerning the analysis of the full set of equations, notice that (10.10), (10.11), (10.12), and (10.13) are uncoupled, while the remaining equations constitute coupled partial differential equations. With respect to the former, it is a straightforward matter to obtain the following solutions: −3a 2 +φ(2λ1 −Ω2 )−Ω2 φ

ˆ (0) e A(0) = A 0

a Ω1

,

(n)

A(n) = A0 e−λ2 φ+φ(2λ3 −λ2 ) e2λ4 fn −2a

(10.14) 2 (n+1) f

n

f n −(Ω3 −λ2 ) f n +(Ω3 −λ2 ) f n

,

2λ5 φ−C1 φφ−Ω4 φ+Ω4 φ ˜ A, A(m) = A(m) 0 e 3a 2 +φ(2λ6 −Ω5 )−Ω5 φ

ˆ (0) e E(0) = E 0

a Ω6

(10.16)

,

(n)

E(n) = E0 e−λ7 φ+φ(2λ8 −λ7 ) e2λ9 f n +2a 2λ8 φ−C2 φφ−Ω9 φ+Ω9 φ ˜ E, E(m) = E(m) 0 e

(10.15)

(10.17) 2 (n−1) f

n

f n −(Ω7 −λ9 ) f n +(Ω7 −λ9 ) f n

,

(10.18) (10.19)

ˆ (0) = A(0) e3a02 , A(n) , A(m) , E ˆ (0) = E(0) e−3a 2 , E(n) , and E(m) denote intewhere A 0 0 0 0 0 0 0 0 ˜ and E ˜ ∼ smp or tmp . It is important to emphasize the use gration constants, while A of φ = φ1 + iφ2 or φ = r eiθ in the process of integration to decouple the physical degrees of freedom encompassed in φ, φ. Notice also that λ1 , λ2 , . . ., and C1 , C2 are further integration/separation constants. The quantities Ω1 , Ω2 , . . ., represent back-reactions of the scalar and fermionic perturbed modes on the homogeneous modes, and are assumed to have very small values [7]. Characteristic features of the no-boundary (Hartle–Hawking) solution [11] are present in the bosonic coefficient E (10.17), (10.18), and (10.19). This state requires 2 |Ω6 | 1 and the term e−na fn f n (n 1) in (10.18) to dominate over the remaining exponential terms. This is equivalent to assuming that the corresponding separation/integration constants in (10.18) and (10.19) are very small. It is also 2 worth noting that a term of the form e−na fn f n is basically what was obtained in bosonic quantum cosmology, but from a Schrödinger-like equation (see Sect. 2.2)

202


and in an adiabatic approximation. It seems that the presence of supersymmetry selects a set of solutions where the no-boundary (Hartle–Hawking) quantum state is mandatory. Since such no-boundary wave functions may lead to a satisfactory spectrum of density perturbations, it thus seems that supersymmetry in the early (quantum) universe intrinsically contains the relevant seeds for structure formation. Finally, it is also important to mention that the states correspond˜ ∼ smp or E˜ ∼ tmp mean that these solutions would represent oneing to E particle or one-antiparticle states, if we adopt the interpretive framework introduced in [8, 9]. Concerning the coefficients B, C, and D, the corresponding equations lead to integral expressions, similar to the ones in Sect. 5.1.4 of Vol. I. However, the terms in f n , f n present in those equations imply that C(n) = 0 is the only possible solution. Hence, we cannot avoid C = 0, which is a particularly interesting result. The obvious question at this juncture is: Do the results presented here contribute to our understanding of the very early universe, and if so, how? Some comments will be advanced here, as a point of departure for future research. SQC is on the way to becoming ‘observational’, even if a vast area remains to be explored and charted, in the sense that it may soon be able to make specific predictions about (observational) cosmological properties from a quantum description. This endorses supersymmetry as a valid, if not mandatory, component of any realistic analysis of a quantum universe. In this context, the answer to the above questions is affirmative, even though some caution is nevertheless advisable. In fact, let us take the bosonic coefficient E [see (10.17), (10.18), and (10.19)], which corresponds to the filled fermionic sector χ C χC ψ D ψ D , when the term 2 e−na fn f n (n 1) in (10.18) dominates over the remaining exponential terms. Then this particular fermionic state implies the following expectation values: !

" ! " f n(1) ∼ f n(2) ∼ n −1 a∗−2 ,

where a∗ would be the value of a when the wavelength of the perturbation modes equals a particular horizon size. Once such conditions have been established, they constitute some of the requirements for the density perturbations δρ/ρ to represent an almost scale-free spectrum of fluctuations, similar to what is found in [7]. In addition, notice that each of the various bosonic amplitudes in (10.9) depends differently on the variables a, φ, φ, f n , f n , and hence each amplitude corresponds to a specific quantum scenario for the very early universe. Supersymmetry thus seems to assign several possible fermionic states with distinct bosonic features, each one leading to different evolution scenarios. In particular, we found the Hartle– Hawking quantum state. Since such a state may lead to a satisfactory spectrum of density perturbations, our results indicate that supersymmetry within a quantum description of the very early universe intrinsically contains the relevant seeds for structure formation.

10.2

Other New Horizons

203

However, as the reader will already have noticed, this model has no potential V (φ, φ; f n , f n ) for the homogeneous and inhomogeneous modes. The presence of such a potential could induce a transition from a quantum supersymmetric (Euclidean) phase to a classical (oscillating) [4] inflationary expansion period. But such potentials would also lead to mixing in the fermionic sectors of Ψ as described in (10.9) and [12]. In other words, it would imply additional (complicated) couplings between the equations to be solved. For the moment, no solutions have yet been found in such a scenario. For just the corresponding homogeneous sector, see [12]. We are thus dealing with a quantum-dominated era of evolution. Eventually, a potential term, e.g., V (φ, φ) ∼ M 2 φφ will be adequately analysed within this programme, and allow us to include a suitable inflationary scenario derived from supergravity. A natural extension will be to expand the tetrad and gravitinos in spherical harmonics as well. This would throw more light on the SQC states associated with the inhomogeneous fermionic modes.

10.2 Other New Horizons As in Vol. I, we present here a list of future research directions. Some of these refer to Vol. I but can and should also be investigated in the context of this volume. Some are simple, while others would require more time and could eventually become part of a graduate or post-graduate project. The reader is welcome to contact the author if any of the following should prove to be of interest: • Investigate why there are no physical states in a locally supersymmetric FRW model with gauged supermatter [13, 14], whereas they can be found in an FRW model with Yang–Mills fields [15–18]. • Include larger gauge groups in supersymmetric FRW models with supermatter. • Study the canonical quantization of Kantowski–Sachs cosmologies (and black holes) in N = 2 and N = 4 supergravity theories [17, 19]. • Perform the canonical quantization of FRW models in N = 3 supergravity. • Analyse the Chern–Simons states in SQC, and find ways to consider other solutions. • Describe the results and features concerning finite probabilities for photon– photon scattering in N = 2 supergravity, but now from a canonical quantization point of view. • Try to obtain a no-boundary (Hartle–Hawking) solution, as well as other solutions corresponding to gravitons (in the same sector) or pairs of gravitinos (in sectors differing by an even fermion number), as quantum states in the full theory. It would also be important to consider the case where supermatter is present. • Another issue of interest is that the action of pure N = 1 supergravity with boundary terms currently used is not fully invariant under supersymmetry transformations. But a particular fully invariant action has been presented in [20–23] for the case of Bianchi class A models. A generalization of this action for the full theory would be most welcome, for a related discussion in the context of

204

•

• • • • • • • • • • • • • • • • • • • •


general relativity. Then proceed with the corresponding quantization and obtain physical states. It would also be particularly interesting to address the following fundamental issues of quantum gravity but now within a supersymmetric scenario: the problem of time and how classical properties may emerge. See [3] and subsequently [24] for an analysis of the problem of time in quantum supergravity. The issue of recovering classical properties from supersymmetric quantum cosmologies was introduced and discussed in [4]. Can SQC become ‘observational’, allowing one to predict the spectrum of fluctuations and vacuum states for matter fields, eventually determining realistic semiclassical descriptions? Can the CFOP claim for finite (nonzero) fermion number be strengthened? Improve on the canonical quantization description for SUGRA with supermatter. In the expressions for the Hamilton–Jacobi equation, provide a proper separation of the terms that correspond to the Hamiltonian and momentum constraints. Improve on the semiclassical description by adding more (super)matter fields, and generalize the DeWitt supermetric. Can SQC be described with the assistance of Killing spinors? Following on from [25], try to obtain a Feynman description of the structure of the SUSY superspace. Can the approaches described in Chaps. 5, 6, 7, and 8 of either volume be shown to be equivalent or at least related in some way? Further investigate the occurrence of anti-de Sitter and possibly de Sitter states in an FRW SUSY minusuperspace. Improve the description of FRW SQC with scalar supermultiplets and a superpotential. Develop the description of SUSY breaking and a superHiggs efect within the SQC framework (possibly involving gaugino condensates). Extend the analysis of Bianchi models in SQC to include a cosmological constant and superpotential (in the case of a scalar supermultiplet). Explore semiclassical SQC (with matter, if possible) within the matrix representation, as well as ‘hidden’ N = 2 SUSY. Re-check the description of the FRW model in the matrix representation of SQC, in particular in the Dirac oscillator framework. Why is there no Teitelboim procedure in the matrix representation of SQC? How can SQC (or the canonical quantization of N = 1 SUGRA) inform us about the chaos scenario for the early universe? How can ‘hidden’ N = 2 SUSY (from SQM to SQC) be extended to an N = 4 framework, in order to relate the contents of Chaps. 5, 6, and 7? How can a conformal transformation eliminate quadratic and quartic fermionic terms? Can we find complex solutions to the SQC Hamilton–Jacobi equation and still have SUSY? Investigate whether the line presented in [26] can be extended to SQC. Explore SQC from the mathematical perspctive that the tools in [27] may provide.

References

205

References 1. Moniz, P.V.: Can the imprint of an early supersymmetric quantum cosmological epoch be present in our cosmological observations? In: COSMO 97: 1st International Workshop on Particle Physics and the Early Universe, Ambleside, England, 15–19 September 1997 197 2. Moniz, P.V.: Origin of structure in a supersymmetric quantum universe. Phys. Rev. D 57, 7071–7074 (1998) 197, 199 3. Kiefer, C.: Quantum Gravity. International Series of Monographs on Physics, vol. 136, 2nd edn., pp. 1–308. Clarendon Press, Oxford (2007) 197, 204 4. Kiefer, C., Luck, T., Moniz, P.: The semiclassical approximation to supersymmetric quantum gravity. Phys. Rev. D 72, 045006 (2005) 198, 203, 204 5. Liddle, A.R., Lyth, D.H.: Cosmological inflation and large-scale structure, 400pp. Cambridge University Press, Cambridge (2000) 198 6. Wess, J., Bagger, J.: Supersymmetry and Supergravity, 259pp. Princeton University Press, Princeton, NJ (1992) 198 7. Halliwell, J.J., Hawking, S.W.: The origin of structure in the universe. Phys. Rev. D 31, 1777 (1985) 198, 199, 201, 202 8. D’Eath, P.D.: Supersymmetric Quantum Cosmology, 252pp. Cambridge University Press, Cambridge (1996) 199, 200, 202 9. D’Eath, P.D., Halliwell, J.J.: Fermions in quantum cosmology. Phys. Rev. D 35, 1100 (1987) 199, 200, 202 10. Esposito, G.: Quantum gravity, quantum cosmology, and Lorentzian geometries. Lect. Notes. Phys. M 12, 1–326 (1992) 199, 200 11. Hartle, J.B., Hawking, S.W.: Wave function of the universe. Phys. Rev. D 28, 2960–2975 (1983) 201 12. Moniz, P.V.: FRW minisuperspace with local N = 4 supersymmetry and self-interacting scalar field. Annalen Phys. 12, 174–198 (2003) 203 13. Cheng, A.D.Y., D’Eath, P.D., Moniz, P.R.L.V.: Quantization of a Friedmann–Robertson– Walker model in N = 1 supergravity with gauged supermatter. gr-qc/9503009 (1995) 203 14. Cheng, A.D.Y., D’Eath, P.D., Moniz, P.R.L.V.: Quantization of a Friedmann–Robertson– Walker model in N = 1 supergravity with gauged supermatter. Class. Quant. Grav. 12, 1343–1354 (1995) 203 15. Moniz, P.V.: FRW model with vector fields in N = 1 supergravity. Helv. Phys. Acta 69, 293–296 (1996) 203 16. Moniz, P.V.: Quantization of a Friedmann–Robertson–Walker model with gauge fields in N = 1 supergravity. gr-qc/9604045 (1996) 203 17. Moniz, P.V.: Supersymmetric quantum cosmology – shaken not stirred. Int. J. Mod. Phys. A 11, 4321–4382 (1996) 203 18. Moniz, P.V.: Wave function of a supersymmetric FRW model with vector fields. Int. J. Mod. Phys. D 6, 465–478 (1997) 203 19. Moniz, P.V.: Why two is more attractive than one . . . or Bianchi class A models and Reissner– Nordstroem black holes in quantum N = 2 supergravity. Nucl. Phys. Proc. Suppl. 57, 307–311 (1997) 203 20. Luckock, H., Oliwa, C.: Quantisation of Bianchi class A models in supergravity and the probability density function of the universe. In: 7th Marcel Grossmann Meeting on General Relativity (MG 7), Stanford, CA, 24–30 July 1994 203 21. Luckock, H.: Boundary conditions for Nicolai maps. J. Phys. A 24, L1057–L1064 (1991) 203 22. Luckock, H.: Boundary terms for globally supersymmetric actions. Int. J. Theor. Phys. 36, 501–508 (1997) 203 23. Luckock, H., Oliwa, C.: The cosmological probability density function for Bianchi class A models in quantum supergravity. Phys. Rev. D 51, 5483–5490 (1995) 203 24. Graham, R., Luckock, H.: Cosmological time in quantum supergravity. Phys. Rev. D 55, 1943–1947 (1997) 204

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25. Barvinsky, A.O., Kiefer, C.: Wheeler–DeWitt equation and Feynman diagrams. Nucl. Phys. B 526, 509–539 (1998) 204 26. Gerhardt, C.: Quantum cosmological Friedmann models with an initial singularity. Class. Quant. Grav. 26, 015001 (2009) 204 27. Schreiber, U.: Supersymmetric homogeneous quantum cosmology, Diploma thesis, University Essen, http://www.theo-phys.uniessen.de/tp/ags/graham_dir/pub_gravi.html (2003) 204

Appendix A

List of Symbols, Notation, and Useful Expressions

In this appendix the reader will find a more detailed description of the conventions and notation used throughout this book, together with a brief description of what spinors are about, followed by a presentation of expressions that can be used to recover some of the formulas in specified chapters.

A.1 List of Symbols κ ≡ κi jk ξ ≡ ξi jk

Contorsion

K

Torsion Kähler function

K IJ ≡ g IJ ≡ G IJ W(#)

Kähler metric Superpotential

V #, # φ, ϕ V (φ)

Vector supermultiplet (Chiral) Scalar supermultiplet Scalar field Scalar potential

χ ≡ γ 0χ †

For Dirac 4-spinor representation

ψ†

Hermitian conjugate (complex conjugate and transposition) Complex conjugate

φ∗ [M]T {, } [, ] θ Jab , J AB q X , X = 1, 2, . . .

Transpose Anticommutator Commutator Grassmannian variable (spinor)

π μαβ

Spin energy–momentum

Lorentz generator (constraint) Minisuperspace coordinatization

Moniz, P.V.: Appendix. Lect. Notes Phys. 804, 207–275 (2010) c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-11570-7

208

A List of Symbols, Notation, and Useful Expressions

S μαβ D [ , ]P ≡ [ , ] [ , ]D F MP V β +, β −

Spin angular momentum Measure in Feynman path integral

ZIJ

Central charges Spacetime line element

ds

Poisson bracket Dirac bracket Superfield Planck mass Minisuperspace potential (Misner–Ryan parametrization)

ds F eμ

Minisuperspace line element Fermion number operator

ea

Orthonormal basis

(3) V

Volume of 3-space

vμ(a) (a) f μν πij

Coordinate basis

Vector field Vector field strength Canonical momenta to h i j

πφ

Canonical momenta to φ

π0

Canonical momenta to N

πi

Canonical momenta to Ni

μ pa ij Pk

Jj Ki Pμ εμνλσ εi jk SA H⊥ Hi G Gs Ψ Ω R

a (tetrad) Canonical momenta to eμ

Canonical momenta to ξikj (torsion) Lorentz rotation generator Lorentz boost generator Translation (Poincaré) generator Four-dimensional permutation Three-dimensional permutation SUSY generator (constraint) Hamiltonian constraint Momentum constraints DeWitt metric DeWitt supermetric Wave function (state) of the universe ADM time Spinor-valued curvature

A.1

List of Symbols

209

βi j (β+ , β− )

Anisotropy matrices (Misner–Ryan representation) in Bianchi cosmologies

pi j HADM Ki j

Canonical momentum to β i j ADM Hamiltonian Extrinsic curvature

(4)R

Four-dimensional spacetime curvature

(3)R

Λ

Three-dimensional spatial curvature Cosmological constant

μ

Γνλ ,

Christoffel connection coefficients Usual derivative (vectors, tensors)

| and (3) D i

Four-dimensional spacetime covariant derivative with respect to the 3-metric h i j , no spin connection and no torsion, with Christoffel term (vectors, tensors)

; and (4) D μ

Four-dimensional spacetime covariant derivative with respect to the 4-metric gμν , no spin connection and no torsion, with Christoffel term (vectors, tensors) Covariant derivative with respect to the 3-metric, including spin connection (vectors, tensors) Covariant derivative with respect to the 4-metric gμν , including spin connection (vectors, tensors) Proper time derivative

' ( Dot over symbol dΩ32

Line element of spatial sections Spatial curvature index

k a(t) ≡ eα(t) N

FRW scale factor Lapse function

Ni μ, ν, . . . a, b, . . . a, b, . . . ˆ . . . = 1, 2, 3, . . . a, ˆ b,

Shift vector World spacetime indices Condensed superindices (either bosonic or fermionic) Local (Lorentz) indices

i, j, k hi j gμν

Spatial indices Spatial 3-metric Spacetime metric

ηab ωμ

≡ A, A

[a]

Local (Lorentz) spatial indices

Lorentz metric {ω0 , ωi }

One-form basis Two-spinor component indices (Weyl representation) Four-spinor component indices, e.g., Dirac representation

210


I, J, . . . J, j M, N Lab

Kähler indices Representation label of SU(2) (spin state) Elements of SL(2,C) Lorentz (transformation) matrix representation

piA A

H A A

Momentum conjugate to eiA A Hamiltonian and momentum constraints condensed (two-spinor components)

S L

Lorentzian action Lagrangian

I S M Σt t

Euclidean action Superspace Four-dimensional spacetime manifold Three-dimensional spatial hypersurface Coordinate time

τ n ≡ {n μ }

Normal to spatial hypersurface

(4) D

Proper time

≡ Dν

ν

Four-dimensional spacetime covariant derivative with respect to the 4-metric gμν , with spin connection and torsion, no Christoffel term (spinors [vectors, tensors]) Four-dimensional spacetime covariant derivative with respect to the 3-metric h i j , with spin connection and torsion, with Christoffel term (spinors [vectors, tensors]) Three-dimensional analogue of Dν (spinors [vectors, tensors])

( and Dν

' (3)D

≡ (3) ∇ j ≡ ∇ j

j

(3s)D

j

A j Three-dimensional analogue of Dν , no torsion (spinors ≡ (3s) ∇ j ≡ ∇ [vectors, tensors])

Mab , M AB

Lagrange multipliers for the Lorentz constraints (Lorentz and 2-spinor indices)

A , ωa , ω A A B B ω Bμ bμ

ψμA nμ,

Spin connections Gravitino field

n

ε AB

σaA A

σ

Three-dimensional analogue of Dν (spinors [vectors, tensors])

A A

Normal to the hypersurfaces Metric for spinor indices (Weyl representation) Pauli matrices with two-component spinor indices (Weyl representation) Infeld–van der Waerden symbols

A.2

Conventions and Notation

ea μ

Tetrad

eâˆ

Spatial tetrad (a, ˆ i = 1, 2, 3)

i

A e A

μ

∼ = (a) Aμ , D ˇ φ , Dˆ μ D Q(a) ε Dφ D %IJK

Tetrad with two-component spinor indices Equality provided constraints are satisfied Gauge group index Gauge group covariant derivatives Gauge group charge operator Infinitesimal spinor Kähler derivative SUSY covariant derivative (SUSY superspace)

R

Kähler connection Kähler curvature

ψ, χ , λ

Spinors

ψ, χ , λ

Conjugate spinors, in the 2-spinor SL(2,C) Weyl representation Minisuperspace potential

U P(φ, φ) T Q Gˆ

(a)

211

SUGRA-induced potential for scalar fields Gauge group generators SUSY supercharge operators Gauge group

T D R

Energy–momentum tensor Minisuperspace dimension Minisuperspace curvature

ˆ D, ˆ D, ˜ D, ˇ ... D, d,

Dimension of space

σ 1, σ A, B, . . .

Axion field Pauli matrices Bosonic amplitudes of SUSY Ψ

EiAB

Densitized spatial triad

A AB j

Ashtekar connection

A.2 Conventions and Notation Throughout this book we employ c = 1 = h¯ and G = 1 = MP−2 , with k ≡ 8π G unless otherwise indicated. In addition, we take: • μ, ν, . . . as world spacetime indices with values 0, 1, 2, 3, • a, b, . . . as local (Lorentz) indices with values 0, 1, 2, 3,

212


• i, j, k as spatial indices with values 1, 2, 3, • A, A as 2-spinor notation indices with values 0, 1 or 0 , 1 , respectively, • [a] as 4-spinor component indices, e.g., the Dirac representation, with values 1, 2, 3, 4. In this book we have also chosen the signature of the 4-metric gμν (or ηab ) to be (−, +, +, +). Therefore, the 3-metric h i j on the spacelike hypersurfaces has the signature (+, +, +), which gives a positive determinant. Thus the 3D totally antisymmetric tensor density can be defined by ε0123 ≡ ε123 = ε123 = +1.

A.3 About Spinors As the fellow explorer may already have noticed (or will notice, if he or she is venturing into this appendix prior to probing more deeply into some of the chapters) there are some idiosyncrasies in the mathematical structures of SUSY and SUGRA which are essentially related to the presence of fermions (and hence of the spinors which represent them). But why is this, and what are the main features the spinors determine? In fact, why are spinors needed to describe fermions? What are spinors and how do they come into the theory? This section is devoted (in part) to introducing this issue. Before proceeding, for completeness let us just indicate that a tetrad formalism is indeed mandatory for introduing spinors.1 The tetrad corresponds to a massless spin-2 particle, the graviton [1], i.e., corresponding to two degrees of freedom.2 Note A.1 Extending towards a simple SUSY framework, with only one generator (labeled N = 1 SUSY), the corresponding (super)multiplet will also contain a fermion with spin 3/2, the gravitino, as described in Chap. 3 of Vol. I, where the reader will find elements of SUSY (not in this appendix).

A.3.1 Spinor Representations of the Lorentz Group Local Poincaré invariance is the symmetry that gives rise to general relativity [2]. It contains Lorentz transformations plus translations and is in fact a semi-direct product of the Lorentz group and the group of translations in spacetime. The Lorentz Tensor representations of the general linear group 4 × 4 matrices GL(4) behave as tensors under the Lorentz subgroup of transformations, but there are no such representations of GL(4) which behave as spinors under the Lorentz (sub)group (see the next section). 1

2 In simple terms, the tetrad has 16 components, but with four equations of motion, plus four degrees of freedom to be removed due to general coordinate invariance and six due to local Lorentz invariance, this leaves 16 − (4 + 4 + 6) = 2.

A.3

About Spinors

213

group has six generators: three rotations Jaˆ and three boosts Kbˆ , a, ˆ bˆ = 1, 2, 3 with commutation relations [3–5]: [Jaˆ , Jbˆ ] = iεaˆ bˆ cˆ Jcˆ ,

[Kaˆ , Kbˆ ] = −iεaˆ bˆ cˆ Jcˆ ,

[Jaˆ , Kbˆ ] = iεaˆ bˆ cˆ Kcˆ .

(A.1)

The generators of the translations are usually denoted Pμ , with3

Pμ , Pν = 0 ,

[Ji , P0 ] = 0 ,

Ji , P j = iεi jk Pk ,

Ki , P j = −iP0 δi j ,

[Ki , P0 ] = −iPi .

(A.2)

(A.3)

Or alternatively, defining the Lorentz generators Lμν ≡ −Lνμ as L0i ≡ Ki and Li j ≡ εi jk Jk , the full Poincaré algebra reads

Pμ , Pν = 0 ,

Lμν , Lρσ = −iηνρ Lμσ + iημρ Lνσ + iηνσ Lμρ − iημσ Lνρ ,

Lμν , Pρ = −iηρμ Pν + iηρν Pμ .

(A.4) (A.5) (A.6)

Let us focus on the mathematical structure of (A.1). The usual tensor (vector) formalism and corresponding representation is quite adequate to deal with most situations of relativistic classical physics, but there are significant advantages in considering a more general exploration, namely from the perspective of the theory of representations of the Lorentz group. The spinor representation is very relevant indeed. For this purpose, it is usual to discuss how, under a general (infinitesimal) Lorentz transformation, a general object transforms linearly, decomposing it into irreducible pieces. To do this, we take the linear combinations J j± ≡

1 (J j ± iK j ) , 2

(A.7)

and the Lorentz algebra separates into two commuting SU(2) algebras: [Ji± , J j± ] = iεi jk Jk± ,

[Ji± , J j∓ ] = 0 ,

(A.8)

i.e., each corresponding to a full angular momentum algebra. A few comments are then in order:

3 Note that we will be using either world (spacetime) indices μ, or local Lorentz indices a, b, . . . , ˆ . . . , which can be related and also world (space) indices i, or local spatial Lorentz indices a, ˆ b, a (see Sect. A.2). through the tetrad eμ

214


• The representations of SU(2) are known, each labelled by an index j, with j = 0, 1/2, 1, 3/2, 2, . . . , giving the spin of the state (in units of h¯ ). The spin-j representation has dimension (2j + 1). • The representations of the Lorentz algebra (A.8) can therefore be labelled by4 (j− , j+ ), the dimension being (2j+ + 1)(2j− + 1), where we find states with j taking integer steps between |j+ − j− | and j+ + j− . – In particular, (0, 0) is the scalar representation, while (0, 1/2) and (1/2, 0) both have dimension two for spin 1/2 and constitute spinorial representations.5 – In the (1/2, 0) representation, J− ≡ {Ji− }i=1,... can be represented by 2 × 2 matrices and J+ = 0, viz., J− = σ) \2, which implies6 J = σ) \2, but K = i) σ \2, where we henceforth use7 the four 2×2 matrices σμ ≡ (1, σ i ), with σ0 usually being the identity matrix and σ) ≡ σi , i = 1, 2, 3, the three Pauli matrices:

σ ≡ 1

01 10

,

σ ≡ 2

0 −i i 0

,

σ ≡ 3

1 0 0 −1

,

(A.9)

satisfying σ i , σ j = 2iεi jk σ k . Note the relation between Pauli matrices with upper and lower indices, namely σ 0 = −σ0 and σ i = σi , using the metric ημν . – In the (0, 1/2) representation we have J = σ \2 but K = −iσ \2. This means that we have two types of spinors, associated respectively with (0, 1/2) and (1/2, 0), which are inequivalent representations. These will correspond in the following to the primed spinor ψ A and the unprimed spinor ψ A , where A = 0, 1 and A = 0 , 1 [6–9]. The ingredients above conspire to make the group8 SL(2,C) the universal cover9 of the Lorentz group. If M is an element of SL(2,C), so is −M, and both produce the same Lorentz transformation. The pairs (j− , j+ ) of the finite dimensional irreducible representations can also be extracted from the eigenvalues j± (j± + 1) of the two Casimir operators J2± (operators proportional to the identity, with the proportionality constant labelling the representation). 4

The representation (1/2, 1/2) has components with j = 1 and j = 0, i.e., the spatial part and time component of a 4-vector. Moreover, taking the tensor product (1/2, 0) ⊗ (0, 1/2), we can get a 4-vector representation.

5

6 It should be noted from J = σ \2 that a spinor effectively rotates through half the angle that a vector rotates through (spinors are periodic only for 4π ). 7

The description that follows is somewhat closer to [10, 9, 11], but different authors use other choices (see, e.g., [3, 5, 12, 13]), in particular concerning the metric signature, σ0 , and some spinorial elements and expressions. See also Sect. A.4. 8

The letter S stands for ‘special’, indicating unit determinant, and the L for ‘linear’, while C denotes the complex number field, whence SL(2,C) is the group of 2 × 2 complex-valued matrices. 9 So the Lorentz group is SL(2,C)/Z , where Z consists of the elements 1 and −1. Note also that 2 2 SU(2) is the universal covering for the spatial rotation group SO(3). There is a two-to-one mapping of SU(2) onto SO(3).

A.3

About Spinors

215

A spinor is therefore the object carrying the basic representation of SL(2,C), fundamentally constituting a complex 2-component object (e.g., 2-component spinors, or Weyl spinors) ψA ≡

ψ A=0 ψ A=1

transforming under an element M according to ψ A −→

ψ A

=M

B

AψB

,

M≡

M1 M2 M3 M4

∈ SL(2,C) ,

(A.10)

with A, B = 0, 1, labelling the components. The peculiar feature is that now the other 2-component object ψ A transforms as ψ A −→ ψ A = M ∗B

A ψ B

,

(A.11)

which is the primed spinor introduced above, while the above ψ A is the unprimed spinor. Note A.2 The representation carried by the ψ A is (1/2, 0) (matrices M). Its complex conjugate ψ A in (0, 1/2) is not equivalent: M and M∗ constitute inequivalent representations, with the complex conjugate of (A.10) associated with (A.11), and ψ A identified with (ψ A )∗ .

Note A.3 There is no unitary matrix U such that N = UMU−1 for matrices N ≡ M∗ . We have instead N = ζ M∗ ζ −1 with ζ =

0 −1 ≡ −iσ2 . 1 0

This follows from the (formal) relation σ2 σ ∗ σ2 = −σ , which suggests writing ζ in terms of a known matrix (appropriate for 2-component spinors), viz., σ2 . It is from this feature applied to a Dirac 4-spinor ψ≡

χ , η

that Lorentz invariants are obtained as (iσ 2 χ )T χ . (This is the Weyl representation, where χ , η are 2-component spinors transforming with M and N, respectively.) To be more concrete, with

216


χ ≡ χA =

χ A=0 χ A=1

,

we put

χA =

χ0 χ1

= iσ 2 χ =

χ1 −χ0

,

whence the invariant is χ A χ A . (Note that χ 0 χ0 + χ 1 χ1 = χ1 χ0 − χ0 χ1 = −χ1 χ 1 − χ0 χ 0 = −χ A χ A , which is quite different from the situation for spacetime vectors and tensors Aμ Aμ = Aμ Aμ !) This can be simplified by introducing new matrices with χ A = ε AB χ B , where (formally!) ε AB ≡ iσ 2 . Likewise for

0

η η ≡ ηA = ,

η1

η A η A , η A = ε A B η B , ε A B ≡ iσ 2 , etc. Of course, ε AB ≡ iσ 2 , ε A B ≡ iσ 2 , are different objects acting on different spinors and spaces. The presence of the matrix representation is formal for computations.

The primed and unprimed index structure10 carries to the generators, e.g., J = σ \2 and K = iσ \2, with the four σμ matrices (σ μ ) A A = {−1, σ i } A A ,

(A.12)

and (σ μ ) A

A

≡ ε A B ε AB (σ μ ) B B ≡ (1, −σ i ) A

A

,

(A.13)

where indices are raised by the antisymmetric 2-index tensors ε AB and ε AB given by11 ε

10

=ε

A B

0 1 = −1 0

,

ε AB = ε A B

0 −1 = 1 0

,

(A.14)

Note that (A.12) and (A.13) can used to convert an O(1,3) vector into an SL(2,C) mixed bispinor

b AB . 11

AB

Note that, in a formal matrix form, ε AB = ε A B = iσ2 , ε AB = ε A B = −iσ 2 .

A.3

About Spinors

217

with which: ψ A = ε AB ψ B ,

ψ A = ε AB ψ β ,

ψ A = ε A B ψ B ,

ψ A = ε A B ψ B . (A.15)

Note A.4 Note the difference between (σ μ ) A A and (σ μ ) A A , i.e., the lower,

upper and respective order of indices, arising from ε AB = ε A B = iσ2 and 2 ε AB = ε A B = −iσ . To be more precise, the (‘natural’) lower (upper) A A

indices on (σ μ ) A A , (σ μ ) A A come from covariance, e.g., analysing how the matrices 0 σμ μ γ ≡ σμ 0 transform under a Lorentz transformation (the Weyl representation here). Further relations [9], in particular between σ μ and σ μ , can be written

σμC D σ μA B = −2δCB δ DA ,

μ Tr σ μ σ ν ≡ σ AB σ ν B A = −2g μν ,

(A.16) (A.17)

constituting two completeness relations. μ In essence, it all involves the statement that σ AB is Hermitian, and that, for a real vector, its corresponding spinor is also Hermitian.

Note A.5

In addition, the following points may be of interest:

• Concerning (A.14) and (A.15), it must be said that this is a matter of convention. • Unprimed indices are always contracted from upper left to lower right (‘ten to four’), while primed indices are always contracted from lower left to upper right (‘eight to two’). This rule does not apply when raising or lowering spinor indices with the ε tensor. • However, other choices can be made (see Sect. A.4) [6, 14, 7–9], and this is what we actually follow from Chap. 4 of Vol. I onwards. In particular, ε

AB

=ε

A B

= ε AB = ε A B

0 1 ≡ −1 0

,

(A.18)

with which [note the position of terms and order of indices, in contrast with (A.14)]:

218


ψ A = ε AB ψ B , ψ A = ψ B ε B A ,

ψ A = ε A B ψ B ,

ψ A = ψ B ε B A . (A.19)

With (A.15) [or (A.19)], scalar products such as ψχ or ψ χ can be employed. Of course, when more than one spinor is present, we have to remember that spinors anticommute.12 Therefore, for 2-component spinors, ψ0 χ1 = −χ1 ψ0 , and also, e.g., ψ0 χ 1 = −χ 1 ψ0 . In more detail, ψχ ≡ ψ A χ A = ε AB ψ B χ A = −ε AB ψ A χ B = −ψ A χ A = χ A ψ A = χ ψ ,

ψ χ ≡ ψ A χ A = . . . = χ A ψ A = χ ψ ,

(A.20) (A.21)

(ψχ )† ≡ (ψ A χ A )† = χ A ψ A = χ ψ = ψ χ ,

(A.22)

as well as μ

ψσ μ χ = ψ A σ AB χ B ,

ψ σ μ χ = ψ A σ μA B χ B ,

(A.23)

from which, e.g., χ σ μ ψ = −ψ σ μ χ ,

(A.24)

χ σ μ σ ν ψ = ψσ ν σ μ χ ,

(A.25)

(χ σ μ ψ)† = ψσ μ χ ,

(A.26)

(χ σ μ σ ν ψ)† = ψσ ν σ μ χ ,

(A.27)

A.3.2 Dirac and Majorana Spinors The above (Weyl) framework will often be used for most of this book [in particular, (A.18) and (A.19)], but if the reader goes on to study the fundamental literature on SUSY and SUGRA, other representations will be met, some of which are also adopted by a few authors in SQC. Let us therefore comment on that (see Chap. 7). In particular, we present the Dirac and Majorana spinors (associated with another representation).

12 The components of two-spinors are Grassmann numbers, anticommuting among themselves. Therefore, complex conjugation includes the reversal of the order of the spinors, e.g., ∗

(ζ ψ)∗ = (ζ A ψ A )∗ = (ψ A )∗ ζ A = ψ A ζ A = ψ ζ .

A.3

About Spinors

219

A 4-component Dirac spinor is made from a 2-component unprimed spinor and a 2-component primed spinor via

ψ[a] ≡

ψA

χA

,

transforming as the reducible (1/2, 0) ⊕ (0, 1/2) representation of the Lorentz group, hence with

ψA 0

and

0

χA

as Weyl spinors.13 But is there any need for a Dirac spinor (apart from in the Dirac equation)? The point is that, under a parity transformation [3, 5], the boost generators K change sign, whereas the generators J do not. It follows that the (j, 0) and (0, j) representations are interchanged under parity, whence 2-component spinors are not sufficient to provide a full description. The Dirac 4-component spinor carries an irreducible representation of the Lorentz group extended by parity. Note A.6

In addition, from

ψ[a] ≡

χA

ηA

† = χ A , η A , ⇒ ψ[a]

(A.28)

the adjoint Dirac spinor is written as (Weyl representation)

A η ψ [a] ≡ −ψ † γ 0 η A , χ A ⇒ ψ T[a] = , χ A

0 −1 . −1 0 (A.29) Note that ψ 1 ψ2 ≡ η1 χ2 + χ 1 η2 where the right-hand side is a Weyl 2-spinor representation, noticing that in the left-hand side we have Dirac 4-spinors, with the bar above either ψ1 or χ1 concerning different configurations and operations, and subscripts 1 and 2 merely labeling different Dirac spinors.

Note A.7

γ0 ≡

The charge conjugate spinor ψ c is defined by

There are also chiral Dirac spinors. These constitute eigenstates of γ 5 and behave differently under Lorentz transformations [see (A.33) and (A.34)].

13

220


ψ c ≡ Cψ T =

−iσ 2 η A iσ 2 χ A

=

ηA

χA

,

(A.30)

where

ε AB = −iσ 2 0 C≡

0 ε A B = iσ 2 Note A.8

.

(A.31)

The Majorana condition ψ = ψ c gives χ = η :

ψ =ψ = c

χA

χA

.

(A.32)

The importance of the Majorana condition is that it defines the Majorana spinor. This is invariant under charge conjugation, therefore constituting a relation (a sort of ‘reality’ condition) between ψ and the (complex) conjugate spinor ψ † . A Majorana spinor is a particular Dirac spinor, namely

ψA

ψA

,

with only half as many independent components. A Majorana spinor is therefore a Dirac spinor for which the Majorana condition (A.32) is implemented: ψ = ψ c , i.e., χ A = ψ A† . Of course, whenever we have (Dirac or Majorana) 4-spinors, a related matrix framework must be used, in the form of the Dirac matrices (here in the chiral or Weyl representation): γμ ≡

0 σμ σμ 0

,

γ5 ≡ iγ 0 γ 1 γ 2 γ 3 =

1 0 0 −1

,

γ5

2

=1,

(A.33) with γ5 inducing the projection operators (1 ± γ5 )/2 for the chiral components ψ A ,

χ A of ψ, determining their helicity as right (positive) or left (negative). But there are many other options [9]: • Standard14 canonical basis: −1 0 γ0 ≡ , 0 1

0 σi γi ≡ −σ i 0

.

(A.34)

14 The standard (Dirac) representation has γ appropriate to describe particles (e.g., plane waves) 0 in the rest frame.

A.3

About Spinors

221

• A Majorana basis, where γi∗ = −γi :

0 −σ 2 γ0 ≡ −σ 2 0 i1 0 γ2 ≡ , 0 −i1

0 iσ 3 , γ1 ≡ , iσ 3 0 0 −iσ 1 . γ3 ≡ −iσ 1 0

(A.35)

• A basis in which the γ matrices are all real and where N = 1 SUGRA allows a real Rarita–Schwinger (gravitino) field (see Sect. 4.1 of Vol. I) [15]:

0 1 , γ0 ≡ −1 0 0 1 γ3 ≡ , 1 0

0 −iσ 2 , γ2 ≡ iσ 2 0 0 iσ 3 . γ 5 ≡ iγ 0 γ 1 γ 2 γ 3 = −iσ 3 0

1 0 γ1 ≡ 0 −1

, (A.36)

Furthermore, 9 μ ν: γ , γ = −2ημν .

(A.37)

Through (A.33), the Lorentz generators then become Lμν −→

i μν γ , 2

where γ

μν

1 μ ν 1 σ μσ ν − σ ν σ μ 0 ν μ ≡ (γ γ − γ γ ) = , σ μσ ν − σ ν σ μ 0 2 2

(A.38)

and {, } and [, ] denote the anticommutator and commutator, respectively, As expec

ted from the representation (1/2, 0) ⊕ (0, 1/2), this determines that the ψ A and χ A μν spinors transform separately. The specific generators are also rewritten iσ for ψ A

and iσ μν for χ A , with 1 μ νC B − (μ ↔ ν) , σ AC σ 4 1 μA C ν ≡ σC B − (μ ↔ ν) . σ 4

(σ μν ) A B ≡ (σ μν ) B A

(A.39) (A.40)

222


A.4 Useful Expressions In this section we present a set of formulas which will help to clarify some properties of SUSY and SUGRA, and help also to simplify some of the expressions in the book. The initial emphasis is on the 2-component spinor, but we will also indicate, whenever relevant, some specific expressions involving 4-component spinors. In a 2-component spinor notation, we can use a spinorial representation for the tetrad. This then allows us to deal with the indices of bosonic and fermionic variables in a suitable equivalent manner. To be more precise, in flat space we therefore

associate a spinor with any vector by the Infeld–van der Waerden symbols σaA A , which are given by [16, 7–9]15 1

σ0A A = − √ 1 , 2

1

σiA A = √ σi . 2

(A.41)

Here, 1 denotes the unit matrix and σi are the three Pauli matrices (A.9). To raise and lower the spinor indices, the different representations of the antisymmetric spinorial

metric ε AB , ε AB , ε A B , and ε A B can be used [see Note A.5 and (A.18)]. Each of them can be written as the same matrix, given by

0 1 −1 0

.

(A.42)

Hence, the spinorial version of the tetrad reads

eμA A = ea μ σaA A ,

ea μ = −σ Aa A eμA A .

(A.43)

Generally, for any tensor quantity T defined in a (curved) spacetime, a correspond

ing spinorial quantity is associated through T → T A A = eμA A T μ , with the inverse

μ relation given by T μ = −e A A T A A . Equipped with this feature, the foliation of spacetime into spatial hypersurfaces Σt from a tetrad viewpoint (spinorial version) is straightforward: • The future-pointing unit normal vector n → n μ has a spinorial form given by

n A A = eμA A n μ .

(A.44)

• The tetrad (A.43) is thereby decomposed into timelike and spatial components

e0A A and eiA A .

15

We henceforth condense and follow the structure in Note A.5 (see Chap. 4 of Vol. I).

A.4

Useful Expressions

223

• Moreover, the 3-metric is written as

h i j = −e A A i e Aj A .

(A.45)

This metric and its inverse are used to lower and raise the spatial indices i, j, k, . . ..

• From the definition of n A A as a future-pointing unit normal to the spatial hypersurfaces Σt , we can further retrieve

n A A eiA A = 0

and

n A A n A A = 1 ,

(A.46)

which allow us to express n A A in terms of eiA A (see Exercise 4.3 of Vol. I). • Using the lapse function N and the shift vector N i , the timelike component of the tetrad can be decomposed according to

e0A A = N n A A + N i eiA A .

(A.47)

• Other useful formulas16 involving the spinorial tetrad eiA A with the unit vector

in spinorial form n A A can be found in Sect. A.4.1. The reader and fellow explorer of SQC should note that these expressions do indeed allow one to considerably simplify, e.g., the differential equations for the bosonic functionals of the wave function of the universe (see, e.g., Chap. 5 of Vol. I).

A.4.1 Metric and Tetrad Using the above definitions [6], we have the following relations for the timelike

normal vector n A A and the tetrad eiA A :

1 B

ε A , 2 1 = εA B , 2 √ 1

= − h i j ε A B − i hεi jk n A A e AB k , 2 1 1

= − h i j ε A B − i √ εi jk n A A e B A k , 2 h

n A A n AB = n A A n B A e A A i e AB j e A A i e Bj A

e A A i eiB B = n A A n B B − ε AB ε A B .

(A.48) (A.49) (A.50) (A.51) (A.52)

16 Equations (A.44), (A.45), (A.46), and (A.47) can be contrasted with (A.60), (A.61), (A.62), (A.63), (A.64), and (A.65).

224


From (A.50) and (A.51), by contracting with εi jl , we obtain i

n A A e AB l = −n AB ekA A = √ εi jl e A A i e AB , j 2 h i

n A A e B A l = −n B A ekA A = − √ εi jl e A A i e Bj A . 2 h

(A.53) (A.54)

In addition, we have

g μν e AμA e Bν B = −ε AB ε A B , A A

(A.55)

B B

gμν = −eμ e ν ε AB ε A B ,

BA = gμν ε A B , 2e A A (μ eν)

(A.56)

(A.57)

AB 2e A A (μ eν) = gμν ε A B .

(A.58)

The normal projection of the expression εilm D BBm j D CA kl , used throughout Chap. 4, is given by [17] −2i

AB i A A

≡ n A A εilm D BBm j D CA kl = √ εilm D B B m j elC D e D D k n D E CB jk A n h 2i

A = √ εilm D B B m j elC D e D

k . h

(A.59)

Now, regarding the 4-spinor framework in Sects. 4.1.1 and 4.1.2 of Vol. I, Ni = e0a ea i ,

(A.60)

N = N j N − e0a e

,

(A.61)

, N = N n a + N m ema ,

(A.62)

j

e0 a = − e0a

a

0

na

γ

n emγ = 0 ,

(A.63) (A.64)

εi jk γs γi = 2iγ ⊥ σ ik ,

(A.65)

and we have the notation − A⊥ = A⊥ = −n μ Aμ = N A0 = ˆ

Ai ≡ Ai −

Ni ⊥ A . N

1 A 0 − N i Ai , N

(A.66) (A.67)

A.4

Useful Expressions

225

A.4.2 Connections and Torsion

In the second order formalism applied to N = 1 SUGRA [13], the tetrad eμ C D and

gravitinos ψμ C , ψν D determine the explicit form of the connections. Four-Dimensional Spacetime We will use

ab [ab] ωμ = ωμ −→ ωμA A B B = ωμAB ε A B + ωμA B ε AB ,

(A.68)

(AB) = sωμAB + κμAB , ωμAB = ωμ

(A.69)

where

while sωμAB is the spinorial version of the torsion-free connection form s ab ωμ (AB)

and κμAB = κμ

b a c = eaν ∂[μ eν] − ebν ∂[μ eν] − eaν ebρ ecμ ∂[ν eρ] ,

(A.70)

is the spinorial version of the contorsion tensor κμab = κμ[ab] , with

κμA A B B = eμA A eνB B κμνρ = κμAB ε A B + κ μA B ε AB .

(A.71)

Furthermore, explicitly, it will in this case relate to the (4D spacetime) torsion through

AA A = −4πiGψ [μ ψν]A , ξμν

(A.72)

whose tensorial version is

ρ

AA . ξ ρ μν = −e A A ξμν

(A.73)

The contorsion tensor κ is defined by κμνρ = ξνμρ + ξρνμ + ξμνρ .

(A.74)

Spatial Representation Within a 3D (spatial surface) representation, the novel ingredient is the expansion

into n A A and eiA A . In more detail, the spin connection is written in the form (3) A A B B

ωi

=

(3)s A A B B

ωi

+

(3) A A B B

κi

,

(A.75)

226


and then decomposed into primed and unprimed parts: (3) A A B B

ωi

=

(3) AB A B

ωi ε

(3) A B AB ωi ε

1 (3) A B B

ω B i , 2

(3) A B

ωi

=

+

.

(A.76)

Using the antisymmetry (3)ωiA A B B = − (3)ωiB B A A , we then obtain the symmetries (3)ω AB = (3)ω B A and (3)ω A B = (3)ω B A and the explicit representations i i i i (3) AB ωi

=

Analogous relations hold for have

(3)sω A A B B

i

(3) AB ωi

(3)κ A A B B . i

and

=(3s)ωiAB +

1 (3) A B B

ω Bi . 2

(3) AB κi

(A.77)

Furthermore, we now

,

(A.78)

where (3s) ωiAB is the spinorial version of the spatial torsion-free connection form (3s) ab ωi

1 a j bk c 1 aj b c 1 a b bj a = e ∂[ j ei] − e e ei ∂ j eck − e n n ∂ j eci − n ∂i n −(a ↔ b) , 2 2 2 (A.79)

and κiAB is the spinorial version of the spatial contorsion tensor17 κiab , with 3

j

κ A A B B i = e A A ekB B 3 κ jki = 3 κ ABi ε A B + 3 κ A B i ε AB .

The three-dimensional torsion-free spin connection

expressed in terms of n A A and eiA A as

(3)sω A A B B

i

(A.81)

can therefore be

(3)sω A A B B = e B B j ∂ e A A

[ j i] i

(A.82)

1 − e A A j e B B k eiCC ∂ j eCC k + e A A j n B B n CC ∂ j eCC i + n A A ∂i n B B 2

B B

−e A A j ∂[ j ei]

1 + e B B j e B B k eiCC ∂ j eCC k + e B B j n A A n CC ∂ j eCC i + n B B ∂i n A A . 2

17

The 3D contorsion is simply obtained by restricting the 4D quantity: (3)

κi jk = κi jk ,

with the spinorial contorsion

(3)κ

A A B B i

=e

A A

je

B B

k

(3)κ

(A.80) jki

= − (3)κ

B B A A i

.

A.4

Useful Expressions

227

A.4.3 Covariant Derivatives

Still referring to a second order formalism, the tetrad eμ C D and the gravitinos

ψμ C , ψν D constitute the action variables, with which a specific covariant derivative Dμ is associated, acting only on spinor indices (not spacetime, with which Γ would be associated). Four-Dimensional Spacetime To be more precise, we use18

A BA Dμ eνA A = ∂μ eνA A + ω Bμ eν + ω BA μ eνAB ,

(A.83)

A Dμ ψνA = ∂μ ψνA + ω Bμ ψνB ,

(A.84)

A the connection form (spinorial representation) as described in Sect. A.4.2 with ω Bμ above.

Spatial Representation Subsequently, we have the spatial covariant derivative indices, where, for a generic tensor in spinorial form,19 (3)

with (3)ω BA and decomposition

D j T AA = ∂ j T AA +

(3)ω A

B

(3) A B A

ωB T

+

(3) D

j

acting on the spinor

(3) A AB

ω B T

,

(A.85)

the two parts of the spin connection [see (A.77)], using the

(3) A A B B

ωi

=

(3)s A A B B

ωi

+

(3) A A B B

κi

.

(A.86)

A.4.4 (Gravitational) Canonical Momenta Related to the presence of (covariant) derivatives in the action of N = 1 SUGRA, we C D and gravitinos ψ C , ψ D . The latter retrieve the momenta conjugate to the tetrad eμ μ ν require the use of Dirac brackets, which are discussed elsewhere (see Appendix B and Chap. 4 of Vol. I), so we present here some elements regarding the former.

18 19

A A = ξ A A , where ξ A A is the spinor version of the torsion. Notice that D[μ eν] μν μν

T

A A ...Z Z

A A

= eμ

Z Z

. . . eν

T μ...ν .

228


For the gravitational canonical momentum, we can write piA A =

δS N =1

δeiA A

p ⊥i = n A A piA A ,

−→ p ji = −e A A j piA A ,

(A.87)

where we can also use (as in the pure gravitational sector, see Sect. 4.2 of Vol. I) p (i j) = −2π i j .

(A.88)

This then leads us to the issue of curvature.

A.4.5 Curvature In fact, we can express the curvature (e.g., the intrinsic curvature or second fundamental form K i j ) in spinorial terms and through the (gravitational) canonical momentum piA A , with the assistance of (A.87) and (A.88). Let us recall that20 h 1/2 - (0)(i j) (0) i j . − trK K h 2k2 . h 1/2 - (i j) − τ (i j) − h i j (K − τ ) , K ∼− 2 2k

πij ∼ −

K i j = −eia ∂ j n a + n a ωab j ebi ,

(A.89) (A.90)

but now emphasizing the influence of the torsion: 1 Ni I j + N j|i − h i j,0 − 2ξ(i j)⊥ , 2N = ξ⊥i j ≡ n μ ξμi j .

K (i j) =

(A.91)

K [i j]

(A.92)

Four-Dimensional Spacetime The spinor-valued curvature is given by AB AB A CB = R(AB) , Rμν μν = 2 ∂[μ ων] + ωC[μ ων] μ

A μ

AB AB R = e A A e BA ν Rμν + e A e AB ν Rμν .

(A.93) (A.94)

p [i j] and additional terms in p ⊥i will appear in the Lorentz constraint Jab ↔ J AB ε A B + J A B ε AB .

20

A.4

Useful Expressions

229

Spatial Representation The components of the 3D curvature in terms of the spin connection read (3) AB Ri j

(3) A B Ri j

= 2 ∂[i (3)ω AB j] +

(3) A (3) C B ωC[i ω j]

= 2 ∂[i (3)ω Aj] B +

,

(3) A (3) C B

ωC [i ω j]

.

(A.95)

Because of the symmetry of (3)ωi[AB] = 0 and (3)ωi[A B ] = 0, the chosen notation (3)ω A and (3)ω A is unambiguous. The horizontal position of the indices does not Bi B i need to be fixed. The scalar curvature is given by (3)

j

R = eiA A e B B

(3) AB A B

Ri j ε

+

(3) A B AB

R

ε

.

(A.96)

The same procedure performed on (3)sωiA A B B leads to the torsion-free scalar curvature: (3s) AB (3)s A (3)s C B Ri j = 2 ∂[i (3)sω AB + ω ω , j] C[i j]

(3s) A B Ri j

= 2 ∂[i (3s)ω Aj] B +

(3s) A (3s) C B

ωC [i ω j]

,

(A.97)

.

(A.98)

and (3s)

j

R = eiA A e B B

(3s) AB A B

Ri j ε

+

(3s) A B AB

R

ε

A.4.6 Decomposition with Four-Component Spinors We now present, in a somewhat summarized manner, a few helpful formulas for the 3 + 1 decomposition of a Cartan–Sciama–Kibble (CSK) theory, and in particular, Einstein gravity with torsion (see Chap. 2 of Vol. I), using 4-component spinors [15]: • The torsion is given by ξμν λ =

1 Γμν λ − Γνμ λ , 2

(A.99)

where eλ are basis vectors, which we will take as a coordinate basis with ei tangent to a spacelike hypersurface and the normal n with components n → n μ = (−N , 0, 0, 0), and where the components of the metric are as in (2.6), (2.7), and (2.8) of Vol. I. Then we can write

230


eμ;ν = eλ Γμν λ ,

(A.100)

(0)λ Γμν λ = Γμν − κμν λ ,

(A.101)

κμν λ = ξμν λ − ξμ λ ν + ξ λ μν ,

(A.102)

(0)λ

with Γμν the Christoffel symbols and κμν λ the contorsion tensor. • The extrinsic curvature K i j can then be computed and retrieved from n; j by [see (2.9) of Vol. I] 1 −h ji,0 + N j|i + Ni| j − κ ji⊥ , (A.103) 2N 1 (A.104) −h ji,0 + N j|i + N j|i + τ( ji) , K ( ji) = 2N τ ji ≡ 2ξ j⊥i , (A.105) K [ ji ] = ξi j⊥ , K ji = −N

(4)

Γ ji0 =

where | denotes the derivative involving the Christoffel symbols for h i j . • The curvature is then obtained from (4) (4)

(3) R − K i j K i j + K 2 − 2(4) R⊥ α ⊥α , γ β = n γ n (α − n β n (γ − K i j K i j + K 2 1 − 2ξ αβ ⊥ n α(β .

R=

R⊥ α ⊥α

(β

(A.106) (A.107)

• Finally, or almost, the Lagrangian density is √

hL = N h

1/2

(3)

R − Ki j K

ij

+K −2 2

β n γ n (α

γ − n β n (γ (β

+ 4ξ

αβ

⊥ n α(β

,

(A.108) from which the first two lines of (4.17) of Vol. I are obtained. From the above, the Hamiltonian and constraints are21 Hm = −2h mi π ik|k ,

21

≡

(A.109)

− , (A.110)

1 (A.111) H⊥ ∼ h −1/2 π i j πi j − π 2 − h 1/2 (3)R + h 1/2 τ (i j) τ(i j) − τ 2 2 √ √ + hξi j⊥ ξ i j ⊥ − 2h 1/2 τ[i j] ξ ji ⊥ + hq k ρk − 2h 1/2 ρ||i i + 2h 1/2 ρ i ρi ,

J

ab

p ka ebk

p kb eak

The reader should notice that, with the above alone, i.e., no gravitino (matter) action, we have P μνλ = 0 for the conjugate to torsion ξμνλ , whose conservation leads to ξμνλ = 0. But if the Rarita–Schwinger field is present in an extended action, with Lagrangian terms, e.g., ελμνρ ψ λ γ5 γμ Dν ψρ , then 1 ξμνλ = − ψ μ γλ ψν , 4 and torsion cannot then be ignored.

A.4

Useful Expressions

231

H = N H⊥ + N i Hi + Mab J ab .

(A.112)

A.4.7 Equations Used in Chap. 4

We employ several derivatives of functionals with respect to the tetrad e AB [17]. j Two of them are: ε4 ilmn A A

δ

δe AB j

(D BBm j D CA kl )

(A.113)

and n AA

δ

δe AB j

DiBj B .

(A.114)

First we need an explicit form for δn A A /δe Bj B . With (4.109) of Vol. I, which

expresses n A A in terms of the tetrad, the relation n A A e A A i = 0 implies 0=e

δn A A e A A i

CC i

δe Bj B

=n

CC

n A A

δn A A

δe Bj B

− εA εA C

C δn

A A

+ eCC j n B B .

δe Bj B

(A.115) In addition, we have δn A A δe Bj B

=

δn CC n CC n A A δe Bj B

= 2n CC n

A A δn

A A

δe Bj B

+

δn A A δe Bj B

,

(A.116)

whence δn A A

= e A A j n B B .

δe Bj B

(A.117)

We then use the derivative of the determinant h of the three-metric: ∂h = hi j h . ∂h i j

(A.118)

Hence, δh δeiA A

= −2heiA A .

(A.119)

Using this and (A.48), (A.49), (A.50), (A.51), and (A.52), we can calculate the expressions

232


δ

n A A εilm

δe AB j

(D BBm j D CA kl )

= −4n A A εilm

δ

1 D

e e D D m n D B elC E e E E k n E A

h Bj

δe AB j

i 2i 1 1

= ε B C δki √ + √ 2eC B i e B B k + eiB B ekC B 1−1+ − 2 2 h h −3i

= √ δki ε B C − 2h i j ε jkl n C B elB B , h

(A.120)

and n

A A

δ

B B

Di j AB δe j

= −2in

A A

δ δe AB j

1

C B

√ e BC j eCC i n h

2i

= − √ n A A n BC e AC i . h

(A.121)

Finally, we can write a transformation rule to express derivatives in terms of the

tetrad eiA A as derivatives in terms of the three-metric h i j . Let F[e] be a functional depending on the tetrad, and note that h i j can be expressed in terms of the

tetrad, since we have the relation h i j = −eiA A e A A j . Moreover, there is of course no inverse relation. We thus restrict the functional F by demanding that it can be written in the form F[h i j ]. Consequently, using the chain rule, we find for the transformation of the functional derivatives: B B CC

δF δeiA A

δe j ek δF δh jk δF = ε BC ε B C

= −

A A δh jk δei δh jk δeiA A =−

δF δF

ε AC ε A C ekCC − ε B A ε B A e Bj B δh ik δh ji

= −2

δF e A A j . δh i j

(A.122)

Using e A A i e A A j = −δ ij , the inverse relation for an arbitrary functional G[h i j ] is simply δG 1 δG

= e A A j A A . δh i j 2 δei Note that it is always possible to rewrite G[h i j ] in the form G[e].

(A.123)

References

233

References 1. Misner, C.W., Thorne, K.S., J.A. Wheeler: Gravitation, 1279pp. Freeman, San Francisco (1973) 212 2. Ortin, T.: Gravity and Strings. Cambridge University Press, Cambridge (2004) 212 3. Bailin, D., Love, A.: Supersymmetric Gauge Field Theory and String Theory. Graduate Student Series in Physics, 322pp. IOP, Bristol (1994) 213, 214, 219 4. Bilal, A.: Introduction to supersymmetry. hep-th/0101055 (2001) 213 5. Muller-Kirsten, H.J.W., Wiedemann, A.: Supersymmetry: An Introduction with Conceptual and Calculational Details, pp. 1–586. World Scientific (1987). Print-86–0955 (Kaiserslautern) (1986) 213, 214, 219 6. D’Eath, P.D.: Supersymmetric Quantum Cosmology, 252pp. Cambridge University Press, Cambridge (1996) 214, 217, 223 7. Penrose, R., Rindler, W.: Spinors and Space–Time. 1. Two-Spinor Calculus and Relativistic Fields. Cambridge Monographs on Mathematical Physics, 458pp. Cambridge University Press, Cambridge (1984) 214, 217, 222 8. Penrose, R., Rindler, W.: Spinors and Space–Time. 2. Spinor and Twistor Methods in Space– Time Geometry. Cambridge Monographs on Mathematical Physics, 501pp. Cambridge University Press, Cambridge (1986) 214, 217, 222 9. Wess, J., Bagger, J.: Supersymmetry and Supergravity, 259pp. Princeton University Press, Princeton, NJ (1992) 214, 217, 220, 222 10. Martin, S.P.: A supersymmetry primer. hep-ph/9709356 (1997) 214 11. West, P.C.: Introduction to Supersymmetry and Supergravity, 425pp. World Scientific, Singapore (1990) 214 12. Srivastava, P.P.: Supersymmetry, Superfields, and Supergravity: An Introduction. Graduate Student Series in Physics, 162pp. Hilger, Bristol (1986) 214 13. Van Nieuwenhuizen, P.: Supergravity. Phys. Rep. 68, 189–398 (1981) 214, 225 14. D’Eath, P.D.: The canonical quantization of supergravity. Phys. Rev. D 29, 2199 (1984) 217 15. Pilati, M.: The canonical formulation of supergravity. Nucl. Phys. B 132, 138 (1978) 221, 229 16. Nilles, H.P.: Supersymmetry, supergravity and particle physics. Phys. Rep. 110, 1 (1984) 222 17. Kiefer, C., Luck, T., Moniz, P.: The semiclassical approximation to supersymmetric quantum gravity. Phys. Rev. D 72, 045006 (2005)

224, 231

Appendix B

Solutions

Problems of Chap. 2 2.1 The Born–Oppenheimer Approximation and Gravitation The Born–Oppenheimer approximation [1–4] was first developed for molecular physics. It was then extended to quantum gravity. In brief, in the form of a recipe, we have the following: • We use the simple Hamiltonian H=

p2 P2 P2 + + V (R, r ) ≡ +h , 2M 2m 2M

with M m. • R, r denote variables for heavy (M) and light (m) particles, respectively. • The purpose is to solve (approximately) the stationary Schrödinger equation H Ψ = EΨ . Then: – Assume (and this is one of the main ingredients!) that the spectrum (in terms of eigenvalues and eigenfunctions) of the light particle is known for each configuration R of the heavy particle, , i.e., h|n; R = ε(R)|n; R. – Carry out the expansion Ψ = k k (R)|k; R. • Returning to the Schrödinger equation, after multiplying by n; R|, obtain an equation for the wave functions n :

h¯ 2 2 ∇ + εn (R) − E n (R) − 2M R =

(B.1)

" h¯ 2 h¯ 2 ! n; R|∇ R k; R ∇ R k (R) + n; R|∇ R2 k; R k (R) . M 2M k

k

235

236

B Soluitions

• Now neglect the off-diagonal terms1 in (B.1): " ! 1 ¯ 2 A2 h¯ 2 2 h 2 [−ih¯ ∇ − h¯ A(R)] − − n;R|∇ R n;R + εn (R) n (R) = En (R). 2M 2M 2M (B.2)

The reader should note that the momentum of the slow particle has been shifted: P → P − h¯ A, where A(R) ≡ −in; R|∇ R n; R. This constitutes a Berry connection.2 Concerning the application of the Born–Oppenheimer expansion to gravitation, let us add the following: 1. An expansion with respect to the large parameter M leads to satisfactory results if the relevant mass scales of non-gravitational fields are much smaller than the Planck mass. 2. Regarding the non-gravitational fields: a. Without them, an M expansion is fully equivalent to an h¯ expansion, i.e., to the usual WKB expansion for the gravitational field. b. If non-gravitational fields are present, the M expansion is analogous to a Born–Oppenheimer expansion: large (nuclear) mass → M, small (electron) mass → mass scale of the non-gravitational field. 2.2 Hamilton–Jacobi Equation and Emergence of ‘Classical’ Spacetime If a solution S0 to the Hamilton–Jacobi equation is known, we can extract πij = M

δS0 , δh i j

(B.3)

from which h˙ i j is recovered in the form h˙ i j = −2N K i j + Ni| j + N j|i ,

(B.4)

with K i j the components of the extrinsic curvature of 3-space, viz., K i j = −16π GGi jkl π kl .

(B.5)

Note that a choice must be made for the lapse function and the shift vector in order for h˙ ab to be specified. Once this and an ‘initial’ 3-geometry have been chosen, (B.4) 1

Neglecting the off-diagonal terms corresponds to a situation in which an interaction with environmental degrees of freedom allows the k (R)|k; R to decohere from each other (see Sects. 2.2.2 and 2.2.3). 2

If the fast particle eigenfunctions are complex, the connection A is nonzero.

Problems of Chap. 2

237

can then be integrated (for a thorough explanation see [5]). A ‘complete’ spacetime will follow, associated with a chosen foliation and a choice of coordinates on each member of this foliation [1]: • Combine all the trajectories in superspace which describe the same spacetime into a sheaf [6]. • The lapse and shift can be chosen in such a way that these curves comprise a sheaf of geodesics. • Superposing expressions in the form of WKB solutions (bearing the gravitational field), one specific trajectory in configuration space, i.e., one specific spacetime, will be described by means of a wave packet. However, there is another point to note here. One must also discuss3 an expansion of the momentum constraints in powers of M. In fact, S0 satisfies the three equations

δS0 δh ab

|b

=0.

(B.6)

The Hamilton–Jacobi equation and (B.6) are equivalent to the G 00 and G 0i parts of the Einstein equations, while the remaining six field equations can be found by differentiating (B.4) with respect to t and eliminating S0 by making use of (B.4) and the Hamilton–Jacobi equation. 2.3 Factor Ordering and Semi-Classical Gravity The terms in (2.19) are independent of the factor ordering chosen for the gravitational kinetic term. Any ambiguity in the factor ordering would have to come from the gravity–matter coupling. But let us elaborate on this [1–4]. Let us introduce a linear derivative term (parametrized by ζi j ) that represents an ambiguity in the factor ordering of the kinetic term. This will then appear as we ‘enter the range’ of the Schrödinger equation. In the Wheeler–DeWitt equation, we now have: δ2 δ h¯ 2 g m + ζi j + MU + H⊥ Ψ = 0 . Gi jkl (B.7) − 2M δh i j δh jk δh i j Note that (2.11) can also be written as Gi jkl

δS0 δK δ2 S0 1 − Gi jkl K=0, δh i j δh kl 2 δh i j δh kl

(B.8)

but with the linear term we have instead 3 Equations (B.6) will follow from the Hamilton–Jacobi equation and the Poisson bracket relations between the constraints H and Hi [7].

238

B Soluitions

δS0 δK δ2 S0 δS0 1 Gi jkl − + ζi j Gi jkl K=0. δh i j δh kl 2 δh i j δh kl δh i j

(B.9)

At order M0 an equation also involving S1 is δS0 δS1 ih¯ − Gi jkl δh i j δh kl 2

δ2 S0 Gi jkl δh i j δh kl

+ Hm = 0 ,

(B.10)

but with the linear term we can write Gi jkl

δS0 δS1 ih¯ − δh i j δh kl 2

Gi jkl

δ2 S0 δS0 + ζi j δh i j δh kl δh i j

+ Hm = 0 .

(B.11)

With regard to (2.15), we now get δS0 δS2 δS1 δS1 1 ih¯ 0 = Gi jkl + Gi jkl − δh i j δh kl 2 δh i j δh kl 2

2 ih¯ δ S2 1 δS1 δS2 − . +√ 2 δφ 2 h δφ δφ

δ2 S1 δS1 + ζi j Gi jkl δh i j δh kl δh i j

(B.12)

If we now further require σ˘ 2 to satisfy

δS0 δσ˘ 2 δK δD δ2 K δK h¯ 2 h¯ 2 Gi jkl − 2 Gi jkl + + ζi j Gi jkl =0, δh i j δh kl δh i j δh kl 2K δh i j δh kl δh i j K we obtain Gi jkl

δS0 δη˘ h¯ 2 = δh i j δh kl 2F

−

δF δK δ2 F δF 2 + Gi jkl + ζi j Gi jkl K δh i j δh kl δh i j δh kl δh i j

ih¯ δ2 η˘ ih¯ δη˘ δF . + √ +√ F h δφ δφ 2 h δφ 2

(B.13)

Hence, the Schrödinger equation with quantum gravitational corrections, including the linear term, is now h2 δΘ m Θ+ = H⊥ Gi jkl ih¯ δτ MF

δF 1 δ2 F 1 1 δK δF − − ζi j K δh i j δh kl 2 δh i j δh kl 2 δh i j

Θ . (B.14)

We decompose derivatives of F into normal and tangential components to the constant S0 hypersurfaces. In more detail (see also Sect. 4.2.6), Gi jkl

δF δF op i ⊥ = − Gi jkl U i j Hm F ⊕ Gmnop $ $kl . δh i j δh mn h¯

(B.15)

Problems of Chap. 2

239

The first term on the right-hand side is the normal component, with

Uij ≡

δS0 δh i j

Gi jkl

δS0 δS0 δh i j δh kl

−1

=−

1 δS0 . 2U δh i j

(B.16)

The second term is the tangential component and $i j is a unit vector tangent to the S0 constant hypersurface, satisfying U i j $i j = 0. Here we use Gmnop

δF op $ $kl ≡ amn . δh mn

We then write (see Sect. 4.2.6), using (B.14), h¯ 2 2MF

−Gi jkl

δ2 F δK δF δF 2 + Gi jkl − ζi j δh i j δh kl K δh i j δh kl δh i j

Θ ≡ Cn ⊕ Ct , (B.17)

where we have

⊥ δS0 1 δU ⊥ δHm ⊥ 2 Cn ≡ − + H F Θ (Hm ) F − ih¯ Gi jkl −F 4MU F δh kl δh i j δh i j m

⊥ δHm 1 δU ⊥ ⊥ 2 =− (Hm ) F − ih¯ −F + H F Θ. 4MU F δτ U δτ m

(B.18)

The reader should note that it is due to the conservation law that there are no ambiguities here, as all terms cancel. Moreover, h¯ 2 Ct ≡ − 4MU F δ − δh kl

2 δK K δh kl

δGi jkl δF op δF op i j $ $ Gmnop $kl + Gmnop $ δh mn h kl δh mn

δF op δF op i j $ $ $ Θ. Gmnop $kl − ζi j Gmnop δh mn δh mn

(B.19)

For the normal components [1], 4π G δ δΘ m m 2 Θ+√ ) Θ + ih¯ 4π G = H⊥ ih¯ (H⊥ δτ δτ h (3) R − 2Λ

m H⊥

√ Θ. h (3) R − 2Λ (B.20)

240

B Soluitions

2.4 Quantum Gravity Corrections and Unitarity vs. Non-Unitarity The second correction term in (2.19) is pure imaginary. This may lead to a ‘complex’ situation. From the Wheeler–DeWitt equation, we get [1, 3] ⎞ ⎛ ⎞ ⎛ ↔ ↔ 1 1 δ ⎝ ∗ δ δ δ ⎝Ψ ∗ Ψ Ψ⎠ + √ Gi jkl Ψ⎠ = 0 . M δh i j δh kl δφ h δφ

(B.21)

Applying the M expansion, at the level of the corrected Schrödinger equation we obtain ⎛ ⎞ ↔ δ δ h δ ¯ ⎝Θ ∗ 0= (Θ ∗ Θ) + √ Θ⎠ δτ δφ 2i h δφ ⎞ ⎤ ⎡ ⎛ ↔ 1 δ δ δ δK ih¯ ⎝Θ ∗ ψ⎠ − Θ∗ Θ⎦ . Gi jkl ⎣ − 2M δh i j δh kl K δh i j δh kl

(B.22)

It is the term between square brackets that is of interest, being proportional to M−1 . Functionally integrating this equation over the field φ and assuming that Θ → 0 for large field configurations: d dt

DφΘ ∗ Θ = 8π G

Dφ

δ d3 x ψ ∗ δτ

m H⊥

√ ψ. h (3) R − 2Λ

(B.23)

A violation of the Schrödinger conservation law thus comes from the abovementioned term in (2.19). 2.5 De Sitter Space and Quantum Gravity Corrections The following is a summary of a rather more detailed explanation presented in [3]: • Apply a conformal transformation and write the 3-metric as h i j = h 1/3 h˜ i j . • The Hamilton–Jacobi equation becomes √

√ 3 h δS0 2 h˜ ik h˜ jl δS0 δS0 + √ − 2 h (3) R − 2Λ = 0 . − √ 16 δ h 2 h δh˜ i j δh˜ kl

(B.24)

√ • Set (3) R = 0 and look for a solution of the form S0 = S0 ( h), viz., 7 S0 = ±8

Λ 3

√

h d3 x ≡ ±8H0

√

h d3 x .

(B.25)

Problems of Chap. 2

241

• The local time parameter4 in configuration space is √ √ δ 3 h δS0 δ δ =− √ √ = 3hΛ √ . δτ 8 δ hδ h δ h

(B.26)

• From the conservation law ‘chosen’ for K, it follows that δK/δτ = 0, i.e., it is constant. Then the functional Schrödinger equation is (setting h¯ = 1) iΘ˙ =

1 δ2 a a3 2 2 2 d x − 3 2 + (∇φ) + m φ Θ . 2 2 2a δφ 3

(B.27)

• Use the Gaussian ansatz5 1 ˇ Θ ≡ Nˇ (t) exp − dk˜ Ω(k, t)χˇ k χˇ −k . 2 Two equations follow, but the relevant one is6 y

+ 2

a

y + (m 2 a 2 + k 2 )y = 0 , a

(B.28)

˜ a where Ωˇ ≡ −ia 3 y˙ /y, and primes denote differentiation with respect to η, conformal time coordinate.7 • To the next order, there are corrections to the Schrödinger equation: ih¯

2π G 2π G δ δΘ m m 2 Θ − √ (H⊥ ) Θ − ih¯ = H⊥ δτ Λ δτ hΛ

m H⊥ √ h

Θ.

(B.29)

The second correction term is a source of non-unitarity. It produces an imaginary contribution to the energy density of the order of magnitude 29G h¯ 2 H03 /480π V c5 (reinstating c), indicating a possible instability of the system whose associated 3 time scale is H0−1 /tPl . The first correction term in (B.29) causes a shift8 √ δ h(x) d y = 3H0 a 3 −→ a(t) = e H0 t . δτ (y)3 d k ikx 5 The momentum representation φ(x) = ˜ k eikx is employed. Gausϕ(k)e ≡ dkϕ (2π )3 sian states are used to describe generalised vacuum states. The special form here is due to the Hamiltonian being quadratic, and the Gaussian form is preserved in time.

2 2 ˙ˇ = Ωˇ − a 3 m 2 + k 6 Proceeding from iΩ . a3 a2 1 7 We have dt = adη˜ −→ a(η) , η˜ ∈ (0, −∞) . ˜ =− H0 η˜ 8 The prediction can be made more concrete through the following elements: 4

∂a 3 This allows one to recover ≡ ∂t

3

242

B Soluitions

!

" ! " ! " 2π G ! ⊥ 2 " 841 ⊥ ⊥ ⊥ ) G h¯ H06 a 3 . Hm −→ Hm − (H −→ Hm − m 2 3 1382400π 3 3a H0 (B.30)

2.6 The Bunch–Davies Vacuum and Quantum Cosmology Take the simplified Gaussian ansatz (see above) ˇ f n2 /2 ˜ = Nˇ (t)e−Ω(t) ,

(B.31)

and insert it in (2.33) to get equations for Nˇ and Ωˇ : N˙ˇ Ωˇ = 3 , 2a Nˇ

(B.32)

n2 Ωˇ 2 iΩ˙ˇ = 3 − a 3 m 2 + 2 . a a

(B.33)

i

Using Ωˇ ≡ −ia 3

y˙ y

≡ −ia 2 , y y

(B.34)

and the conformal time η, with dt = adη, we get 2 y − y + η

m2 2 +n y =0, H 2 η2

(B.35)

where we take a(η) = −1/H η. We can obtain [3]

( ⊥ 2) 2 (x), i.e., a four-point correlation function. – Note that (Hm ) is divergent, rather like Tμν – But use the Bunch–Davies vacuum state (an adiabatic vacuum state) to compute the expectation value [8]. The Bunch–Davies vacuum arises as a particular solution of (B.28). It √ is de Sitter SO(3,1) invariant, reducing to the Minkowski vacuum at early times, i.e., Ωˇ → k 2 + m 2 as ˙ˇ Then t → −∞, where the metric is essentially static and one can put Ω˙ = 0 and a = 1 in iΩ. !

"

⊥ Hm

29h¯ H04 a 3 . 960π 2

The fluctuation of the Bunch–Davies state will be (quantum mechanically) zero, and ! " ! "2 ⊥ 2 ⊥ (H m ) ≈ Hm .

Problems of Chap. 3

Ωˇ −

η2 H 2

243

i i −→ −

2 Z ν−1 3 − 2ν Z ν−1 m 2 2 +n η H +n 2η Zν Zν 3ηH 2 −→

n2a2 m2a3 (n + ia H ) + i , (B.36) 3H n2 + a2 H 2

where we have used the usual inflationary limit m 2 /H 2 9/4, ν 3/2−m 2 /3H 2 , not choosing a real Bessel function for Z so that the Gaussian can be normalized. We take complex Hankel functions. If we choose a de Sitter invariant, i.e., invariant ˇ under √ SO(3,1), this reduces to the Minkowski vacuum at early times, i.e., Ω tends 2 2 to n + m as t → −∞, where the metric is essentially static and one can put Ω˙ˇ = 0 and a = 1. This is the Bunch–Davies vacuum.

Problems of Chap. 3 3.1 SUSY Breaking Conditions Take a state | f for which (see Chap. 3 of Vol. I) f | P0 | f ∼

B2 1B 1 B B 'S A | f '2 + BS †A | f B ≥ 0 . 4 4

(B.37)

If | f is invariant under all SUSY generators, i.e., S A | f = 0, then necessarily f | P0 | f = 0. Conversely, if f | P0 | f > 0, not all S A and S †A can annihilate the state | f . This non-invariance of the vacuum state implies that a spontaneous breaking of the (super)symmetry has occurred. 3.2 No-Scale SUGRA An interesting Kähler class of potentials are those for which we obtain no-scale SUGRA (extracted from [9–12] within the SUGRA limits of string theory), by which we mean that the effective potential for the hidden sector is flat (i.e., constant), wherein the Planck mass is the only input mass, to fix features such as the hierarchy problem, with subsequent non-gravitational radiative corrections to fix the degeneracy: • A trivial situation is described by G = −3 ln φ + φ ∗ ⇒ V = 0 ,

∀φ ,

where φ is the hidden sector scalar. This simple solution has a simple flat potential (no scale) model, where the potential V vanishes identically, making the vacuum expectation value φ arbitrary, with an arbitrary value for the mass

244

B soluitions

of the gravitino (and the cosmological constant). These models arise naturally in (super)string compactifications, where the size of the compact dimensions is not fixed [13]. Changing them thus costs no energy and a moduli space can be defined. They correspond to Reφ. In a (SUGRA) extension of the standard model, the visible sector (of the chiral multiplets) is connected to a SUSY breaking (hidden) sector. This is all very model dependent. • An extension is G = −3 ln φ + φ ∗ − ϕ r ∗ ϕr + ln |W|2 ,

W ∼ d pqr ϕ p ϕq ϕr ,

where φ is now in the visible sector and ϕ is in the hidden sector. With da ≡ Gr Tar s ϕs , this leads to −2 ∂W 2 1 V ∼ φ + φ ∗ − ϕ r ∗ ϕr ∂ϕ + 2 Re(da db ) , r whose minimum requires ∂W = da = 0 , ∂ϕr

∀ r, a

⇒ Vmin = 0 .

3.3 Finite Energy State Degeneracy On the one hand, from Exercise 3.1, the energy vanishes only if S| f = S † | f = 0. On the other hand, let us take a state f , associated with a given energy. Since S commutes with the Hamiltonian and cannot change the energy of such a state, with S| f = $| f , we have 0 = S 2 | f = $S| f = $2 | f leading to S| f . Hence, if a state | f is unique, it must satisfy S| f = S † | f = 0, have zero energy, and necessarily be the ground state (see Note 3.6 and [14–19]). 3.4 Pair States Notice that the energy eigenvalues of both H1 and H2 are positive semi-definite, (1,2) ≥ 0 [see (3.40) and [20]]: i.e., E n E = ψ|H|ψ ∼ Sψ|Sψ + S † ψ|S † ψ ≥ 0 . The Schrödinger equation for H1 is H1 ψn(1) = A† Aψn(1) = E n(1) ψn(1) .

(B.38)

Multiplying both sides from the left by A, we get H2 Aψn(1) = AA† Aψn(1) = E n(1) Aψn(1) .

(B.39)

Problems of Chap. 3

245

Similarly, the Schrödinger equation for A2 , viz., A2 ψn(2) = AA† ψn(2) = E n(2) ψn(2) ,

(B.40)

H1 A† ψn(2) = A† AA† ψn(2) = E n(2) A† ψn(2) .

(B.41)

implies

We may argue as follows: (1) • If Aψ0 = 0, the argument goes through for all the states including the ground state ,and hence all the eigenstates of the two Hamiltonians are paired, i.e., they are related by

E n(2) = E n(1) > 0 ,

(B.42)

−1/2 ψn(2) = E n(1) Aψn(1) ,

(B.43)

−1/2 A† ψn(2) , ψn(1) = E n(2)

(B.44)

where n = 0, 1, 2, . . .. (1) (1) • If Aψ0 = 0, then E 0 = 0 and this state is unpaired, while all other states of the two Hamiltonians are paired. It is then clear that the eigenvalues and eigenfunctions of the two Hamiltonians H1 and H2 are related by (1) E n(2) = E n+1 ,

E 0(1) = 0 ,

(B.45)

(1) −1/2 (1) Aψn+1 , ψn(2) = E n+1

(B.46)

−1/2 (1) A† ψn(2) , ψn+1 = E n(2)

(B.47)

where n = 0, 1, 2, . . .. Note also the following points: • Adequate state wave functions must vanish at x = ±∞. From the above discus(1) (2) sion and (3.42), one cannot have both9 Aψ0 = 0 and A† ψ0 = 0. Only one of the two ground state energies can be zero. 9

Note that one would have ψ0(2) exp

√

2m h¯

x

W(y)dy

.

(B.48)

246

B Soluitions

• For Aψ0(1) = 0, since the ground state wave function of H1 is annihilated by the operator A, this state has no SUSY partner. Knowing all the eigenfunctions of H1 , we can determine the eigenfunctions of H2 using the operator A. And conversely, using A† , we can reconstruct all the eigenfunctions of H1 from those of H2 except for the ground state. • For (3.42), the normalizable zero energy ground state is associated with H1 , and W is positive (negative) for large positive (negative) x, implying a zero fermion number. • If we cannot find normalizable functions like (3.42) and (B.48), then H1 does not have a zero eigenvalue, and SUSY is broken. For W of the form wx n , if w is positive and n odd, we get a normalizable state, but not the reverse, i.e., for w negative and n even. With W± ≡ limx→±∞ W, if sign(W+ ) = sign(W− ), SUSY is broken, whereas if sign(W+ ) = −sign(W− ), SUSY remains intact, i.e., we can also write Δ=

1 sign(W+ ) − sign(W− ) . 2

• The Witten index does not depend on the details of the superpotential being invariant under deformations that maintain their asymptotic behavior, something that is further discussed in the context of topological invariance. If SUSY remains intact, and if the potential V (x) provides an exactly solvable situation with n bound states and ground state energy E 0 , one can extend to a V1 (x) = V (x) − E 0 setting, whose ground state energy is zero. One can obtain all the n − 1 eigenstates of H2 , then obtain all the n − 2 eigenstates of H3 . In this way, new exactly solvable settings are possible.

Problems of Chap. 4 4.1 N=1 SUGRA Action, SUSY Transformations, and Time Gauge Let us for this discussion rewrite the N = 1 SUGRA as (see [21–23] for more details) 1 d4 x e R , S2 = 2 2k M

S3/2 = S3/2 + S 3/2

(B.49)

Problems of Chap. 4

247

1

d4 x εμνρσ ψ A μ e A A ν Dρ ψ A σ − ψ A μ e A A ν Dρ ψ A σ ,(B.50) 2 M 1 d3 x eK ˆ , (B.51) S2B = 2 k ∂M 1

B d3 x εi jk ψ A i e A A j ψ A k , (B.52) S3/2 = 2 ∂M =

where e = det [ea μ ], eˆ = det [eaˆ i ], aˆ = 1, 2, 3, and the integration is over a spacetime manifold M with boundary ∂M. The boundary terms, labelled by the subscript B, are necessary for dealing with quantum amplitudes of transitions and also to obtain the correct amplitudes, with prescribed data, leading to classical solutions, i.e., with δS = 0 providing the classical field equations [24, 25]. Recall that the curvature scalar R is given in terms of the tetrad field e and the spin connection ωab μ by R(e, ω) = ea μ eb ν R ab μν (ω) ,

(B.53)

that is, R ab μν (ω) = ∂μ ωab ν − ∂ν ωab μ + ωac μ ωc b ν − ωac ν ωc b μ .

(B.54)

The variation is then δS e, ψ, ψ,ω(e, ψ, ψ) = δe

δS δe

δS + δω

ψ,ψ,ω

e,ψ,ψ

+ δψ

δS δS + δψ δψ e,ψ,ω δψ e,ψ,ω

(B.55)

δω(e, ψ, ψ) δω(e, ψ, ψ) δω(e, ψ, ψ) δψ + δe δψ + δψ δe δψ

with the help of the chain rule. It is useful to recall that ωab μ satisfies its own field equations, so one can drop the last term in (B.55) after inserting ωab μ (e, ψ, ψ) into the action. Let us then vary the SUGRA action with respect to the tetrad and the gravitinos. In list form, we have the following points: • Take δe = eδ ln det [ea μ ] = eδ Tr ln [ea μ ] = eea μ δea μ ,

(B.56)

δea ν = −(δeb μ )eb ν ea μ .

(B.57)

where

,

248

B Soluitions

• It follows that δ(e R) = (δe)R + e(δR) = e ea μ (δea μ )R + δ(ea μ eb ν )R ab μν = e (δea μ )ea μ R + 2ea μ (δeb ν )R ab μν = eδea μ (ea μ R − 2ea ν eb μ ec ρ R cb ρν ) .

(B.58)

• In addition, ˆ δ(eK ˆ ) = eˆ δK + eˆ−1 (δe)K ˆ = eˆ ω0aˆ i δeaˆ i + ebˆ j (δeb j )ω0aˆ i eaˆ i ˆ

ˆ (e j e i − e i e j )δeb . = eω ˆ a0 j i aˆ bˆ aˆ bˆ

(B.59)

• Note that we have used the time gauge here, and we are assuming it to be preserved under the SUSY transformations10 (discussed at length in Exercises 5.1–5.3).11 • Moreover12 1

δS 3/2 = d4 x εμνρσ (δψ A μ )e A A ν Dρ ψ A σ + ψ A μ (δe A A ν )Dρ ψ A σ 2 M 1

= d4 x εμνρσ (2k−1 Dμ ε A + Λ A B ψ B μ )e A A ν Dρ ψ A σ 2 M +a volume integral 1

d4 x εμνρσ 2k−1 ∂μ (ε A e A A ν Dρ ψ A σ ) − 2k−1 ε A Dμ (e A A ν Dρ ψ A σ ) = 2 M +a volume integral , 1

d3 x εi jk ε A e A A i D j ψ A k + a volume integral , = k ∂M

(B.60)

1 μνρσ 1

ε [Dρ , Dσ ]ε A = εμνρσ R A B ρσ ε B . 2 2 11 In brief, the usual SUGRA action is extended by the boundary terms, namely (B.52). The new

feature are also the Λ BA and Λ BA terms in (B.60) and (B.61), which correspond to the Lorentz transformation terms. 10 ε μνρσ D

12

ρ Dσ ε

A

=

Dμ agrees with ∂μ when it acts on objects with no free spinor indices.

Problems of Chap. 4

δS 3/2 = − =−

1 2 1 2

M

249

d4 x εμνρσ ψ A μ (δe A A ν )Dρ ψ A σ + ψ A μ e A A ν Dρ (δψ A σ )

M

d4 x εμνρσ ψ A μ e A A ν Dρ (2k−1 Dσ ε A + Λ A

B ψ

B

σ)

+a volume integral 1

d4 x εμνρσ ψ A μ e A A ν Dρ (Λ A B ψ B σ ) + a volume integral =− 2 M 1

=− d4 x εμνρσ ∂ρ (ψ A μ e A A ν Λ A B ψ B σ ) − Dρ (ψ A μ e A A ν )Λ A B ψ B σ 2 M +a volume integral 1

d3 x εi jk ψ A i e A A j Λ A B ψ B k + a volume integral . =− 2 ∂M

(B.61)

• Let us address two questions the reader may be wondering about: – Note that we are taking left-handed SUSY transformations [25], i.e., ε = 0, which could be the setup for the amplitude to go from prescribed data

(eiA A , ψ iA )I on an initial surface to data (eiA A , ψ Ai )F on a final surface.

The gravitinos filling in between (eiA A , ψ iA )I on an initial surface and data

μ (eiA A , ψ Ai )F on a final surface would now be independent of (ψ A , ψ Aν ). – With n μ the future-pointing unit vector normal to the boundary ∂M, if we assume13 that the three spacelike axes of the tetrad lie in ∂M, so that n a = eaμ n μ = (1, 0, 0, 0) ,

(B.62)

the extrinsic curvature has the simple form K = h i j n a ωab j ebi = ω0aˆ i eaˆ i .

(B.63)

The corresponding action of N = 1 SUGRA is subsequently invariant, up to a boundary term, if local supersymmetry transformations become

δe A A μ ≡ −ik(ε A ψ A μ + ε A ψ A μ ) + Λ A B e B A μ + Λ A δψ A μ ≡ 2k−1 Dμ ε A + Λ A B ψ B μ ,

δψ A μ ≡ 2k−1 Dμ ε A + Λ A

13

B ψ

This gauge choice is known as the time gauge.

B

μ

B e

AB

μ

,(B.64) (B.65)

,

(B.66)

250

B Soluitions

where

Λ A B ≡ −2ikn A A n CC εC ψ C i e B A i ,

(B.67)

and its Hermitian conjugate ΛA

B

≡ −2ikn A A n CC εC ψ C i e AB i ,

(B.68)

and where ε A and ε A are spacetime-dependent anticommuting fields. The inclusion of the Lorentz terms in (B.64) ensures the preservation of the gauge defined in (B.62) (see Exercises 5.1–5.3). The reader should note that the variation of the tetrad does not depend on Dμ ε or Dμ ε [see (B.64)]. Thus the volume Einsten–Hilbert sector does not involve any derivatives of ε or ε, and so cannot be integrated by parts to give a surface term. Hence, since the theory is supersymmetric, its variation must cancel with the volume terms arising from the variation of the gravitino action. From the remarks following (B.64), we are only interested in surface terms, that is, we will integrate by parts to extract all surface terms in δS3/2 which involve Dμ ε or Dμ ε. Collecting equations (B.59), (B.60), (B.61), with the volume integral terms vanishing in this SUSY context, the total variation under these local supersymmetry transformations is 1 1

d3 x εi jk ε A e A A i (D j ψ A k ) − d3 x εi jk Λ A B ψ B i e A A j ψ A k δS = k ∂M 2 ∂M 1 ˆ (e j e i − e i e j )δebˆ , + 2 d3 x h 1/2 ωa0 (B.69) j i aˆ bˆ aˆ bˆ k ∂M provided that the gauge condition (B.62) is preserved. Finally, note the generic expression [25] retrieved under left-handed SUSY transformations: 2

d3 x εi jk ε A e A A i (3s) D j ψ A k , (B.70) δS = k ∂M where (3s) D j is the torsion-free spatial covariant derivative. 4.2 S0 Must Depend on the Gravitino ψ A i With the SUSY constraints and the ansatz Ψ = exp i(S0 G −1 + S1 + S2 G + · · · )/h¯ , obtain to lowest order G 0 ,

Problems of Chap. 4

251

[Ψ ]−1 S A Ψ

O(G 0 )

=

εi jk e A A i (3s) D j ψkA + 4π iψiA

δS0

= 0 . δe A A i

(B.71)

This must hold for arbitrary fields ψiA and eiA A . Then S0 must at least depend on

eiA A . Otherwise we would get the condition εi jk e A A i (3s) D j ψkA = 0, which cannot hold for all fields. Assume then that S0 does not depend on the gravitino field ψiA . Integrating (B.71)

over space with an arbitrary continuous spinorial test function ε A (x) leads to

I0 ≡

3

d xε

A

ε

i jk

e A A i

(3s)

D j ψkA

+ 4π iψiA

δS0 δeiA A

=0.

(B.72)

With the replacement ψiA & → ψiA exp [φ(x)] and ε A (x) & → ε A (x) exp [−φ(x)], this becomes [26] I0 ≡

d3 xε

A

exp(−φ) i jk e A A i (3s) D j ψkA exp φ + 4π iψiA

δS0 δeiA A

=0.

exp φ (B.73)

Now for "I0 ≡ I0 − I0 = 0, "I0 =

d3 x εi jk eiA A (x)ε A (x)ψ Ak (x)∂ j φ(x) = 0 ,

(B.74)

which cannot hold for all fields. To higher orders, the calculation for n ≥ 1 leads to [Ψ ]−1 S A Ψ

O(G n )

=

−4πiG n ψiA

δSn

= 0 . δe A A i

(B.75)

So can we choose Sn not to depend on the bosonic field eiA A ? This would be very restrictive: no proper bosonic limit would exist. We therefore dismiss this hypothesis. Hence, to satisfy (B.75), it is necessary to introduce a dependence on the gravitino field at each order. This means that we must have S n ≡ S n [e, ψ] for all n [27]. 4.3 Obtaining Equation (4.13) Write the fermionic part of the last line as

252

B Soluitions

1 − ih¯ 2

(3s)

Di ε

i jk

ψAj D

B

A lk

δ δψlB

1 = εi jk (3s) Di (ψ A j ψ A k ) 2 - . 1 = εi jk (3s) Di ψ A j ψ A k + ψ A j (3s) Di ψ A k 2 . 1 i jk - (3s) = ε . D j ψ Ak ψ A i − ψ Ai (3s) D j ψ A k 2 (B.76)

Comparing the above with the third line written in the form . 1 i jk - (3s) D j ψ Ak ψ A i + ψ Ai (3s) D j ψ A k ε , 2

(B.77)

the terms containing (3s) D j (ψ A k ) will cancel out. Finally, the normal projection of the remaining term containing (3s) D j ψkA is given by

n A A ih¯ εi jk

(3s)

δ δ h¯ i jk BC A (3s)

l D j ψ Ak D B A li = e e D ψ . ε √ j Ak C i δψlB δψlB h (B.78)

4.4 Decomposition of the Hamilton–Jacobi Equation into Bosonic and Fermionic Sectors For the WKB wave functional, write

Ψ [e, ψ] ≡ exp

i i B0 G −1 exp (F0 G −1 + S1 + · · · ) . h¯ h¯

(B.79)

Then select the pure bosonic part B0 satisfying

4πin A A DiBj B

δB0

δB0

δe AB j

δeiB A

+U = 0 ,

(B.80)

i.e., corresponding to the Hamilton–Jacobi equation (4.22). This solution B0 can then be used to determine the condition for the part F0 involving fermions: 0 = 4π i ψiB −n

A A

δF0 δe AB j

DiBj B

δF0

AB i E CB jk

δF0

δψkC

δF0

δe AB δeiB A j

+ ψiB

−n

A A

δB0

δe AB j

DiBj B

AB i E CB jk

δF0

δF0 δψkC

δB0

δe AB δeiB A j

δF i

0 + √ i jk eiBC e A C l (3s) D j ψ Ak . δψlB h

−n

A A

DiBj B

δB0

δF0

δe AB δeiB A j

(B.81)

Problems of Chap. 4

253

If a solution S0 of (4.20) is to be found [omitting the term (4.26)], together with F0 = S0 − B0 , then the above equation is satisfied. (s) 4.5 The DeWitt Supermetric Gab Should Contain the DeWitt Metric G i jkl

Take arbitrary ‘vectors’ va ≡

j

B AB

,

FDl

v˜ b ≡

B˜ Bi A

F˜ k

,

(B.82)

C

and obtain (see Chap. 4)

j (s) Gab v a v˜ b = −4πi n A A DiBj B + n B B D Aji A B AB B˜ Bi A

+4πin B B ψ Cj ε jkm DC A mi D D B lk FDl B˜ Bi A

j +4πin A A ψiB εilm D BBm j D CA kl B AB F˜Ck .

(B.83)

Let j

B AB ≡

δa

δe AB j

,

B˜ Bi A ≡

δb δeiB A

,

where a[e] and b[e] are two arbitrary functionals (that can also be written as a[h i j ] and b[h i j ]). Then for the first line on the right-hand side of (B.83) (see Appendix A),

DiBj B

δa

δb

δe AB j

δeiB A

+n

B B

D Aji A

4.6 Towards a Conservation Law? Condition (4.34) can be rewritten as [27]

δa

δb

δe AB j

δeiB A

δa δb . δh jk δh il (B.84) Therefore, for quantities that can be written in terms of the three-metric h i j and the gravitino, the block B contains the usual DeWitt metric. To be more precise, it is not exactly the usual DeWitt metric due to the change of the fundamental bosonic

field from h i j to eiA A . Rather it is a tetrad version of it. 4πi n

A A

= −32πGil jk

254

nAA

B Soluitions

δ δe AB j

δS0 A A δS0 D B B δK DiBj B

K = n

ij B A δei δe AB δeiB A j

−ψiB

δS0 δe AB j

AB i EC B jk

δK

δψkC

− ψiB εilm D BBm j D CA kl

3i δS0 − √ ψkC + ψ B j ε jkl n C B elB B

. h δψkC

δS0 δψkC

K

(B.85)

If the right-hand side of (B.85) is equal to zero, then it can be interpreted as a conservation law (see [1] and Sect. 2.1, and also [28–31, 2, 32, 3, 4, 34]). But here, the physical context is SuperRiem(Σ). A conservation law can only be established in the highly unrealistic and special case of (i) a vanishing dependence of S0 and K on the gravitino and (ii) the assumption that S0 [e] and K[e] can be expressed in the form S0 [h i j ] and K[h i j ]. In fact, (B.85) then implies the conservation equation n

A A

δ δe AB j

DiBj B

δS0 δeiB A

K

−2

=0.

(B.86)

However, as explained in Sect. 4.2.1 (see also Sect. 4.1), S0 ought to depend on the gravitino [27].

Problems of Chap. 5 5.1 Time Gauge and SUSY Transformations The variation of the supergravity action can induce boundary terms. Focusing on extracting a finite quantum amplitude, we aim at exact invariance by adding boundary correction terms. This has so far been impossible for full supergravity, but for homogeneous Bianchi A models, the precise invariance of the action under the subalgebra can be restored by adding an appropriate boundary correction to the action; more precisely, under the subalgebra of left-handed SUSY generators (see [35, 21, 36, 22, 23]). Let us take a non-diagonal Bianchi class A ansatz in the time gauge. In order to preserve this gauge, the transformation laws must be modified by adding a Lorentz generator term to the supersymmetry generators. In fact, the time gauge represents a boundary fixing property (see Exercise 4.1). To preserve it, all bosonic generators must be symmetries of the boundaries, i.e., not induce translations normal to it. Any anticommutator of any fermionic generators should induce bosonic generators tangential to the boundary. Consequently, the tetrad axes must remain within a 3D boundary, whence the gauge condition e0 i = 0 will be preserved under a supersymmetry transformation. The general SUSY transformations will not preserve this gauge. One way forward is through a combination of SUSY transformations added to appropriate Lorentz transformations:

Problems of Chap. 5

255

δe A A μ = −ik ε A ψ A μ + ε A ψ A μ + Λ A B e B A μ + Λ A B e AB μ ,

(B.87)

for some parameters Λ which are linear in ε A and ε A , such that δe0 i = 0 or

n A A δe A A i = 0. A solution is

Λ A B = −2ikn A A n CC εC ψ C i e B A i ,

(B.88)

where we have Λ A A = 0 and Λ AB = Λ B A . Thus the extended supersymmetry transformations are

δe A A μ = −ik(ε A ψ A μ + ε A ψ A μ ) + Λ A B e B A μ + Λ A δψ A μ = 2k−1 Dμ ε A + Λ A B ψ B μ , δψ

A

μ

= 2k−1 Dμ ε

A

+Λ

A

B ψ

B

μ

B e

AB

μ

, (B.89) (B.90)

.

(B.91)

The addition of the Lorentz term to the SUSY transformations subsequently induces new SUSY generators in terms of the Lorentz rotation generators J BC , J B C , given by S A(new) ≡ S A + φ A BC J BC , S A (new) ≡ S A + φ A

B C

(B.92)

J B C ,

(B.93)

where specifically here

φC AB ≡ φC (AB) = ik2 n A A n CC ψ C i e B A i ,

(B.94)

and φ C A B is the Hermitian conjugate of φC AB . 5.2 Variation of N=1 SUGRA Action, Bianchi Models, and Boundary Terms Let us now see what the action of SUGRA (or rather, its variation for left-handed SUSY transformations) becomes for Bianchi A models. The restriction to lefthanded SUSY transformations means also that ε and ε are no longer Hermitian

conjugates of each other: e A A μ is no longer required to be Hermitian and ψ A μ and

ψ A μ become independent quantities [35, 21, 36, 22, 23]. For spatially homogeneous cosmologies, (B.69) takes the form 1 1

d3 x ε pqr ε A e A A p (Dq ψ A r ) − d3x ε pqr Λ A B ψ B p e A A q ψ A r δS = k ∂M 2 ∂M 1 ˆ (e q e p − e p e q )δebˆ , + 2 d3 x h 1/2 ωa0 (B.95) q p aˆ bˆ aˆ bˆ k ∂M

256

B Soluitions

where eaˆ p , e A A p , ψ A p , and ψ A

p

are defined by

eaˆ i ≡ eaˆ p (t)ω p i ,

e A A i ≡ e A A p (t)ω p i ,

ψ A i ≡ ψ A p (t)ω p i ,

ψ A i ≡ ψ A p (t)ω p i ,

(B.96)

(B.97)

with ω˜ p = ω p i dx i . Now, after a rather lengthy calculation using eaμ ≡

N

0

N i eai ˆ eai ˆ

,

we obtain the following: • For the torsion-free spin connection14 ˆ

ˆ

ˆˆ

(s) aˆ bˆ ω

p

ˆ cˆd ≡ 2Q cˆ [b εa] edˆ p − Q cˆˆ εaˆ bd ecˆ p ,

(s) a0 ωˆ

p

≡

(B.98)

d

1 aˆ N q aˆ bˆ ˆ ˆ ˆ ˆ bˆ e p e˙bq (e˙ p + eaq Q εbˆ cˆdˆ ecˆ p ed q − Q bcˆ εaˆ cˆd ebˆ p edq . ˆ )+ ˆ 2N N (B.99)

• For the contorsion,

ˆ ˆ ˆ ˆ br ˆ A ψ A + 1 eaq A ψ A

κ aˆ b p ≡ ik2 e[bq σ a] ψ e e ψ , (B.100) q] r] [p [q AA p AA 2

q ik2 1 aˆ aq ˆ ˆ A ψ A − N σ aˆ A

A

≡ ψ + e n σ κ a0 [p p 0] AA AA A A ψ [ p ψ q] 2 N N 1 ˆ N r aq

+ eaq e A A p ψ A [q ψ A 0] − e ˆ e A A p ψ A [q ψ A r ] . (B.101) N N • Subsequently, (s) AB ω

14

ˆ

p

1 A

ˆ ˆ B A = n A σaˆ iN eaˆ p Q bˆ − 2ebˆ p Q aˆ b + b N ˆ ˆ ˆ ˆ +N q Q aˆ b εbˆ cˆdˆ ecˆ p ed q − Q bcˆ εaˆ cˆd ebˆ p edq ˆ

ˆ

ˆ

Q aˆ b ≡ Q (aˆ b) ≡ eaˆ p m pq eb q (det [eaˆ p ])−1 .

1 aˆ ˆ bˆ e p e˙bq (e˙ p + eaq ˆ ) 2 ,

(B.102)

Problems of Chap. 5

κ AB

p

257

ik2 N

= N eC A q ε(AC ψ A [ p ψ B) q] + e A A q e B A r e D D p ψ D [q ψ D r ] 2N 2

−n (A A ψ A [ p ψ B) 0] − N r eCC p ψ C +eCC p ψ C

C A B A q [q ψ 0] n A e

C A B A q [q ψ r ] n A e

+ N q n (A A ψ A [ p ψ B) q] .

(B.103)

• And last but not least, δeaˆ p = −σ aˆ A A δe A A

p ˆ

A

= ikσ aˆ A A ε ψ A p + kεaˆ bcˆ ebˆ q ecˆ p n A A ε A ψ A q .

(B.104)

• Substituting (B), (B.103), and (B.104) into (B.95), we obtain δS =

ςdet [eaˆ i 4kN i + 2kN

+

p ]ε

A

ψA

s

1 ˆ σaˆ A A ebˆ s Q aˆ b k

e A A r e˙aˆ r eaˆ s + e˙ A A t h ts − 2e A A s e˙aˆ q eaˆ q

ˆ

ˆ

N p σ cˆ A A ebˆ s ed p Q aˆ b εaˆ cˆdˆ

ˆ ˆ +N p σcˆ A A ebs ed p Q aˆ cˆ εaˆ bˆ dˆ

k 1 + n A A χ pqr h pr h qs − e A A q χ pq0 h ps 2 2 k + e A A s χ p0q h pq − e A A p χq0r h s(r h q) p 2N kN p + e A A s χ pqr h qr + e A A r χt pq h s(t h q)r , 2N (B.105) where ς≡

d3 x det [ω p i ]

and

χ pqr ≡ ψ A [ p ψ A q] e A A r ,

χ p0q ≡ ψ A [ p ψ A 0] e A A q ,

χ pq0 ≡ ψ A [ p ψ A q] n A A .

(B.106)

258

B Soluitions

5.3 Invariance of N=1 SUGRA Action, Bianchi Models, and Boundary Terms It is possible to restore the invariance of the action of N = 1 SUGRA restricted to the Bianchi class A setting, under a subalgebra of (left!) SUSY generators [35, 21, 36, 22, 23]. If we add S

B

d ≡i dt

1 Wg + Wψ 2

(B.107)

to the supergravity Lagrangian, where Wg ≡

1 ςm pq h pq , k2

Wψ ≡ iς ε pqr ψ A p ψ A q e A A r ,

(B.108)

then invariance of the N = 1 SUGRA action is restored under left-handed SUSY transformations.

To check this, besides the transformation for e A A p and eaˆ p , we also need [23] δψ A

p

=

2i B A A

ˆ ˆ ε n σaˆ AB eaˆ p Q bˆ − 2ebˆ p Q aˆ b b k 1 B A A

ˆ bˆ − ε n σaˆ AB e˙aˆ p + eaq e p e˙bq ˆ kN 2

ˆ ˆ cˆ aˆ bˆ dˆ − N q ε B n A A σaˆ AB eb p ecˆ q Q aˆ d εdˆbˆ cˆ + ecˆ p ebq ˆ Q dˆ ε kN 1

−ikε B e AC r ε(A C ψ B ) [ p ψ A r] − ikε B e A A r e AB t e D D p ψ D [r ψ D t] 2 ik ik

+ ε B n A (A ψ B ) [ p ψ A 0] − N q ε B n A (A ψ B ) [ p ψ A q] N N ik

ik

− ε B n A A e AB q eCC p ψ C [q ψ C 0] + N q ε B n A A e AB r eCC p ψ C [r ψ C q] N N

−2ikεC n A A n CC ψ C q e AB q ψ B

p

.

(B.109)

The significance of the above is that the existence of a non-trivial, boundarypreserving (sub)algebra allows the theory to be SUSY invariant and admits a Nicolai map.15 But the addition of the boundary term also presents a significant bonus which underlies the (perhaps crucial) developments in Sect. 5.2.3 of Vol. I and Sect. 4.1.1

15 Each bosonic component of the wave function will evolve under supersymmetry according to a type of Fokker–Planck equation. In Bianchi class A cosmologies, for the quantum states that permit Nicolai maps, just two states appear, corresponding to the empty and filled fermion states (see Sects. 3.3 and 3.4).

Problems of Chap. 5

259

of this volume: it simplifies the canonical formulation of the theory compared with the formulation presented in Chap. 4 of Vol. I. In particular, it leads to simplifications in the new Dirac brackets, whereupon the quantization provides a wider range of potential tools [37–40]. 5.4 Quantization of Bianchi Models and Boundary Terms The exact invariance of the N = 1 SUGRA Bianchi A theory under the action of the left-handed generators is due to (B.107). For right-handed SUSY transformations (ε = 0), the correction term would have been the Hermitian conjugate. Note that Wψ is just the fermion number F, and recall that only the Euclidean version of a supersymmetric theory is known to admit a Nicolai map. Hence we take

SEB ≡

SB E

d dτ

d ≡− dτ

1 Wg + Wψ 2

1 Wg + Wψ 2

,

(B.110)

.

(B.111)

With SEB = −S B E , this can be written [23] L −→ L ± ≡ L ±

d dτ

1 1

ςm pq h pq + iς ε pqr ψ A p ψ A q e A A r 2 2 k

.

(B.112)

Now with I is the Euclidean action introduced in Sect. 2.7 of Vol. I, define

1 I± ≡ I ± Wg + Wψ , 2

(B.113)

from which there follows an expression for the new momentum p± A A p : p ± A A p ≡

∂ I± ∂e A A

p

1 2 = −i p A A p ± iς ε pqr ψ A q ψ Ar ∓ 2 ς m pq e A A q . 2 k

(B.114)

Then the following Dirac brackets become possible among the variables e A A p ,

p ± A A p , ψ A p , and ψ A p :

260

B Soluitions

-

eAA p, eB B

-

.

q

e A A p , p± B B q

9 9

p + A A p , p + B B q

,

(B.116)

(B.118)

: 9 A ψ p, ψ Bq D = 0 ,

(B.119)

ψ A p, ψ B

q

:

.

eAA p, ψ B

D

. D

. D

.

q

p + A A p , ψ B q

p − A A p , ψ B

p + A A p , ψ B

9

q B δ p

=0,

e AA p, ψ B q

-

D

-

-

:

= εABεA

(B.117)

ψ A p, ψ A q

9

D

(B.115)

=0,

-

-

.

=0,

D

p − A A p , p − B B q

-

D

D

: D

. q

D

. q

p − A A p , ψ B q

=0, =

(B.120)

1 A A

D pq , σ

(B.121)

=0,

(B.122)

=0,

(B.123)

=0,

(B.124)

=0,

(B.125)

D

: D

= −iε pr s ψ A s D A B rq ,

(B.126)

= iε pr s ψ As D B A qr ,

(B.127)

where (B.117), (B.118), (B.124), and (B.125) are indeed much simpler then the ones without (B.113). The Hamiltonian becomes (see Exercise 5.1)

H ≡ −e A A 0 H A A + ψ A 0 S A(new) + S A (new) ψ A 0

− ω AB0 + ψ C 0 φC AB J AB − ω A B 0 + ψ C 0 φ C A B J A B , (B.128)

Problems of Chap. 5

261

where we now have

J AB = ie(A A p p − B)A p , J A B = ie

A

(A

p +

p

AB ) p

,

(B.129) (B.130)

together with 1

S A (new) = − k2 ψ A p p + A A p + φ A B C J B C , 2 S A(new) =

1 2 A − k ψ p p A A p + φ A BC J BC . 2

(B.131) (B.132)

The reader can indeed confirm that the resulting SUSY constraints become simpler. Note also that, in H A A ≡ −n A A H⊥ + e A A p H p , the form of H A A is complicated, but we can write 2 - (new) (new) . + terms proportional to J and J S A , S A

D k2

k4 1 −

= p AB p p + B A q D B B q p + iε pqr p + B A s D B B sq ψ Ar ψ B p 4 ς

H A A

1

+ ς h 1/2 n B B ψ A p ψ B [q ψ Ap] ψ Bq 2 1

pqr s B B

+ iς ε e B B ψ A r ψ q ψ A[ p ψ s] . (B.133) 2

+iε pqr p − AB s D B B

qs ψ A r ψ

B

p

Quantum mechanically [23], we can obtain an explicit representation of the differential operator16 C S A (new) as ∂ ∂ 1 1

C S A (new) + = h¯ k2 ψ A p − h¯ k2 h 1/2 n C A ψ C q D AC q p , (B.134)

A A AC 2 2 ∂e p ∂e p while the differential operator SCA(new) − is represented by ∂ ∂ 1 1 2 1/2

C A q SCA(new) − = − h¯ k2 ψ A p , (B.135) ¯ k h n AC ψ C q D p

+ h

A A C 2 ∂e p 2 ∂e A p S A (new) ± are corresponding to the ± in the action in (B.113). So SCA(new) ± and C S A (new) by the following transformarelated to the original operators SCA(new) and C tions (see Sect. 3.3): 16 In the following, τ should be considered as a mere parameter, although in a path integral presentation, it would be the Euclidean time.

262

B Soluitions

SCA(new) + ≡ e−Wg /h¯ SCA(new) eWg /h¯ ,

(B.136)

C S A (new) + ≡ e−Wg /h¯ C S A (new) eWg /h¯ ,

(B.137)

SCA(new) − ≡ eWg /h¯ SCA(new) e−Wg /h¯ ,

(B.138)

C S A (new) − ≡ eWg /h¯ C S A (new) e−Wg /h¯ .

(B.139)

The transformed wave functions Ψ + , Ψ − are therefore associated with wave functions Ψ (e, ψ, τ ), Ψ (e, ψ, τ ) by Ψ + (e, ψ, τ ) = e−Wg /h¯ Ψ (e, ψ, τ ) ,

(B.140)

Ψ − (e, ψ, τ ) = eWg /h¯ Ψ (e, ψ, τ ) .

(B.141)

In summary, we may take the (Euclidean) representations of the operators p + A A p ,

ψ A p acting on a wave function Ψ (e A A p , ψ A p ) by means of C p + A A p = −h¯

∂

∂e A A

,

(B.142)

p

∂ C A = h¯ D A A

ψ , p qp ς ∂ψ A q

(B.143)

where

C C S A (new) + = C S A + + φ A B C J B C ,

(B.144)

SCA(new) − = SCA − + φ A BC JCBC ,

(B.145)

∂ 1 C S A + = h¯ k2 ψ A p ,

A 2 ∂e A p

(B.146)

∂ 1

SCA − = − h¯ k2 ψ A p .

A 2 ∂e A p

(B.147)

and

Problems of Chap. 6 6.1 Spinor Symmetry

Since e A A i n A A = 0, it follows that (2h)1/2 ie AA i n B A is symmetric.

Problems of Chap. 6

263

6.2 FRW with Cosmological Constant The (dimensionally) reduced Lagrangian and Hamiltonian can be expressed, after integrating over the spatial coordinates, as L = ωζ ˙ + θ˙A η A − H ω, ζ ; θ A , η A ,

(B.148)

√ γ 2 1 ζ − $ γ θ A θ A H = 4 12Vζ 1/2 iω − ω2 + 12 4 √ γ

+ 4 12Vζ −1/2 η A 2 (i − ω) θ A + η A , 3$

(B.149)

where the gauge A AB0 = ψ A0 = N i = 0, N = 1 was used. The classical evolution is given by [41, 24, 42, 43] γ 2 √ dω 12Vζ 1/2 , = [ω, H ] = dt 3

(B.150)

√ dθ A = −8 12Vζ −1/2 (i − ω) θ A , dt √ dσ = 4 12V −σ 1/2 (i − 2ω) θ A + 2ζ −1/2 η A θ A , dt √ dη 1 A = 4 12Vζ −1/2 $ γ θ − 2 (i − ω) η A , dt 2

(B.151) (B.152) (B.153)

with the relations (from the Hamilton–Jacobi equation) ζ = ηA =

12 ∂Y = 2 ∂ω γ

1 −iω + ω2 + $ γ θ A θ A 4

,

(B.154)

∂Y 6$

= − (i − ω) θ A , ∂θ A γ

Y =−

12 γ 2

ω3 1 iω2 − + $ γ (i − ω) θ A θ A 2 3 4

(B.155) ,

(B.156)

√ where $ , γ are rescalings of Λ = −6Υ 2 and $ = 2/3. A comment is now in order. In (B.150) and (B.152), the presence of the Grassmann variables suggests that the solution process begins with the bosonic sector, i.e., obtaining an anti-de Sitter background, namely, an H 4 hyperboloid, for a negative cosmological constant (see Sect. 5.1.3 of Vol. I). The following step (to the

264

B Soluitions

next order) is to solve (B.151), (B.153) which are linear in fermions, providing the evolution of the gravitino in the background (recall the analysis and discussion in Sect. 4.2). As expected, we consistently insert these results into (B.150), (B.152), thereby obtaining fermionic corrections to ω and ζ . For more details, the reader should consult [24, 44, 42, 43]. 6.3 N=2 Hidden SUSY and Ashtekar’s Variables Another element that emerges is the (hidden) N = 2 supersymmetrization of Bianchi cosmologies straight from bosonic general relativity, using Ashtekar’s new variables (see Chap. 6). In the following, we summarize the main features of [45]. Restricting to a minisuperspace at a pure gravity level, the constraint equations (not with a 2-spinor form) are: J i = Da Eai ,

(B.157)

Cb = Eai Fiab ,

(B.158)

˜k , H = εi jk Eai Ebj R ab

(B.159)

where Riab is the curvature of Aia , and Da the covariant derivative formed with Aia . Restricting to Bianchi cosmologies, we introduce a basis of vectors X ia [see (6.54)– (6.57)], and then expand into new variables: E˜ aj ≡ Eˇ ij X ia ,

(B.160)

ˇ j χi . ˜ aj = A A i a

(B.161) j

Dropping the diacritic to simplify the notation, the quantities Eij and Ai only depend on time. Inserting these substitutions into the constraint equations [41], we obtain17 J i ≡ C k jk Ei j + εi jk Am j Em k ,

(B.162)

j

Hk = −Ei Aim C m jk + εimn Eij A jm Akn ,

(B.163) j

j

H = εi jk C p mn Eim Enj A pk + Eim Enj (Aim An − Ain Am ) .

(B.164)

Moreover, the constraint equations can be written in a more compact and rather interesting fashion. Introducing the variables Q i j ≡ Eik Akj ,

(B.165)

the Hamiltonian constraint for all Bianchi class A models can be written as 17

Further restricting to diagonal models, Eij = diag(E1 , E2 , E3 ) and Aij = diag(A1 , A2 , A3 ).

Problems of Chap. 6

265

H ≡ Q ∗i k Q k i − Q ∗i i Q j j .

(B.166)

All the dynamics of all class A diagonal models can be further summarized by H = Q1 Q2 + Q1 Q3 + Q2 Q1 + Q2 Q3 + Q3 Q1 + Q3 Q2 =G

ij

Qi Q j ,

(B.167)

where i, j = 1, . . . , 3, and ⎡

Gi j

⎤ 0 1 1 1⎢ ⎥ = ⎣1 0 1⎦ . 2 1 1 0

(B.168)

Using the Misner–Ryan parametrization with βi j given by βi j = diag(β+ +

√

3β− , β+ −

√

3β− , −2β+ ) ,

it is possible to find solutions of the form [46]

Ψ [α, β± ] = C[α, β± ] exp −I (α, β± ) . This involves requiring I to be the solution of the Euclidean Hamilton–Jacobi equation for this particular model. The relation with the Ashtekar variables is that, if we have a wave function Ψ A given in terms of the Ashtekar variables [47], we can reconstruct the wave function in terms of the Misner–Ryan variables by choosing / ΨMR = exp(±iI A )Ψ A , where I A is given by I A = 2i Eia Γ i a d3 x, with I A equal to ∓iI for the Bianchi IX case. Γ i a is the corresponding (compatible) spin connection. But where is the (‘hidden’ N = 2) SUSY? The above Hamiltonian suggests (see Chap. 8 of Vol. I) introducing the expressions S ≡ ψ i Qi ,

(B.169)

S ≡ ψ i Q i∗ ,

(B.170)

ψ i ψ j + ψ i ψ j = Gi j ,

(B.171)

for the SUSY constraints, with

and G as given above. The Hamiltonian constraint becomes H=

1 SS + SS . 2

In quantum mechanical terms, one realization is Ψ [A, η], with

(B.172)

266

B Soluitions

Ei Ψ [A] =

∂ Ψ [A] , ∂Ai

(B.173)

Ai Ψ [A] = Ai Ψ [A] ,

(B.174)

i

together with ψ i = ηi , ψ = G i j ∂/∂η j . The equations become SΨ [A, η] = Q i ηi Ψ [A, η] ,

(B.175)

∂ SΨ [A, η] = Qˆ i∗ i Ψ [A, η] , ∂η

(B.176)

and using the reality conditions, we get ∂ Ψ [A, η] = 0 , ∂Ai

(B.177)

∂ ki ∂ G Ψ [A, η] = 0 . ∂Ak ∂ηi

(B.178)

ηi Ai (Ak − 2 Γ k )

The only solution admitted by this system of equations is Ψ [ A] = constant, but it has a nontrivial form in terms of the Misner–Ryan variables.

Problems of Chap. 7 7.1 An Avant Garde (Non-SUSY) Square Root Formulation Take the corresponding Hamiltonian constraint to be − p2Ω + p2+ + p2− = 0 ,

(B.179)

with the convention in [48] (see Sect. 7.2 for more details on the expressions used here). With the quantum correspondence pΩ −→ −ih¯

∂ , ∂Ω

p± −→ −ih¯

∂ , ∂β±

(B.180)

we may write ∂ 2Ψ ∂ 2Ψ ∂ 2Ψ = + . 2 2 ∂Ω 2 ∂β+ ∂β−

(B.181)

Problems of Chap. 7

267

Now ‘linearize’ the square operator ∂ 2Ψ ∂ 2Ψ + , 2 2 ∂β+ ∂β− to obtain 1/2 ∂Ψ ∂Ψ ∂Ψ ∂ 2Ψ ∂Ψ ∂ 2Ψ i + −→ i − iα− , =± = iα+ 2 2 ∂Ω ∂Ω ∂β ∂β ∂β+ ∂β− + −

(B.182)

2 = 1 and which anticommute! The minimal where α± are matrices satisfying α± rank for such matrices is two. Then taking α± to be the Pauli matrices, namely α+ ≡ σ1 , α− ≡ σ2 , Ψ must be a two-component ‘vector’ (spinor). Writing

Ψ =

Ψ+ Ψ−

,

(B.183)

we obtain plane wave solutions

Ψ± ∼ exp i (p+ β+ + p− β− − EΩ) ,

(B.184)

where ± (p+ β+ + p− β− )1/2 = E and p+ , p− , Ω are constants. The two spinor degrees of freedom were described in [48] as ‘disturbing’. In fact, the two solutions correspond to expanding (E > 0) and contracting (E < 0) universes, but how should one then interpret the ‘spin’ states Ψ± in terms of physical attributes of the universe? The reader should remember that this was in 1972! Researchers had to wait for SUGRA in order to obtain a clearer view (see Sect. 7.2). The above is a possible representation within SQC, i.e., SUGRA as applied to cosmology, using the matrix representation for fermionic momenta! It is altogether astonishing how this emerged back in 1972, before the full development of SUSY and SUGRA. 7.2 From the Lorentz Constraints to the Lorentz Conditions We find that ΨIII = − (J12 )−1 J13 ΨIV . Then show that J12 ΨII = J23 ΨIV , whence J23 = J12 (J13 )−1 J23 (J12 )−1 J13 , J12 = J23 (J13 )−1 J12 (J23 )−1 J13 , J13 = J23 (J12 )−1 J13 (J23 )−1 J12 . Using J A = ε0ABC J BC /2, the result follows.

(B.185)

268

B Soluitions

7.3 From the Lorentz Constraints to the Non-diagonal ‘Missing’ Equations First obtain the equations of motion and Lorentz constraints for the diagonal Bianchi type IX cosmological model [49]. Then, e.g., substitute the J 01 and the J 23 component expressions in the (0, 1) component of the Einstein–Rarita–Schwinger equation. This reduces to

−3β˙+ +

√ ˙ ψˇ 2 γ 2 ψˇ 1 = 0 . 3β_

(B.186)

At the quantum level (in this matrix representation approach!), (B.186) yields the same result as the Lorentz constraints. Now with this result, we can analyze the (2, 3) component of the Einstein–Rarita–Schwinger equation. The last term is once more ψˇ 2 γ 2 ψˇ 1 and the first and second terms cancel out. The first of the three remaining terms is quadratic in the gravitino field components. The other two terms can also be reduced to quadratic terms in ψˇ i , multiplied by linear combinations of ˙ with the help of the equations for ψˇ i . The same procedure is applied ˙ β˙+ , and β_, Ω, to, e.g., the (0, 2) and (0, 3) components of the Einstein–Rarita–Schwinger equation. Substituting into all the i = j equations, and the ψˇ i equations are used once more. The quadratic expressions resulting from this procedure for all the equations i = j are: ψˇ 1 γ 0 γ 1 γ 2 φ1 ,

ψˇ 2 γ 0 γ 2 γ 3 ψˇ 1 ,

φ 1 γ 0 γ 1 γ 3 φ1 ,

φ 1 γ 0 γ 1 γ 3 ψˇ 2 ,

ψˇ 1 γ 0 γ 2 γ 3 φ1 ,

φ 3 γ 0 γ 2 γ 3 ψˇ 1 ,

ψˇ 2 γ 0 γ 1 γ 2 φ2 ,

φ 3 γ 1 γ 2 γ 3 ψˇ 1 ,

ψˇ 2 γ 0 γ 1 γ 3 φ2 ,

φ 1 γ 0 γ 1 γ 2 ψˇ 3 ,

ψˇ 2 γ 0 γ 2 γ 3 φ2 ,

φ 3 γ 0 γ 1 γ 2 ψˇ 2 ,

ψˇ 3 γ 0 γ 1 γ 2 φ3 ,

φ 2 γ 0 γ 1 γ 3 ψˇ 3 ,

ψˇ 3 γ 0 γ 1 γ 3 φ3 ,

φ 3 γ 1 γ 2 γ 3 ψˇ 2 ,

ψˇ 3 γ 0 γ 2 γ 3 φ3 ,

φ 1 γ 1 γ 2 γ 3 ψˇ 2 .

(B.187)

These expressions at the quantum level (i.e., proceeding from the variables ψˇ i to χi ) lead to the same wave function that has been obtained by applying the J ab constraints. Therefore the equations i = j yield the same result as the Lorentz constraints J ab .

Problems of Chap. 8

269

Problems of Chap. 8 8.1 Invariance of the Action Under N=2 Conformal SUSY For the the action (8.39), we get (see also [50]) δS =

1 2

· f G X Y (q Z )q˙ X q˙ Y + f N U (q Z ) dt , N

(B.188)

under the reparametrization t → t + f (t) with δq X (t) = f (t)q˙ X ,

δN (t) = ( f N )· .

(B.189)

Hence, we get a total derivative. We are interested in the action (8.41), whose variation is δSgrav =

GX Y Dη LDη Dη Q X Dη QY + W(Q Z ) 2N

GX Y X Y Z dηdηdt , Dη Q Dη Q + W(Q ) +Dη LDη 2N i 2

(B.190)

under (8.7), (8.8), (8.12), and i i δQ = LQ˙ + Dη LDη Q + Dη LDη Q . 2 2

(B.191)

This is a total derivative, so the action is indeed invariant, as claimed. 8.2 Dirac Brackets in N=2 Conformal SUSY From ψ(t), χ (t), and λ(t), we have the following second class constraints: i pψ ≡ πψ + ψ = 0 , 3

(B.192)

i pψ ≡ πψ + λ = 0 , 3

(B.193)

p I (χ ) ≡ π I (χ ) −

i G χJ = 0 , 2κ 2 J I

(B.194)

p I (χ ) ≡ π I (χ ) −

i G χJ = 0 , 2κ 2 I J

(B.195)

270

B Soluitions

Π I (λ) ≡ π I (λ) −

i G λJ = 0 , 2κ 2 J I

(B.196)

p I (λ) ≡ π I (λ) −

i G χJ = 0 , 2κ 2 I J

(B.197)

˙ π I (χ ) = dL/dχ˙ I , and π I (λ) = dL/dλ˙ I are the momenta where πψ = dL/dψ, conjugate to the anticommuting variables ψ(t), χ (t), and λ(t), respectively. Then write the matrices Cψψ =

2 i, 3

C I J (χ ) = −

i G , k2 I J

C I J (λ) = −

i G , k2 I J

(B.198)

and their inverses 3 C ψψ = (Cψψ )−1 = − i , 2

C I J (χ ) = ik2 G I J ,

C I J (λ) = ik2 G I J . (B.199)

The Dirac brackets [ , ]D are (recall Chap. 4 of Vol. I)

[ f, g]D = [ f, g] − f, pi (C −1 )ik pk , g ,

(B.200)

and allow elimination of the momenta conjugate to the fermionic variables, leaving us with the following non-zero Dirac bracket relations: [a, πa ]D = [a, πa ]D = −1 , = φ I , πφJ = −δ IJ , φ I , πφJ

D

φ I , πφJ = φ I , πφJ = −δ IJ , D

χI,χJ

λI , λ J

ψ, ψ

D

D

D

(B.201) (B.202) (B.203)

= −ik2 G I J ,

(B.204)

= −ik2 G I J ,

(B.205)

=

3 i. 2

8.3 FRW Wave Function in N=2 Conformal SUSY With k = 0, 1, take the action to be (see [51, 52])

(B.206)

Problems of Chap. 8

271

S = SFRW + Smat , √ k 2 1 A SFRW = 6 − 2 Dη ADη A + 2 A dηdηdt , 2k N 2k 3 1A 3 Dη ΦDη Φ − 2A W(#) dηdηdt , Smat = 2 N

(B.207)

where we use # = φ(t) + iηχ (t) + iηχ (t) + F (t)ηη

(B.208)

for the component of the (scalar) matter superfields #(t, η, η), with #+ = #. As often indicated, after the integration over the Grassmann complex coordinates η and η, and making the following redefinition of the ‘fermion’ fields (Grassmann variables) ψ(t) −→

1 −1/2 (t)ψ(t) , a 3

χ (t) −→ a −3/2 (t)χ (t) ,

we find the Lagrangian in which the field F(t) is auxiliary, and they can be eliminated with the help of their equations of motion. In addition, we use the Taylor expansion for the superpotential: W(#) = W(φ) +

∂W 1 ∂ 2W (# − φ)2 + · · · , (# − φ) + ∂φ 2 ∂φ 2

with # as given above.18 In terms of components of the superfields A, N, and #, the Lagrangian then reads (somewhat more simply than in Chap. 8) L=−

√ k 1/2 3 a(Da)2 2 + + iψDψ a (ψ 0 ψ − ψ0 ψ) 3 k k2 N

√ 1 3k a 3 (Dφ)2 + Na −1 kψψ + 2 Na + − iχDχ 3 2 N k √ 3√ kNa −1 χχ − k2 NW(φ)ψψ − 6 kNW(φ)a 2 − 2 3 ik −Na 3 V (φ) + k2 NW(φ)χ χ + Dφ(ψχ + ψχ) 2 2 −2N

∂W(φ) k2 −3/2 ∂ 2 W(φ) χ χ − kN − ψχ ) + (ψ0 ψ − ψ 0 ψ)χχ (ψχ a ∂φ 4 ∂φ 2

−ka 3/2 (ψ 0 ψ − ψ0 ψ)W(φ) + a 3/2

18

∂W(φ) (ψ 0 χ − ψ0 χ ) , ∂φ

(B.209)

This series expansion ends at this point, in the second term, since (# − φ) is nilpotent.

272

B Soluitions

where Da = a˙ −

ik −1/2 (ψ0 ψ + ψ 0 ψ) , a 6

i Dφ = φ˙ − a −3/2 (ψ 0 χ + ψ0 χ) , 2

are the supercovariant derivatives, and i Dχ = χ˙ − V χ , 2

i Dψ = ψ˙ − V ψ , 2

are the U(1) covariant derivatives. The potential for the homogeneous scalar fields is V (φ) = 2

∂W(φ) ∂φ

2 − 3k2 W2 (φ) ,

(B.210)

and consists of two terms. One is the potential for the scalar field in the case of global supersymmetry. The potential (B.210) is not positive semi-definite, in contrast with what happens in standard supersymmetric quantum mechanics. Indeed, the present model describing the minisuperspace approach to supergravity coupled to matter, allows supersymmetry breaking when the vacuum energy is equal to zero V (φ) = 0. Quantization requires for the Dirac brackets

ψ, ψ

D

=

3 i, 2

[χ , χ ]D = −i ,

(B.211)

{χ , χ } = 1 ,

(B.212)

φ, πφ = −i .

(B.213)

whence : 3 ψ, ψ = − , 2

9

[a, πa ] = −i ,

Antisymmetrizing, i.e., writing a bilinear combination in the form of the commutators, e.g., χ χ −→

1 [χ, χ ] , 2

this leads eventually to eigenstates of the Hamiltonian with four components: ⎡

Ψ1 (a, φ)

⎤

⎢ Ψ (a, φ) ⎥ ⎥ ⎢ 2 Ψ (a, φ) = ⎢ ⎥ , ⎣ Ψ3 (a, φ) ⎦ Ψ4 (a, φ)

(B.214)

References

273

using instead ψS − ψS Ψ and χS − χ S Ψ , with a matrix representation for ψ, ψ, χ, and χ. Then only Ψ1 or Ψ4 have the right behaviour when a → ∞, because the other components are infinite as a → ∞. We thus get the partial differential equations −a

−1/2

−a −1/2

√ 2 1/2 3 −3/2 ∂ 3/2 − 6W(φ)a + 6 k MPl a + a Ψ4 = 0 , (B.215) ∂a 4 ∂ 3 ∂W(φ) + 2a Ψ4 = 0 , (B.216) ∂φ ∂φ √ 2 1/2 3 −3/2 ∂ a + a + 6W(φ)a 3/2 − 6 k MPl Ψ1 = 0 , (B.217) ∂a 4 ∂ 3 ∂W(φ) − 2a Ψ1 = 0 , (B.218) ∂φ ∂φ

with solutions √ 2 2 Ψ −→ Ψ4 (a, φ) a 3/4 exp −2w(φ) a 3 + 3 k MPl a ,

(B.219)

√ 2 2 Ψ −→ Ψ1 (a, φ) a 3/4 exp 2w(φ) a 3 − 3 k MPl a .

(B.220)

These are eigenstates of the Hamiltonian with zero energy and also with zero fermionic number.

References

1. Kiefer, C.: Quantum Gravity. International Series of Monographs on Physics, vol. 136, 2nd edn., pp. 1–308. Clarendon Press, Oxford (2007) 235, 237, 239, 240, 254 2. Kiefer, C.: How does quantum gravity modify the Schrödinger equation for matter fields? Class. Quant. Grav. 9, S147–S156 (1992) 235, 237, 254 3. Kiefer, C.: The semiclassical approximation to quantum gravity. gr-qc/9312015 (1993) 235, 237, 240, 242, 25 4. Kiefer, C., Singh, T.P.: Quantum gravitational corrections to the functional Schrödinger equation. Phys. Rev. D 44, 1067–1076 (1991) 235, 237, 254 5. Gerlach, U.H.: Derivation of the ten Einstein field equations from the semiclassical approximation to quantum geometrodynamics. Phys. Rev. 177, 1929–1941 (1969) 237 6. DeWitt, B.S.: Quantum theories of gravity. Gen. Rel. Grav. 1, 181–189 (1970) 237 7. Teitelboim, C.: How commutators of constraints reflect the space-time structure. Ann. Phys. 79, 542–557 (1973) 237 8. Birrell, N.D., Davies, P.C.W.: Quantum Fields in Curved Space, 340pp. Cambridge University Press, Cambridge (1982) 242 9. Kaku, M.: Introduction to Superstrings and M-Theory, 587pp. Springer, Heidelberg (1999) 243 10. Kaku, M.: Strings, Conformal Fields, and M-Theory, 531pp. Springer, Heidelberg (2000) 243

274

B Soluitions

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Index

A Ashtekar connection, 112 Ashtekar–Jacobson formulation, 116 N = 1 SUGRA, 116 B Bianchi I matrix representation, 133 Bianchi IX matrix representation, 138 Born–Oppenheimer semiclassical (quantum) gravity, 14 Boundary conditions, 5, 19, 21 infinite wall proposal, 5, 28 no boundary, 5, 19, 21 tunneling, 5, 19, 21 C Cartan–Sciama–Kibble N = 1 SUGRA, 229 torsion, 229 Chiral superfields, 35 Classical SUSY spin 1/2 particle Dirac equation, 130 quantization, 130 Constraints, 4 Contorsion decomposing 3D connection, 227 Conventions and notation, 211 Cosmological constant SUSY breaking, 41 Covariant derivative, 227 minisuperspace, 20 Curvature, 228 extrinsic, 228 intrinsic, 228 spinor valued, 228

D Decoherence quantum cosmology, 21 SQC, 74 DeWitt metric, 71 DeWitt supermetric, 71, 74, 75, 78 N = 1 SUGRA, 71 F Fayet–Iliopoulos potential, 35 Fermionic differential operator representation, 87, 96, 101 supersymmetry breaking, 87 Fermionic momentum differential operator representation, 6, 7 matrix operator representation, 87 matrix representation, 7 FRW model matrix representation, 147 G General relativity constraints DeWitt metric, 71 Goldstino, 38 Grassmann numbers, 218 Gravitational canonical momenta, 227 Gravitino mass, 41 H Hamiltonian minisuperspace, 20 Hamiltonian constraint N = 1 SUGRA, 67 SQC FRW, 92 generic gauge supermatter FRW, 92 scalar supermultiplet FRW, 89

277

278 Hamiltonian formulation N = 1 SUGRA, 6, 7 general relativity, 5 supergravity, 6, 7 Hamilton–Jacobi equation, 15 N = 1 SUGRA, 68, 70 body and soul, 70 bosonic and fermionic parts, 69–71, 75 spacetime emergence, 70 supermanifolds, 71 superRiemaniann geometries, 71 SUSY spacetime, 70 quantum cosmology, 15 quantum gravity, 15 Hidden supersymmetries, 7 SQC, 7 I Infeld–van der Waerden symbols, 222 Inflationary cosmology, 28 K Kähler geometry, 90, 91 metric, 90, 91 potential, 90, 91 L List of symbols, 207 Loop quantum cosmology, 111 Loop quantum gravity, 111 Ashtekar connection, 112 Barbero–Immirzi parameter, 112 connection quantization, 114 Hamiltonian formulation, 114 reality conditions, 112 Lorentz conditions matrix representation, 152 rest frame solution, 153 Lorentz constraint matrix representation, 147, 155 SQC Bianchi, 96 FRW, 92 generic gauge supermatter FRW, 92 Lorentz constraint trivialization matrix representation, 153 Lorentz group, 214 representations, 214 spinors, 214 M Matrix representation, 127 fermionic momenta, 127

Index Lorentz constraint, 128 Minisuperspace, 4, 20 covariant derivative, 20 curvature, 20 D’Alembert equation, 20 dimension, 20 factor ordering problem, 20 Hamiltonian, 20 Laplace–Bertrami, 20 minisupermetric, 20 validity, 20 Wheeler–DeWitt equation, 20 perturbations, 20 WKB approximation, 21 Momentum constraints N = 1 SUGRA, 67 M-theory, 4 N Nicolai maps, 55 Nilpotency, 70 P Pauli matrices, 222 spinors, 222 Q Quantum constraints N = 1 SUGRA, 67 Quantum cosmology, 4, 6, 7 decoherence, 21 Hamilton–Jacobi equation, 15 historical, 4 inflation, 13, 19, 21, 28 vacuum, 21 minisuperspace, 6, 19, 21 beyond, 19 Hamiltonian, 20 perturbations, 19, 21 vacuum state, 21 validity, 20 WKB, 21 observational, 13, 19, 66 predictions, 13, 19 vacuum state, 21 Schrödinger equation, 15 semiclassical, 13, 19, 66 semiclassical limit, 7 structure formation, 19 vacuum state, 21 summary and review, 31 superstring, 5, 28 infinite wall proposal, 5, 28 landscape problem, 5, 28

Index Tomonaga–Schwinger equation, 15, 74 vacuum state structure formation, 21 Quantum FRW solutions fermionic differential operator representation generic gauge supermatter, 94 Quantum gravity, 4 Hamilton–Jacobi equation, 15 observational context, 4 Schrödinger equation, 15 semiclassical, 13, 66 superspace, 4, 19, 21, 61, 70 Tomonaga–Schwinger equation, 15 Quantum gravity corrections, 17 Schrödinger equation, 17 R Reality conditions, 112 N = 1 SUGRA, 116 S Schrödinger equation, 15, 74 quantum cosmology, 15 quantum gravity, 15 quantum gravity corrections, 17 N = 1 SUGRA, 75, 78, 79 normal component, 79 SQC and observation, 81 tangential component, 79 SQC, 74 Semiclassical expansion N = 1 SUGRA, 66 Semiclassical (quantum) gravity, 13 Born–Oppenheimer, 13 Hamilton–Jacobi equation, 15 observational, 13 Schrödinger equation, 15, 16 corrections, 17 time functional, 15 time functional, 15 Tomonaga–Schwinger equation, 15, 16 time functional, 15 Semiclassical N = 1 SUGRA, 61, 68 Born–Oppenheimer, 68 observational, 68 Schrödinger equation, 74 time functional, 74 time functional, 74 Tomonaga–Schwinger equation, 74 time functional, 74 SL(2,C), 214 Spacetime foliation SUGRA, 222

279 Spinors, 212 Dirac, 218 Infeld–van der Waerden symbols, 222 Lorentz (group algebra), 213 Lorentz (group representations), 212 Lorentz transformations, 212 Majorana, 218 Pauli matrices, 222 primed, 214, 216 tetrad, 212 unprimed, 214, 216 Weyl, 218 SQC, 5, 87 achievements, 7 Ashtekar–Jacobson formulation, 111, 118, 120 Bianchi class A models, 120 FRW with cosmological constant, 118 Bianchi, 96, 101 Ansatze, 96, 101 Lorentz constraints, 96 Bianchi I Matrix representation, 133 Bianchi IX Matrix representation, 138 Bianchi minisuperspace reduction, 96, 101 Lagrangian, 96 N = 2 SUGRA, 101 Bianchi wave function of the universe, 96, 101 connection/loop variables, 7 connection quantization, 111 decoherence, 74 fermionic differential operator representation, 6, 7, 87, 96, 101 fermionic momenta matrix representation, 127 FRW, 88 constraints, 92 gauge fields ansatze, 91 generic gauge supermatter, 91 Hamiltonian constraint, 92 Lorentz constraints, 92 SUSY constraint, 92 FRW generic gauge supermatter sector quantum states, 96 FRW model matrix representation, 147 FRW scalar supermultiplet sector quantum states, 89 FRW wave function of the universe generic gauge supermatter, 94

280 further exploration, 87 generic gauge supermatter FRW constraints, 92 Hamiltonian constraint, 92 Lorentz constraints, 92 SUSY constraint, 92 wave function of the universe, 94 hidden N = 2 SUSY, 7 hidden supersymmetries, 7 loop quantization, 111 Lorentz conditions matrix representation, 152 Lorentz constraint matrix representation, 128, 147 Lorentz constraint trivialization matrix representation, 153 matrix fermionic operator representation, 7, 87 matrix representation, 127 Bianchi I, 133 Bianchi IX, 138 FRW, 147 historical background, 129 Lorentz boost components, 155 Lorentz constraint, 128 Taub, 143 non-supersymmetric square root, 129 observational, 61, 66, 75 observational insights, 81 open issues, 6, 7, 19, 21 observational context, 6, 21 SUSY breaking, 6 rest frame solution matrix representation, 153 scalar supermultiplet Bianchi quantum states, 96, 101 scalar supermultiplet Bianchi class A quantum states, 96 scalar supermultiplet FRW constraints, 89 Hamiltonian constraint, 89 quantum states, 91 wave function of the universe, 89 Schrödinger equation, 74 semiclassical, 61, 66, 75 N = 1 SUGRA, 87 N = 2 SUGRA, 87, 101 Taub model matrix representation, 143 wave function of the universe, 6, 89, 94, 96, 101, 106 SQM SUSY breaking, 45

Index Witten index, 45 SUGRA spacetime foliation, 222 summary and review, 56 N = 1 SUGRA, 72, 74, 75 bosonic and fermionic parts, 74, 75 conservation law, 253 simplified form, 73 N = 1 SUGRA, 3, 4 affinely connected spaces, 229 Ashtekar–Jacobson formulation, 116 Ashtekar–Jacobson Hamiltonian formulation, 116 Ashtekar–Jacobson quantization, 117 Bianchi class A models, 120 FRW with cosmological constant, 118 bosonic states, 62 Cartan–Sciama–Kibble, 229 CFOP argument, 61 DeWitt supermetric, 71, 75, 78 Schrödinger equation, 74, 75 finite fermion number states, 63 gravitino mass super Higgs effect, 41 Hamilton–Jacobi equation, 68, 70 body and soul, 70 bosonic and fermionic parts, 69–71, 75 DeWitt supermetric, 71, 75 spacetime emergence, 70 supermanifolds, 71 superRiemaniann geometries, 71 SUSY spacetime, 70 Hamiltonian constraint normal projection, 67 Hamiltonian formulation, 6, 7 infinite fermion number states, 63 matrix representation, 130 momentum, 67 physical states, 61 bosonic states, 62 CFOP argument, 61 D’Eath’s claims, 61 finite fermion number, 63 infinite fermion number, 63 quantum constraints, 67 Hamiltonian, 67 quantum states, 61 Riemann–Cartan, 229 torsion, 229 Schrödinger equation bosonic and fermionic parts, 74, 75 using DeWitt supermetric, 74, 75 Schrödinger equation, 72, 74, 75

Index

N N

N

N

N

conservation law, 253 DeWitt supermetric, 78 normal component, 79 quantum gravity corrections, 75, 78 simplified form, 73 SQC and observation, 81 tangential component, 79 using DeWitt supermetric, 74 second-order formalism, 225 semiclassical expansion, 66 semiclassical formulation, 61 semiclassical limit, 7 SQC, 87 Bianchi ansatze, 96 FRW, 88 FRW gauge fields ansatze, 91 generic gauge supermatter, 91 super DeWitt metric, 74 super Higgs effect, 39 gravitino mass, 41 supermatter gaugino condensate, 40 SUSY breaking, 39 superRiem(Σ), 61, 70, 78 normal part, 78 tangential part, 78 SUSY breaking, 39 cosmological constant, 41 gaugino condensate, 40 gravitino mass, 41 super Higgs effect, 39 Tomonaga–Schwinger equation, 74 Wheeler–DeWitt equation, 67 = 2 (local) conformal supersymmetry, 163 = 2 SQM, 43 Nicolai Maps, 55 quantum states, 51 Fock space, 51 Topology and vacuum states, 52 = 2 SUGRA, 3, 7 SQC, 87, 101 Bianchi ansatze, 101 = 2 SUGRA (SQC) Bianchi quantum states vector field sector, 106 Bianchi wave function of the universe, 106 vector field sector Bianchi class A global O(2), 103 Bianchi class A local O(2), 105 Bianchi class A quantum states, 101 Bianchi quantum states, 106 = 2 conformal supersymmetry (complex) scalar fields, 172

281 Hamiltonian, 177 Bianchi class A, 169 FRW universe, 164 Hamiltonian formulation, 168 quantum cosmology, 181 supersymmetry breaking, 183 time reparametrization, 164 N = 2 Supersymmetric quantum mechanics, 43 N = 2 supersymmetric quantum mechanics Fock space, 51 Nicolai maps, 55 quantum states, 51 Fock space, 51 sigma model, 48 quantum states, 51 N = 8 SUGRA, 3 Superfield description, 163 N = 2 conformal supersymmetry, 163 Superfields, 35, 47 chiral, 35 vector, 35 Supergravity Hamiltonian formulation, 6, 7 square root, 5, 6 Supergravity (SUGRA), 6, 7 SuperRiem(Σ), 61, 70, 78 normal part, 78 tangential part, 78 Superspace quantum gravity, 19, 21 Superspace (N = 1 SUGRA) canonical quantization, 67 Dirac quantization, 67 Superspace (gravity), 61, 70 Superspace (SUSY), 47 interpretation, 47 supertranslations, 47 Superstring, 4 cosmological models, 7 dualities, 5 quantum cosmology, 5, 28 infinite wall proposal, 5, 28 landscape problem, 5, 28 Supersymmetric quantum cosmology (SQC), 5, 87 Bianchi Ansatze, 96, 101 FRW, 88 gauge fields ansatze, 91 generic gauge supermatter, 91 generic gauge supermatter sector FRW quantum states, 96

282 scalar supermultiplet Bianchi quantum states, 96, 101 scalar supermultiplet Bianchi class A quantum states, 96 scalar supermultiplet FRW quantum states, 91 scalar supermultiplet sector FRW quantum states, 89 Supersymmetric quantum mechanics (SQM), 7, 35, 43, 88 quantum states, 51 Fock space, 51 SUSY, 6, 7, 35 breaking, 35, 36, 183 conditions, 36 cosmological constant, 41 fermionic masses, 39 gaugino condensate, 40 goldstino, 38 gravitino mass, 41 mechanisms, 37 N = 1 SUGRA, 39 super Higgs effect, 39, 41 chiral superfields, 35 Fayet–Iliopoulos potential, 35 goldstino, 38 Grassmannian variables, 47 Nicolai maps, 55 N = 2 SQM, 43 Fock space, 51 quantum states, 51 summary and review, 56 superfields, 35, 47 supermultiplet, 35, 90 chiral, 35, 90 vector, 35, 90 superspace, 43 supersymmetric quantum mechanics, 35, 43, 88 quantum states, 51 supertranslations, 47 vector superfields, 35 SUSY Bianchi fermionic differential operator representation quantum states, 96 scalar supermultiplet quantum solutions, 96 SUSY Bianchi class A fermionic differential operator representation, 96, 101 scalar supermultiplet sector, 96 N = 2 SUGRA, 101

Index vector field sector, 101 SUSY breaking SQM, 45 N = 1 SUGRA super Higgs effect, 42 SUSY constraint SQC FRW, 92 generic gauge supermatter FRW, 92 SUSY FRW fermionic differential operator representation gauge fields sector, 91 generic gauge supermatter, 91 supersymmetry breaking, 87 generic gauge supermatter sector quantum solutions, 96 scalar supermultiplet sector quantum solutions, 89 supersymmetry breaking, 87 T Taub model matrix representation, 143 Tetrad, 223 metric, 223 tetrad decomposition spinorial version, 222 Time functional semiclassical N = 1 SUGRA, 74 semiclassical (quantum) gravity, 15 Tomonaga–Schwinger equation, 15, 74 quantum cosmology, 15, 74 quantum gravity, 15 N = 1 SUGRA, 74 Torsion, 225 Cartan–Sciama–Kibble, 229 connections, 225 U Unit normal vector spinorial form, 222 Useful expressions, 222 V Vector superfields, 35 W Wave function of the universe, 4, 6, 19–21, 89, 94, 96, 101, 106 perturbations, 20 SQC, 6, 89, 94, 96, 101, 106 Bianchi, 96, 101 generic gauge supermatter FRW, 94

Index scalar supermultiplet FRW, 89 N = 2 SUGRA (SQC) Bianchi, 106 WKB approximation, 21 Wheeler–DeWitt equation, 6, 20 minisuperspace, 20 perturbations, 20 perturbations, 20 polynomial structure, 111

283 N = 1 SUGRA, 67 semiclassical limit, 67 Witten index, 45 SQM, 45 WKB approximation minisuperspace, 21 quantum cosmology, 21 wave function of the universe, 21

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