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PAT H I N T E G R A L S A N D H A M I LT O N I A N S
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PAT H I N T E G R A L S A N D H A M I LT O N I A N S
Providing a pedagogical introduction to the essential principles of path integrals and Hamiltonians, this book describes cuttingedge quantum mathematical techniques applicable to a vast range of fields, from quantum mechanics, solid state physics, statistical mechanics, quantum field theory, and superstring theory to financial modeling, polymers, biology, chemistry, and quantum finance. Eschewing use of the Schrödinger equation, the powerful and flexible combination of Hamiltonian operators and path integrals is used to study a range of different quantum and classical random systems, succinctly demonstrating the interplay between a system’s path integral, state space, and Hamiltonian. With a practical emphasis on the methodological and mathematical aspects of each derivation, this is a perfect introduction to these versatile mathematical methods, suitable for researchers and graduate students in physics and engineering. B e l a l E . Ba a q u i e is a Professor of Physics at the National University of Singapore, specializing in quantum field theory, quantum mathematics, and quantum finance. He is the author of Quantum Finance (2004), Interest Rates and Coupon Bonds in Quantum Finance (2009), and The Theoretical Foundations of Quantum Mechanics (2013), and coauthor of Exploring Integrated Science (2010).
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PATH INTEGRALS AND H A M I LTO N I A N S Principles and Methods B E L A L E . BA AQ U I E National University of Singapore
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University Printing House, Cambridge CB2 8BS, United Kingdom Published in the United States of America by Cambridge University Press, New York Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107009790 © Belal E. Baaquie 2014 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2014 Printed in the United Kingdom by CPI Group Ltd, Croydon CR0 4YY A catalog record for this publication is available from the British Library Library of Congress Cataloging in Publication Data ISBN 9781107009790 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or thirdparty internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
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This book is dedicated to the memory of Kenneth Geddes Wilson (19362013). Intellectual giant, visionary scientist, exceptional educator, altruistic spirit.
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“9781107009790AR” — 2013/11/5 — 21:26 — page vii — #7
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Contents
Preface Acknowledgements 1
Synopsis
Part one
Fundamental principles
page xv xviii 1 5
2
The mathematical structure of quantum mechanics 2.1 The Copenhagen quantum postulate 2.2 The superstructure of quantum mechanics 2.3 Degree of freedom space F 2.4 State space V(F) 2.4.1 Hilbert space 2.5 Operators O(F) 2.6 The process of measurement 2.7 The Schrödinger differential equation 2.8 Heisenberg operator approach 2.9 Dirac–Feynman path integral formulation 2.10 Three formulations of quantum mechanics 2.11 Quantum entity 2.12 Summary: quantum mathematics
7 7 10 10 11 14 14 18 19 22 23 25 26 27
3
Operators 3.1 Continuous degree of freedom 3.2 Basis states for state space 3.3 Hermitian operators 3.3.1 Eigenfunctions; completeness 3.3.2 Hamiltonian for a periodic degree of freedom 3.4 Position and momentum operators xˆ and pˆ 3.4.1 Momentum operator pˆ
30 30 35 36 37 39 40 41
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3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14
Weyl operators Quantum numbers; commuting operator Heisenberg commutation equation Unitary representation of Heisenberg algebra Density matrix: pure and mixed states Selfadjoint operators 3.10.1 Momentum operator on finite interval Selfadjoint domain 3.11.1 Real eigenvalues Hamiltonian’s selfadjoint extension 3.12.1 Delta function potential Fermi pseudopotential Summary
43 46 47 48 50 51 52 54 54 55 57 59 60
4
The Feynman path integral 4.1 Probability amplitude and time evolution 4.2 Evolution kernel 4.3 Superposition: indeterminate paths 4.4 The Dirac–Feynman formula 4.5 The Lagrangian 4.5.1 Infinite divisibility of quantum paths 4.6 The Feynman path integral 4.7 Path integral for evolution kernel 4.8 Composition rule for probability amplitudes 4.9 Summary
61 61 63 65 67 69 70 70 73 76 79
5
Hamiltonian mechanics 5.1 Canonical equations 5.2 Symmetries and conservation laws 5.3 Euclidean Lagrangian and Hamiltonian 5.4 Phase space path integrals 5.5 Poisson bracket 5.6 Commutation equations 5.7 Dirac bracket and constrained quantization 5.7.1 Dirac bracket for two constraints 5.8 Free particle evolution kernel 5.9 Hamiltonian and path integral 5.10 Coherent states 5.11 Coherent state vector 5.12 Completeness equation: overcomplete 5.13 Operators; normal ordering
80 80 82 84 85 87 88 90 91 93 94 95 96 98 98
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5.14
ix
Path integral for coherent states 5.14.1 Simple harmonic oscillator Forced harmonic oscillator Summary
99 101 102 103
Path integral quantization 6.1 Hamiltonian from Lagrangian 6.2 Path integral’s classical limit → 0 6.2.1 Nonclassical paths and free particle 6.3 Fermat’s principle of least time 6.4 Functional differentiation 6.4.1 Chain rule 6.5 Equations of motion 6.6 Correlation functions 6.7 Heisenberg commutation equation 6.7.1 Euclidean commutation equation 6.8 Summary
105 106 109 111 112 115 115 116 117 118 121 122
5.15 5.16 6
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Part two Stochastic processes
123
7
125 127 128 129 132 134 136 137 138 140 142 143 144 145 148 149 151 153 156 158
Stochastic systems 7.1 Classical probability: objective reality 7.1.1 Joint, marginal and conditional probabilities 7.2 Review of Gaussian integration 7.3 Gaussian white noise 7.3.1 Integrals of white noise 7.4 Ito calculus 7.4.1 Stock price 7.5 Wilson expansion 7.6 Linear Langevin equation 7.6.1 Random paths 7.7 Langevin equation with potential 7.7.1 Correlation functions 7.8 Nonlinear Langevin equation 7.9 Stochastic quantization 7.9.1 Linear Langevin path integral 7.10 Fokker–Planck Hamiltonian 7.11 PseudoHermitian Fokker–Planck Hamiltonian 7.12 Fokker–Planck path integral 7.13 Summary
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Part three
Discrete degrees of freedom
159
8
Ising model 8.1 Ising degree of freedom and state space 8.1.1 Ising spin’s state space V 8.1.2 Bloch sphere 8.2 Transfer matrix 8.3 Correlators 8.3.1 Periodic lattice 8.4 Correlator for periodic boundary conditions 8.4.1 Correlator as vacuum expectation values 8.5 Ising model’s path integral 8.5.1 Ising partition function 8.5.2 Path integral calculation of Cr 8.6 Spin decimation 8.7 Ising model on 2×N lattice 8.8 Summary
161 161 163 164 165 167 168 169 171 171 172 173 175 176 179
9
Ising model: magnetic field 9.1 Periodic Ising model in a magnetic field 9.2 Ising model’s evolution kernel 9.3 Magnetization 9.3.1 Correlator 9.4 Linear regression 9.5 Open chain Ising model in a magnetic field 9.5.1 Open chain magnetization 9.6 Block spin renormalization 9.6.1 Block spin renormalization: magnetic field 9.7 Summary
180 180 182 183 184 185 189 190 191 195 196
10
Fermions 10.1 Fermionic variables 10.2 Fermion integration 10.3 Fermion Hilbert space 10.3.1 Fermionic completeness equation 10.3.2 Fermionic momentum operator 10.4 Antifermion state space 10.5 Fermion and antifermion Hilbert space 10.6 Real and complex fermions: Gaussian integration 10.6.1 Complex Gaussian fermion 10.7 Fermionic operators
198 199 200 201 203 204 204 206 207 209 211
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10.8 10.9 10.10 10.11 10.12 10.13
Fermionic path integral Fermion–antifermion Hamiltonian 10.9.1 Orthogonality and completeness Fermion–antifermion Lagrangian Fermionic transition probability amplitude Quark confinement Summary
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211 214 216 217 219 220 222
Part four Quadratic path integrals
223
11
Simple harmonic oscillator 11.1 Oscillator Hamiltonian 11.2 The propagator 11.2.1 Finite time propagator 11.3 Infinite time oscillator 11.4 Harmonic oscillator’s evolution kernel 11.5 Normalization 11.6 Generating functional for the oscillator 11.6.1 Classical solution with source 11.6.2 Source free classical solution 11.7 Harmonic oscillator’s conditional probability 11.8 Free particle path integral 11.9 Finite lattice path integral 11.9.1 Coordinate and momentum basis 11.10 Lattice free energy 11.11 Lattice propagator 11.12 Lattice transfer matrix and propagator 11.13 Eigenfunctions from evolution kernel 11.14 Summary
225 226 226 227 230 230 233 234 234 236 239 240 241 243 243 245 246 249 250
12
Gaussian path integrals 12.1 Exponential operators 12.2 Periodic path integral 12.3 Oscillator normalization 12.4 Evolution kernel for indeterminate final position 12.5 Free degree of freedom: constant external source 12.6 Evolution kernel for indeterminate positions 12.7 Simple harmonic oscillator: Fourier expansion 12.8 Evolution kernel for a magnetic field 12.9 Summary
251 252 253 254 256 260 261 264 267 270
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Part five
Action with acceleration
271
13
Acceleration Lagrangian 13.1 Lagrangian 13.2 Quadratic potential: the classical solution 13.3 Propagator: path integral 13.4 Dirac constraints and acceleration Hamiltonian 13.5 Phase space path integral and Hamiltonian operator 13.6 Acceleration path integral 13.7 Change of path integral boundary conditions 13.8 Evolution kernel 13.9 Summary
273 273 275 277 280 283 286 289 291 293
14
PseudoHermitian Euclidean Hamiltonian 14.1 PseudoHermitian Hamiltonian; similarity transformation 14.2 Equivalent Hermitian Hamiltonian HO 14.3 The matrix elements of e−τ Q 14.4 e−τ Q and similarity transformations 14.5 Eigenfunctions of oscillator Hamiltonian HO 14.6 Eigenfunctions of H and H† 14.6.1 Dual energy eigenstates 14.7 Vacuum state; eQ/2 14.8 Vacuum state and classical action 14.9 Excited states of H 14.9.1 Energy ω1 eigenstate 10 (x, v) 14.9.2 Energy ω2 eigenstate 01 (x, v) 14.10 Complex ω1 , ω2 14.11 State space V of Euclidean Hamiltonian 14.11.1 Operators acting on V 14.11.2 Heisenberg operator equations 14.12 Propagator: operators 14.13 Propagator: state space 14.14 Many degrees of freedom 14.15 Summary
294 295 297 298 301 304 305 307 309 312 313 314 315 317 318 319 321 322 324 326 328
15
NonHermitian Hamiltonian: Jordan blocks 15.1 Hamiltonian: equal frequency limit 15.2 Propagator and states for equal frequency 15.3 State vectors for equal frequency 15.3.1 State vector ψ1 (τ ) 15.3.2 State vector ψ2 (τ )
330 331 331 334 334 335
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15.4 15.5 15.6 15.7
15.8 15.9
Completeness equation for 2 × 2 block Equal frequency propagator Hamiltonian: Jordan block structure 2×2 Jordan block 15.7.1 Hamiltonian 15.7.2 Schrödinger equation for Jordan block 15.7.3 Time evolution Jordan block propagator Summary
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336 337 339 340 342 343 344 344 347
Part six Nonlinear path integrals
349
16
The quartic potential: instantons 16.1 Semiclassical approximation 16.2 A onedimensional integral 16.3 Instantons in quantum mechanics 16.4 Instanton zero mode 16.5 Instanton zero mode: Faddeev–Popov analysis 16.5.1 Instanton coefficient N 16.6 Multiinstantons 16.7 Instanton transition amplitude 16.7.1 Lowest energy states 16.8 Instanton correlation function 16.9 The dilute gas approximation 16.10 Ising model and the double well potential 16.11 Nonlocal Ising model 16.12 Spontaneous symmetry breaking 16.12.1 Infinite well 16.12.2 Double well 16.13 Restoration of symmetry 16.14 Multiple wells 16.15 Summary
351 352 353 355 362 364 368 370 371 372 373 374 376 377 380 381 381 381 383 383
17
Compact degrees of freedom 17.1 Degree of freedom: a circle 17.1.1 Poisson summation formula 17.1.2 The S1 Lagrangian 17.2 Multiple classical solutions 17.2.1 Large radius limit 17.3 Degree of freedom: a sphere 17.4 Lagrangian for the rigid rotor
385 386 387 388 388 391 391 393
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17.5 17.6 17.7 17.8 17.9
Cancellation of divergence Conformation of DNA DNA extension DNA persistence length Summary
References Index
395 397 399 401 403 405 409
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Preface
Quantum mechanics is undoubtedly one of the most accurate and important scientific theories in the history of science. The theoretical foundations of quantum mechanics have been discussed in depth in Baaquie (2013e), where the main focus is on the interpretation of the mathematical symbols of quantum mechanics and on its enigmatic superstructure. In contrast, the main focus of this book is on the mathematics of path integral quantum mechanics. The traditional approach to quantum mechanics has been to study the Schrödinger equation, one of the cornerstones of quantum mechanics, and which is a special case of partial differential equations. Needless to say, the study of the Schrödinger equation continues to be a central task of quantum mechanics, yielding a steady stream of new and valuable results. Interestingly enough, there are two other formulations of quantum mechanics, namely the operator approach of Heisenberg and the path integral approach of Dirac–Feynman, that provide a mathematical framework which is independent of the Schrödinger equation. In this book, the Schrödinger equation is never directly solved; instead the Hamiltonian operator is analyzed and path integrals for different quantum and classical random systems are studied to gain an understanding of quantum mathematics. I became aware of path integrals when I was a graduate student, and what intrigued me most was the novelty, flexibility and versatility of their theoretical and mathematical framework. I have spent most of my research years in exploring and employing this framework. Path integration is a natural generalization of integral calculus and is essentially the integral calculus of infinitely many variables, also called functional integration. There is, however, a fundamental feature of path integration that sets it apart from functional integration, namely the role played by the Hamiltonian in the formalism. All the path integrals discussed in this book have an underlying linear structure that is encoded in the Hamiltonian operator and its linear vector state space. It is this
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combination of the path integral and its underlying Hamiltonian that provides a powerful and flexible mathematical machinery that can address a vast variety and range of diverse problems. Path integration can also address systems that do not have a Hamiltonian and these systems are not studied. Instead, topics have been chosen that can demonstrate the interplay of the system’s path integral, state space, and Hamiltonian. The Hamiltonian operator and the mathematical formalism of path integration make them eminently suitable for describing quantum indeterminacy as well as classical randomness. In two chapters of the book, namely Chapter 7 on stochastic processes and Chapter 17 on compact degrees of freedom, path integrals are applied to classical stochastic and random systems. The rest of the chapters analyze systems that have quantum indeterminacy. The range and depth of subjects that come under the sway of path integrals are unified by a common thread, which is the mathematics of path integrals. Problems seemingly unrelated to indeterminacy such as the classification of knots and links or the mathematical properties of manifolds have been solved using path integration. The applications of path integrals are almost as vast as calculus, ranging from finance, polymers, biology, and chemistry to quantum mechanics, solid state physics, statistical mechanics, quantum field theory, superstring theory, and all the way to pure mathematics. The concepts and theoretical underpinnings of quantum mechanics lead to a whole set of new mathematical ideas and have given rise to the subject of quantum mathematics. The groundbreaking and pioneering book by Feynman and Hibbs (1965) laid the foundation for the study of path integrals in quantum mechanics and is always worth reading. More recent books such as those by Kleinert (1990) and ZinnJustin (2005) discuss many important aspects of path integration and cover a wide range of applications. Given the complex theoretical and mathematical nature of the subject, no single book can conceivably cover the gamut of worthwhile topics that appear in the study of path integration and there is always a need for books that break new ground. The topics chosen in this book have a minimal overlap with other books on path integrals. A major field of theoretical physics that is based on path integrals is quantum field theory, which includes the Standard Model of particles and forces. The study of quantum field theory leads to the concept of nonlinear gauge fields and to the concept of renormalization, both of which are beyond the scope this book. The purpose of the book is to provide a pedagogical introduction to the essential principles of path integrals and of Hamiltonians, as well as to work out in full detail some of the varied methods and techniques that have proven useful in actually performing path integrations. The emphasis in all the derivations is on the methodological and mathematical aspect of the problem – with matters of interpretation
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Preface
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being discussed only in passing. Starting from the simplest examples, the various chapters lay the ground work for analyzing more advanced topics. The book provides an introduction to the foundations of path integral quantum mechanics and is a primer to the techniques and methods employed in the study of quantum finance, as formulated by Baaquie (2004) and Baaquie (2010).
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Acknowledgements
I would like to acknowledge and express my thanks to many outstanding teachers, scholars, and researchers whose work motivated me to study path integral quantum mechanics and to grapple with its mathematical formalism. I had the singular privilege of doing my Ph.D. thesis under the guidance of Nobel Lanreate Kenneth G. Wilson; his visionary conception of quantum mechanics and of quantum field theory – rooted in the path integral – greatly enlightened and inspired me, and continues to do so today. As an undergraduate I had the honor of meeting and conversing a number of times with Richard P. Feynman, the legendary discoverer of the path integral, and this left a permanent impression on me. I thank Frederick H. Willeboordse for his consistent support and Wang Qinghai, Kang Hway Chuan, Zahur Ahmed, Duxin and Cao Yang for many helpful discussions. I thank my wife Najma for being a wonderful companion and for her uplifting approach to family and professional life. I thank my precious family members Arzish, Farah, and Tazkiah for their delightful company and warm encouragement. They have made this book possible. I am deeply indebted to my late father Muhammad Abdul Baaquie for being a life long source of encouragement and whose virtuous qualities continue to be a beacon of inspiration.
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1 Synopsis
This book studies the mathematical aspect of path integrals and Hamiltonians – which emerge from the formulation of quantum mechanics. The theoretical framework of quantum mechanics provides the mathematical tools for studying both quantum indeterminacy and classical randomness. Many problems arising in quantum mechanics as well as in vastly different fields such as finance and economics can be addressed by the mathematics of quantum mechanics, or quantum mathematics in short. All the topics and subjects in the various chapters have been specifically chosen to illustrate the structure of quantum mathematics, and are not tied to any specific discipline, be it quantum mechanics or quantum finance. The book is divided into the following six parts, in accordance with the Chapter dependency flowchart given below. • Part one addresses the Fundamental principles of path integrals and (Hamiltonian) operators and consists of five chapters. Chapter 2 is on the Mathematical structure of quantum mechanics and introduces the mathematical framework that emerges from the quantum principle. Chapters 3 to 6 discuss the mathematical pillars of quantum mathematics, starting from the Feynman path integral, summarizing Hamiltonian mechanics and introducing path integral quantization. • Part two is on Stochastic processes. Stochastic systems are dissipative and are shown to be effectively modeled by the path integral. Chapter 7 is focused on the application of quantum mathematics to classical random systems and to stochastic processes. • Part three discusses Discrete degrees of freedom. Chapters 8 and 9 discuss the simplest quantum mechanical degree of freedom, namely the double valued Ising spin. The Ising model is discussed in some detail as this model contains all the essential ideas that unfold later for more complex degrees of freedom. The general properties of path integrals and Hamiltonians are discussed in the context of the Ising spin. Chapter 10 on Fermions introduces a degree of freedom
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Synopsis Chapter dependency flowchart 2. Mathematical structure of quantum mechanics
3. Operators
7. Stochastic systems
4. Feynman path integral
8. Ising model
10. Fermions
5. Hamiltonian mechanics
11. Simple harmonic oscillator
6. Path integral quantization
16. Quartic potential: instantons
17. Compact degrees of freedom
9. Ising model: magnetic field 12. Guassian path integral
13. Acceleration Lagrangian
14. PseudoHermitian Euclidean Hamiltonian
15. NonHermitian Hamiltonian: Jordan blocks
that is essentially discrete – but is represented by fermionic variables that are distinct from real variables. The calculus of fermions and the key structures of quantum mathematics such as the Hamiltonian, state space, and path integrals are discussed in some detail. • Part four covers of Quadratic path integrals. Chapter 11 is on the simple harmonic oscillator – one of the prime exemplars of quantum mechanics – and it is studied using both the Hamiltonian and path integral approach. In Chapter 12 different types of Gaussian path integrals are evaluated using techniques that are useful for analyzing and solving path integrals. • Part five is on the Acceleration action. An action with an acceleration term is defined for Euclidean time and is shown to have a novel structure not present in usual quantum mechanics. In Chapter 13, the Lagrangian and path integral are
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analyzed and shown to be equivalent to a constrained system. The Hamiltonian is obtained using the Dirac constraint method. In Chapter 14, the acceleration Hamiltonian is shown to be pseudoHermitian and its state space and propagator are derived. Chapter 15 examines a critical point of the acceleration action and the Hamiltonian is shown to be essentially nonHermitian, being block diagonal and with each block being a Jordan block. • Part six is on Nonlinear path integrals. Chapter 16 studies the nonlinear quartic Lagrangian to illustrate the qualitatively new features that nonlinear path integrals exhibit. The double well potential is studied in some detail as an exemplar of nonlinear path integrals that can be analyzed using the semiclassical expansion. And lastly, in Chapter 17 degrees of freedom are analyzed that take values in a compact manifold; these systems have a nonlinearity that arises from the nature of the degree of freedom itself – rather than from a nonlinear piece in the Lagrangian. Semiclassical expansions of the path integral about multiple classical solutions, classified by a winding number and path integrals on curved manifolds, are briefly touched upon.
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Part one Fundamental principles
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2 The mathematical structure of quantum mechanics
An examination of the postulates of quantum mechanics reveals a number of fundamental mathematical constructs that form its theoretical underpinnings. Many of the results that are summarized in this Chapter will only become clear after reading the rest of the book and a rereading may be in order. The dynamical variables of classical mechanics are superseded by the quantum degree of freedom. An exhaustive and complete description of the indeterminate degree of freedom is given by its state function, which is an element of a state space that, in general, is an infinite dimensional linear vector space. The properties of the indeterminate degree of freedom are extracted from its state vector by the linear action of operators representing experimentally observable quantities. Repeated applications of the operators on the state function yield the average value of the operator for the state [Baaquie (2013e)]. The conceptual framework of quantum mechanics is discussed in Section 2.1. The concepts of degree of freedom, state space and operators are briefly reviewed in Sections 2.3–2.5. Three distinct formulations of quantum mechanics emerge from the superstructure of quantum mechanics and these are briefly summarized in Sections 2.7–2.9. 2.1 The Copenhagen quantum postulate The Copenhagen interpretation of quantum mechanics, pioneered by Niels Bohr and Werner Heisenberg, provides a conceptual framework for the interpretation of the mathematical constructs of quantum mechanics and is the standard interpretation that is followed by the majority of practicing physicists [Stapp (1963), Dirac (1999)]. The Copenhagen interpretation is not universally accepted by the physics community, with many alternative explanations being proposed for understanding quantum mechanics [Baaquie (2013e)]. Instead of entering this debate, this book
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The mathematical structure of quantum mechanics
is based on the Copenhagen interpretation, which can be summarized by the following postulates: • The quantum entity consists of its degree of freedom F and its state vector ψ(t, F). The foundation of the quantum entity is its degree of freedom, which takes a range of values and constitutes a space F. The quantum degree of freedom is completely described by the quantum state ψ(t, F), a complex valued function of the degree of freedom that is an element of state space V(F). • The quantum entity is an inseparable pair, namely, the degree of freedom and its state vector. • All physically observable quantities are obtained by applying Hermitian operators O(F) on the state ψ(t, F). • Experimental observations collapse the quantum state and repeated observations yield Eψ [O(F)], which is the expectation value of the operator O(F) for the state ψ(t, F). • The Schrödinger equation determines the time dependence of the state vector, namely of ψ(t, F), but does not determine the process of measurement. It needs to be emphasized that the state vector ψ(t, F) provides only statistical information about the quantum entity; the result of any particular experiment is impossible to predict.1 The organization of the theoretical superstructure of quantum mechanics is shown in Figure 2.1. The quantum state ψ(t, F) is a complex number that describes the degree of freedom and is more fundamental than the observed probabilities, which are always real positive numbers. The scheme of assigning expectation values to operators, such as Eψ [O(F)], leads to a generalization of classical probability to quantum probability and is discussed in detail in Baaquie (2013e). To give a concrete realization of the Copenhagen quantum postulate, consider a quantum particle moving in one dimension; the degree of freedom is the real line, namely F = = {xx ∈ (−∞, +∞)} with state ψ(t, ). Consider the position operator O(x);2 a measurement projects the state to a point x ∈ and collapses the quantum state to yield, after repeated measurements +∞ 2 P (t, x) ≡ Eψ [O(x)] = ψ(t, x) , P (t, x) > 0, dxP (t, x) = 1. (2.1) −∞
Note from Eq. 2.1 that P (t, x) obeys all the requirements to be interpreted as a probability distribution. A complete description of a quantum system requires 1 There are special quantum states called eigenstates for which one can exactly predict the outcome of some
experiments. But for even this special case the degree of freedom is indeterminate and can never be directly observed. 2 The position projection operator O(x) = xx; see Chapter 3.
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2.1 The Copenhagen quantum postulate
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EV [Oˆ ()] Oˆ () V() Quantum Entity
Figure 2.1 The theoretical superstructure of quantum mechanics; the quantum entity is constituted by the degree of freedom F and its state vector, which is an element of state space V(F ); operators O(F ) act on the state vector to extract information about the degree of freedom and lead to the final result EV [O(F )]; only the final result, which is furthest from the quantum entity, is empirically observed.
specifying the probability P (t, x) for all the possible states of the quantum system. For a quantum particle in space, its possible quantum states are the different positions x ∈ [−∞, +∞]. The position of the quantum particle is indeterminate and P (t, x) = ψ(t, x)2 is the probability of the state vector collapsing at time t and at O(x) – the projection operator at position x. The moment that the state ψ(t, ) is observed at specific projection operator O(x), the state ψ(t, ) instantaneously becomes zero everywhere else. The transition from ψ(t, ) to ψ(t, x)2 is an expression of the collapse of the quantum state. It needs to be emphasized that no classical wave undergoes a collapse on being observed; the collapse of the state ψ(t, ) is a purely quantum phenomenon. The pioneers of quantum mechanics termed it as “wave mechanics” since the Newtonian description of the particle by its trajectory x(t) was replaced by the state ψ(t, ) that looked like a classical wave that is spread over (all of) space . Hence the term “wave function” is used by many physicists for denoting ψ(t, ). The state ψ(t, F) of a quantum particle is not a classical wave; rather, the only thing it has in common with a classical wave is that it is sometimes spread over space. However, there are quantum states that are not spread over space. For example, the up and down spin states of a quantum particle exist at a single point; such quantum states are described by a state that has no dependence on space and hence is not spread over space.
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In the text, the terms state, quantum state, state function, or state vector are henceforth used for ψ(t, F), as these are more precise terms than the term wave function. The result given in Eq. 2.1 is an expression of the great discovery of quantum theory, namely, that behind what is directly observed – the outcome of experiments from which one can compute the probabilities P (t, x) = ψ(t, x)2 – there lies an unobservable world of the probability amplitude that is fully described by the quantum state ψ(t, F).
2.2 The superstructure of quantum mechanics The description and dynamics of a quantum entity require an elaborate theoretical framework. The quantum entity is the foundation of the mathematical superstructure that consists of five main constructs: • The quantum degree of freedom space F. • The quantum state vector ψ(F), which is an element of the linear vector state space V(F). • The time evolution of ψ(F), given by the Schrödinger equation. • Operators O(F) that act on the state space V(F). • The process of measurement, with repeated observations yielding the expectation value of the operators, namely Eψ [O(F)]. The five mathematical pillars of quantum mechanics are shown in Figure 2.2.
2.3 Degree of freedom space F Recall that in classical mechanics a system is described by dynamical variables, and its time dependence is given by Newton’s equations of motion. In quantum mechanics, the description of a quantum entity starts with the generalization of the classical dynamical variables and, following Dirac (1999), is called the quantum degree of freedom.
Degree of freedom
State space V()
Dynamics ∂ ψ (t, ) ∂t
Operators
Observation
Oˆ ()
Eψ [Oˆ ()]
Figure 2.2 The five mathematical pillars of quantum mechanics.
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2.4 State space V(F )
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The degree of freedom is the root and ground on which the quantum entity is anchored. The degree of freedom embodies the qualities and properties of a quantum entity. A single quantum entity, for example the electron, can simultaneously have many degrees of freedom, such as spin, position, angular momentum and so on that all, taken together, describe the quantum entity. The symbol F is taken to represent all the degrees of freedom of a quantum entity. A remarkable conclusion of quantum mechanics – validated by experiments – is that a quantum degree of freedom does not have any precise value before it is observed; the degree of freedom is inherently indeterminate and does not have a determinate objective existence before it is observed. One interpretation of the degree of freedom being intrinsically indeterminate is that it simultaneously has a range of possible values; the collection of all possible values of the degree of freedom constitutes a space that is denoted by F; the space F is time independent. The entire edifice of quantum mechanics is built on the degree of freedom and, in particular, on the space F.
2.4 State space V(F ) In the quantum mechanical framework, a quantum degree of freedom is inherently indeterminate and, metaphorically speaking, simultaneously has a range of possible values that constitutes the space F. Consider an experimental device designed to examine and study the properties of a degree of freedom. For a quantum entity that has spin , the degree of freedom consists of 2 + 1 discrete points. A device built for observing a spin system needs to have 2 + 1 possible distinct outcomes, one for each of the possible values of the degree of freedom. The experiment needs to be repeated many times due to the indeterminacy of the quantum degree of freedom. The outcome of each particular experiment is completely uncertain and indeterminate, with the degree of freedom inducing the device to take any one of its (the device’s) many possible values.3 However, the cumulative result of repeated experiments shows a pattern – for example, with the device pointer having some positions being more likely to be observed than others. How does one describe the statistical regularities of the indeterminate and uncertain outcomes of an experiment carried out on a degree of freedom? As mentioned in Section 2.3, the subject of quantum probability arose from the need to describe quantum indeterminacy. A complex valued state vector, also called the state function and denoted by ψ, is introduced to describe the observable properties of the 3 It is always assumed, unless stated otherwise, that a quantum state is not an eigenstate.
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degree of freedom. The quantum state ψ maps the degree of freedom space F to the complex numbers C, namely ψ : F → C. In particular, for the special case of coordinate degree of freedom x ∈ = F the state vector ψ is a complex function of x and hence x ∈ ⇒ ψ(x) ∈ C.
Noteworthy 2.1 Dirac’s formulation of the quantum state. • The foundation of the quantum entity is the degree of freedom F; the quantum state (state, state vector, and state function) provides an exhaustive description of the quantum entity. • The term state or state vector refers to the quantum state considered as a vector in state space V(F ), usually denoted by ψ(t, F). • In Dirac’s bracket notation, a state vector is denoted by ψ(t, F) or ψ in short, and is called a ket vector. • The dual to the ket vector is denoted by ψ(t, F) or ψ in brief and is called a bra vector. • The scalar product of two state vectors χ , ψ is a complex number ∈ C and is denoted by the full bracket, namely χ ψ. • The term state function refers to the components of the state vector and is denoted by xψ(t, F) ≡ xψt ≡ ψ(t, x), where x ∈ F, namely x is a representation of the degree of freedom F. For degrees of freedom taking discrete values, Dirac’s bra and ket vectors are nothing except the row and column vectors of a finite dimensional linear vector space, with the bracket of two state vectors being the usual scalar product of two vectors. When the degree of freedom becomes continuous, Dirac’s notation carries over into functional analysis and allows for studying questions of the convergence of infinite sequences of state vectors that go beyond linear algebra.
One of the most remarkable properties of the quantum state vector ψ is that it is an element of a state space V that is a linear vector space. The precise structure of the linear vector space V depends on the nature of the quantum degree of freedom F. From the simplest quantum system consisting of two possible states, to a system having N degrees of freedom in four dimensional spacetime, to quantum fields having an infinite number of degrees of freedom, there is a linear vector space V and a state vector defined for these degrees of freedom.
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2.4 State space V(F )
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Euclidean space N is a finite dimensional linear vector space; the linear vector spaces V that occur in quantum mechanics and quantum field theory are usually state spaces that are an infinite dimensional generalization of N . Infinite dimensional linear vector spaces arise in many applications in science and engineering, including the study of partial differential equations and dynamical systems and many of their properties, such as the addition of vectors and so on, are the generalizations of the properties of finite dimensional vector spaces. The state vector is an element of a time independent normed linear vector space, namely ψ ∈ V(F). The following are some of the main properties of a vector space V: 1. Since they are elements of a linear vector space, a state vector can be added to other state vectors. In particular, ket vectors ψ and χ are complex valued vectors of V and can be added as follows η = aψ + bχ,
(2.2)
where a, b are complex numbers ∈ C, and yield another element η of V. Vector addition is commutative and associative. 2. For every ket vector ψ ∈ V, there is a dual (bra) vector ψ that is an element of the dual linear vector space VD . The dual vector space is also linear and yields the following η = a ∗ ψ + b∗ χ . The collection of all (dual) bra vectors forms the dual space VD . 3. More formally, VD is the collection of all linear mappings that take elements of V to C by the scalar product. In mathematical notation VD : V → C. The vector space and its dual are not necessarily isomorphic.4 4. For any two ket ψ and bra η vectors belonging to V and VD , respectively, the scalar product, namely ηψ, yields a complex number and has the following property: ηψ = ψη∗ ,
4 Two spaces are isomorphic if there is an invertible mapping that maps each element of one space to a
(unique) element of the other space.
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where ∗ stands for complex conjugation. The scalar product is linear and yields ηζ = a ∗ ψζ + b∗ χ ζ . In particular, ψψ ≡ ψ2 is a real number – a fact of far reaching consequence in quantum mechanics. 5. One of the fundamental properties of quantum states is that two states are distinct if they are linearly independent. In particular, two states ψ and χ are completely distinct if and only if they are orthogonal, namely χ ψ = 0 : orthogonal.
(2.3)
2.4.1 Hilbert space Starting in the 1900s, Hilbert space was studied by David Hilbert, Erhard Schmidt, and Frigyes Riesz as belonging to the class of infinite dimensional function space. The main feature that arises in a Hilbert space is the issue of convergence of an infinite sequence of elements of Hilbert space, something that is absent in a finite dimensional vector space. To allow for the probabilistic interpretation of the state vector ψ, all state vectors that represent physical systems must have unit norm, that is ψψ ≡ ψ2 = 1 : unit norm. The restriction of the linear vector space V to be a normed vector space defines a Hilbert space. For a Hilbert space, the dual state space is isomorphic to the Hilbert space, namely V VD , shown in Figure 2.3. The state space of quantum entities is a Hilbert space. However, there are classical random systems, for example that occur in finance and for quantum dissipative processes, where the state space is not a Hilbert space and in particular leads to a dual state space: VD is not isomorphic to the state space V [Baaquie (2004)]. For the continuous degree of freedom F = , an element of ψ of Hilbert space has unit norm and hence yields +∞ 2 ψψ ≡ ψ = dxψ(x)2 = 1 : unit norm. −∞
2.5 Operators O(F ) The connection of the quantum degree of freedom with its observable and measurable properties is indirect and is always, of necessity, mediated by the process of
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2.5 Operators O(F ) V = State Space
ψ
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ψ
Figure 2.3 Hilbert space is a unit norm state space with V VD .
measurement. A consistent interpretation of quantum mechanics requires that the measurement process plays a central role in the theoretical framework of quantum mechanics. In classical mechanics, observation and measurement of the physical properties plays no role in the definition of the classical system. For instance, a classical particle is fully specified by its position and velocity at time t and denoted by x(t), v(t); it is immaterial whether a measurement is performed to ascertain the position and velocity of the classical particle; in other words, the position and velocity of the classical particle x(t), v(t) exist objectively, regardless of whether its position or velocity is measured or not. In contrast to classical mechanics, in quantum mechanics the degree of freedom F, in principle, can never be directly observed. All the observable physical properties of a degree of freedom are the result of a process of measurement carried out on the state vector ψ. Operators, discussed in Chapter 3, are mathematical objects that represent physical properties of the degree of freedom F and act on the state vector; the action of operators on the state vector is a mathematical representation of the process measuring the physical properties of the quantum entity. The degree of freedom F and its measurable properties – represented by the operators Oi – are separated by the quantum state vector ψ(t, F) [Baaquie (2013e)]. An experiment can only measure the effects of the degree of freedom – via the state vector ψ(t, F) – on the operators Oi . Furthermore, each experimental device is designed and tailor made to measure a specific physical property of the degree of freedom, represented by an operator Oi .
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The mathematical structure of quantum mechanics V
V
Oˆ O ψ
ψ
Hilbert Space
Figure 2.4 An operator O acting on element ψ of the state space V and mapping it to Oψ.
Every degree of freedom F defines a state space V and operators O that act on that state space. All operators O are mathematically defined to be linear mappings of the state space V into itself, shown in Figure 2.4, and yield, for constant a, b O : ψ → Oψ ⇒ O : V → V O aψ1 + bψ2 = aOψ1 + bOψ2 : linear. Operators are the generalization of matrices; an arbitrary element of an operator O is given by ˆ χ Oψ with ψ ∈ V, χ  ∈ VD . The diagonal matrix element of an operator is given by ˆ ψOψ with ψ ∈ V, ψ ∈ VD . Important physical quantities associated with a particle such as its position, momentum, energy, angular momentum, and so on are physical observables that are represented by Hermitian operators, discussed in Section 3.3. Physical quantities are indeterminate; the best that we can do in quantum mechanics is to measure the average value of a physical quantity, termed as its expectation value. For example, a quantum particle, in general, has no fixed value for its observable properties, but only has an average value. For example, the expectation value (average value) of the particle’s position xˆ is given by (2.4) E[x] ˆ = dxxP (x) = dx x ψt (x)2 .
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2.5 Operators O(F )
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The quantum particle’s average value of the position is interpreted as the diagonal values of the position operator xˆ since Eqs. 2.4 and 3.29 yield the following: E[x] ˆ = dxψt xxxψt = ψt xψ ˆ t . All (Hermitian) operators are linear mappings of V onto itself. Let O be an observable, which could be the position operator x, ˆ or momentum operator p, ˆ or the Hamiltonian operator H and so on. Generically, for an operator Oˆ we have Oˆ : V → V. Hence, an operator is an element of the space formed by the outer product of V with its dual VD , that is Oˆ ∈ V ⊗ VD .
(2.5)
A fundamental postulate of quantum mechanics that follows from Eq. 2.1 is the following: on repeatedly measuring the value of the observable O in some state χ , the expectation value (average value) of the observable is given by E[O] ≡ χ Oχ.
(2.6)
In other words, the expectation value of the observable is the diagonal value of the operator O for the given state χ. The expected value of a physical quantity is always a real quantity, and this is the reason for representing all observables by Hermitian operators. Consider some physical quantity, such as a particle’s position, and let it be represented by a Hermitian operator Oˆ with eigenvalues λi and eigenstates ψi defined by ˆ i = λi χi , χi χj = δi−j , Oχ
(2.7)
where, for Hermitian operators the eigenvalues λi are all real. A typical physical state can always be expressed as a superposition of the eigenstates of a Hermitian operator with amplitude ci and can hence be written as ψ = ci χi . i
The result of measuring the physical quantity Oˆ for the state ψ(x) always results in the state function ψ(x) “collapsing” (being projected), with probability ci 2 , to ˆ say χi – whose eigenvalue λi is then one of the eigenstates of the operator O, physically observed.
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After repeated measurements on the system – each made in an identical manner and hence represented by ψ – the average value of Oˆ is given by ˆ ˆ i = = ci 2 χi Oχ ci 2 λi . (2.8) Eψ [O] = ψOψ i
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The measured values of the position, energy, momentum, and so on of a quantum particle are always real numbers. Hence, all physical quantities such as the average position, momentum, energy, and so on must correspond to operators that have only real eigenvalues, namely, for which all λi are real; this is the reason why all physical quantities are represented by Hermitian operators.
2.6 The process of measurement Ignore for the moment details of what constitutes an experimental device. What is clear from numerous experiments is that the experimental readings obtained by observing a quantum entity by the experimental device cannot be explained by deterministic classical physics and, in fact, require quantum mechanics for an appropriate explanation. Consider a degree of freedom F; the existence of a range of possible values of the degree of freedom is encoded in its state vector ψ(F). Let physical operators O(F) represent the observables of the quantum degree of freedom. Recall the degree of freedom cannot be directly observed; instead, what can be measured is the effect of the degree of freedom on the operators mediated by the state vector ψ(F). The preparation of a quantum state yields the quantum state ψ(F), which is then subjected to repeated measurements. Operators O(F) are the mathematical basis of measurement theory. The experimental device is designed to measure the properties of the operator O(F). Measurement theory requires knowledge of special quantum states, namely the eigenstates χn of the operator O(F), which are defined in Eq. 2.7. The process of measurement ascertains the properties of the degree of freedom by subjecting it to the experimental device. The measurement is mathematically represented by applying the operator O(F) on the state of the system ψ(F) and projecting it to one of the eigenstates of O(F), namely ψ(F) → measurement = O(F)ψ(F) → χn : collapse of state ψ(F). Applying O(F) on the state vector causes it to collapse to one of O(F)’s eigenstates. The projection of the state vector ψ to one of the eigenstates χn of the operator O(F) is discontinuous and instantaneous; it is termed as the collapse of the state vector ψ. The result of a measurement has to be postulated to lead to
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2.7 The Schrödinger differential equation
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the collapse of the state vector and is a feature of quantum mechanics that is not governed by the Schrödinger equation. Unlike classical mechanics, where the same initial condition yields the same final outcome, in quantum mechanics the same initial condition leads to a wide range of possible final states. The result of identical quantum experiments is inherently uncertain.5 For example, radioactive atoms, even though identically prepared, decay randomly in time precisely according to the probabilistic predictions of quantum mechanics. After many repeated observations performed on state ψ(F), all of which in principle are identical to each other, the experiment yields the average value of the physical operator O(F), namely O → measurements on ψ(F) → Eψ [O(F)]. The process of measurement cannot be modeled by the Schrödinger equation, and this has long been a point of contention among physicists. Many theorists hold that the fundamental equations of quantum mechanics should determine both the evolution of the quantum state as well as the collapse of the state caused by the process of measurement. As of now, there has been no resolution of this conundrum.
2.7 The Schrödinger differential equation The discussion so far has been kinematical, in other words, focused on the framework for describing a quantum system. One of the fundamental goals of physics is to obtain the dynamical equations that predict the future state of a system. This requirement in quantum mechanics is met by the Schrödinger partial differential equation that determines the future time evolution of the state function ψ(t, F), where t parameterizes time. The Schrödinger equation is time reversible. To exist, all physical entities must have energy; hence, it is reasonable that the Hamiltonian operator H should enter the Schrödinger equation. The Hamiltonian operator H represents the energy of a quantum entity; H determines the form and numerical range of the possible allowed energies of a given quantum entity. Furthermore, energy is the quantity that is conjugate to time, similar to position being conjugate to momentum and one would consequently expect that H should play a central role in the state vector’s time evolution. However, in the final analysis, there is no derivation of the Schrödinger equation from any underlying principle and one has to simply postulate it to be true. The Schrödinger equation is expressed in the language of state space and operators and determines the time evolution of the state function ψ(t), with t being 5 Except, as mentioned earlier, for eigenstates.
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the time parameter. One needs to specify the degrees of freedom of the system in question, that in turn specifies the nature of the state space V; one also needs to specify the Hamiltonian H . The celebrated Schrödinger equation is given by ∂ψ(t) = H ψ(t). (2.9) i ∂t For the case of the degree of freedom being all the possible positions of a quantum particle, F = , in the position basis x, the state vector is −
xψ(t) = ψ(t, x) and the Schrödinger equation given in Eq. 2.9, yields the following ∂ − x ψ(t) = xH ψ(t) i ∂t ∂ ∂ψ(t, x) = H (x, )ψ(t, x), (2.10) ⇒ − i ∂t ∂x where we note that the Hamiltonian operator acts on the dual basis. For a quantum particle with mass m moving in one dimension in a potential V (x), the Hamiltonian is given by ∂2 + V (x) 2m ∂x 2 and yields Schrödinger’s partial differential equation H =−
(2.11)
∂ψ(t, x) ∂ 2 ψ(t, x) =− + V (x)ψ(t, x). i ∂t 2m ∂x 2 A variety of techniques has been developed for solving the Schrödinger equation for a wide class of potentials as well as for multiparticle quantum systems [Gottfried and Yan (2003)]. Let ψ be the initial value of the state vector at t = 0 with ψψ = 1. Equation 2.9 can be integrated to yield the following formal solution −
ψ(t) = e−itH / ψ = U (t)ψ.
(2.12)
Similar to the momentum operator translating the state vector in space, as in Eq. 3.39, the Hamiltonian H is an operator that translates the initial state vector in time, as in Eq. 2.12. The evolution operator U (t) is defined by U (t) = e−itH / , U † (t) = eitH / and is unitary since H is Hermitian; more precisely U (t)U † (t) = I.
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2.7 The Schrödinger differential equation
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The unitarity of U (t), and by implication the Hermiticity of H , is crucial for the conservation of probability. The total probability of the quantum system is conserved over time since unitarity of U (t) ensures that the normalization of the state function is timeindependent; more precisely ψ(t)ψ(t) = ψU † (t)U (t)ψ = ψψ = 1. The operator U (t) is the exponential of the Hamiltonian H that in many cases, as is the case given in Eq. 2.11, is a differential operator. The Feynman path integral is a mathematical tool for analyzing U (t) and is discussed in Chapter 4. The Schrödinger equation given in Eq. 2.9 is a linear equation for the state function ψ(t). Consider two solutions ψ1 (t) and ψ2 (t) of the Schrödinger equation; then their linear combination yields yet another solution of the Schrödinger equation given by ψ(t) = αψ1 (t) + βψ2 (t),
(2.13)
where α, β are complex numbers. The quantum superposition of state vectors given in Eq. 2.13 is of far reaching significance and in particular leads to the Dirac– Feynman formulation of quantum mechanics discussed in Section 2.9. The mathematical reason that state vector ψ(t) is an element of a normed linear vector space is due to the linearity of the Schrödinger equation and yields the result that all state vectors ψ(t) are elements of a linear vector space V. The fact that ψ(t) is an element of a linear vector space leads to many nonclassical and unexpected phenomena such as the existence of entangled states and the quantum superposition principle [Baaquie (2013e)]. The Schrödinger equation has the following remarkable features: • It is a first order differential equation in time, in contrast to Newton’s equation of motion that is a second order differential equation in time. At t = 0, the Schrödinger equation requires that the initial state function be specified for all values of the degree of freedom, namely ψ(), whereas in Newton’s law, only the position and velocity at the starting point of the particle are required. • At each instant, Schrödinger’s equation specifies the state function for all values of the indeterminate degree of freedom. In contrast, Newton’s law of motion specifies only the determinate position and velocity of a particle. • The state vector ψ(t) is complex valued. In fact, the Schrödinger equation is the first equation in natural science in which complex numbers are essential and not just a convenient mathematical tool for representing real quantities. Quantum mechanics introduces a great complication in the description of Nature by replacing the dynamical variables x, p of classical mechanics, which consist of only six real numbers for every instant of time, by an entire space F of the
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indeterminate degree of freedom; a description of the quantum entity requires, in addition, a state vector ψ that is a function of the space F. According to Dirac (1999), the great complication introduced by quantum indeterminacy is “offset” by the great simplification due to the linearity of the Schrödinger equation.
2.8 Heisenberg operator approach Every physical property of a degree of freedom is mathematically realized by a Hermitian operator O. Generalizing Eq. 2.8 to time dependent state vectors and from Eq. 2.12, the expectation value of an operator at time t, namely O(t), is given by Eψ [O(t)] = ψ(t)Oψ(t) = ψeitH / Oe−itH / ψ = tr O(t)ρ) : ρ = ψψ.
(2.14)
The density matrix ρ is a timeindependent operator that encodes the initial quantum state of the degree of freedom. From Eq. 2.14, the timedependent expectation value has two possible interpretations: the state vector is evolving in time, namely, the state vector is ψ(t) and the operator O is constant, or equivalently, the state vector is fixed, namely ψ and instead, the operator is evolving in time and is given by O(t). The timedependent Heisenberg operators O(t) are given by O(t) = eitH / Oe−itH / ∂O(t) = [O(t), H ]. i ∂t
(2.15)
In the Heisenberg formulation of quantum mechanics, quantum indeterminacy is completely described by the algebra of Hermitian operators. All physical observables of a quantum degree of freedom are elements of the Heisenberg operator algebra, and so are the density matrices that encode the initial quantum state of the degree of freedom. Quantum indeterminacy is reflected in the spectral decomposition of the operators in terms of its eigenvalues and projection operators (eigenvectors), as given in Eq. 3.21. For example, the single value of energy for a classical entity is replaced by a whole range of eigenenergies of the Hamiltonian operator for a quantum degree of freedom, with the eigenfunctions encoding the inherent indeterminacy of the degree of freedom. The time dependence of the state vector given by the Schrödinger equation is replaced by the time dependence of the operators given in Eq. 2.15. All expectation values are obtained by performing a trace over this operator algebra, namely by tr ρO(t)) as given in Eq. 2.14.
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2.9 Dirac–Feynman path integral formulation
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From the aspect of quantum probability, Heisenberg’s operator formulation goes far beyond just providing a mathematical framework for the mechanics of the quantum, but instead, also lays the foundation of the quantum theory of probability [Baaquie (2013e)].
2.9 Dirac–Feynman path integral formulation The time evolution of physical entities is fundamental to our understanding of Nature. For a classical entity evolving in time, its trajectory exists objectively, regardless of whether it is observed or not, with both its position x(t) and velocity v(t) having exact values for each instant of time t. We need to determine the mode of existence of quantum indeterminacy for the case of the time evolution of a quantum degree of freedom. Consider a quantum particle with degree of freedom x ∈ = F. Suppose that the particle is observed at time ti , with the position operator finding the particle at point xi and a second observation is at time tf , with the position operator finding the particle at point xf . To simplify the discussion, suppose there are Nslits between the initial and final positions, located at positions x1 , x2 , . . . , xN , as shown in Figure 2.5. There are two cases for the quantum particle making a transition from xi , ti to xf , tf , namely when the path taken at an intermediate time t is observed and when it is not observed. For the case when the path taken at an intermediate time t is observed, one simply obtains the classical result. Time xf
tf
t
x1
x3
x2
xN
ti xi Space
Figure 2.5 A quantum particle is observed at first at initial position xi at time ti and a second time at final position xf at time tf . The quantum particle’s path being indeterminate means that the single particle simultaneously exists in all the allowed paths.
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What is the description of the quantum particle making a transition from xi , ti to xf , tf when it is not observed at an intermediate time t? The following is a summary of the conclusions: • The quantum indeterminacy of the degree of freedom, together with the linearity of the Schrödinger equation, leads to the conclusion that the path of the quantum particle is indeterminate. • The indeterminacy of the path is realized by the quantum particle by existing in all possible paths simultaneously; or metaphorically speaking, the single quantum particle simultaneously “takes” all possible paths. • For the case of N slits between the initial and final positions shown in Figure 2.5, the quantum particle simultaneously exists in all the Npaths. The concept of the probability amplitude, which is a complex number, is used for describing the indeterminate paths of a quantum system. To start with, a probability amplitude is assigned to each determinate path. In the case of no observation being made to determine which path was taken, all the paths are indistinguishable and hence the particle’s path is indeterminate, with the particle simultaneously existing in all the Npaths, as shown in Figure 2.5. The probability amplitude for the quantum particle having an indeterminate path is obtained by combining the probability amplitudes for the different determinate paths using the quantum superposition principle. Let probability amplitude φn be assigned to the determinate path going through a slit at xn with n = 1, 2, . . . N, as shown in Figure 2.5, and let φ(xf , tf xi , ti ) be the net probability amplitude for a particle that is observed at position xi at time ti and then observed at position xf at later time tf . The probability amplitude φ(xf , tf xi , ti ) for the transition is obtained by superposing the probability amplitudes for all indistinguishable determinate paths and yields φ(xf , tf xi , ti ) =
N
φn : indistinguishable paths.
(2.16)
n=1
Once the probability amplitude is determined, its modulus squared, namely φ2 yields the probability for the process in question. For the Nslit case 2 φ(xf , tf xi , ti ) = P (xf , tf xi , ti ), dxf P (xf , tf xi , ti ) = 1, where P (xf , tf xi , ti ) is the conditional probability that a particle, observed at position xi at time ti , will be observed at position xf at later time tf . Quantum mechanics can be formulated entirely in terms of indeterminate paths, a formulation that is independent of the framework of the state vector and the
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2.10 Three formulations of quantum mechanics
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Schrödinger equation; this approach, known as the Dirac–Feynman formulation, is discussed in Chapter 4.
2.10 Three formulations of quantum mechanics In summary, quantum mechanics has the following three independent, but equivalent, mathematical formulations for describing quantum indeterminacy: • The Schrödinger equation for the state vector postulates that the quantum state vector encodes all the information that can be extracted from a quantum degree of freedom. The degree of freedom forever remains indeterminate since all measurements only encounter the quantum state vector, causing it to collapse to an observed manifestation. • The Heisenberg operator formalism. The state vector is completely dispensed with and instead a density matrix, which is an operator, represents the quantum entity. All observations consist of detecting the collapse of the density matrix, which makes a transition from the pure to a mixed density matrix; the detection of the mixed density matrix by projection operators results in the experimental determination of the probability of the various projection operators detecting the quantum entity. Quantum probability assigns probabilities to projection operators. The indeterminate nature of the degree of freedom is reflected in that it is never detected by any of the operators. The violation of the Bellinequality shows that the quantum indeterminacy cannot be explained by classical probability theory; in particular, the degree of freedom has no determinate value before an observation – and hence no objective existence – showing its indeterminate nature.6 • The Dirac–Feynman path integral formulation. The path integral is the sum over all the indeterminate (indistinguishable) paths, from the initial to the final state, and reflects quantum indeterminacy which is at the foundation of quantum mechanics. The state vector appears as initial and final conditions for the indeterminate paths that are being summed over. In the path integral approach, the quantum degrees of freedom appear as integration variables and provide the clearest representation of the indeterminate nature of the degree of freedom. An integration variable has no fixed value but, rather, takes values over its entire range; for the degree of freedom this means that the entire degree of freedom space F is integrated over. The freedom to change variables for path integration is equivalent to changing the representation 6 Quantum probability is fundamentally different from classical probability. The difference was crystallized by
the groundbreaking work of Bell (2004) and is discussed in detail by Baaquie (2013e).
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chosen for the degree of freedom, and is similar to the freedom in choosing basis states for Hilbert space. Each framework has its own advantages, throwing light on different aspects of quantum mechanics that would otherwise be difficult to express. For example, the Schrödinger equation is most suitable for studying the bound sates of quantum entities such as atoms and molecules; the Heisenberg formulation is most suitable for studying the measurement process; and the Feynman path integral is most appropriate for studying the indeterminate quantum paths.
2.11 Quantum entity In light of the mathematical superstructure of quantum mechanics, what is a quantum entity? A careful study of what is an entity, a thing, an object leads to the remarkable conclusion that the quantum entity is intrinsically indeterminate and its description requires a framework that departs from our classical conception of Nature. The quantum entity’s foundation is its degree of freedom F and quantum indeterminacy is due to to the intrinsic indeterminacy of the degree of freedom. A landmark step, taken by Max Born, was to postulate that quantum indeterminacy can be described by a state vector ψ(F) that obeys the laws of quantum probability. The state vector is inseparable from the degree of freedom and encodes all the information that can be obtained from the indeterminate degree of freedom, and is illustrated in Figure 2.6. The state vector ψ(F) encompasses the degree of freedom, but does not do so in physical space; rather, Figure 2.6 illustrates the fact that all observations carried
ψ( )
Figure 2.6 A quantum entity is constituted by its degree of freedom F and the state vector ψ(F) that permanently encompasses and envelopes its degrees of freedom.
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out on the degree of freedom always encounter the state vector and no observation can ever come into direct “contact” with the degree of freedom itself. All “contact” of the measuring device with the degree of freedom is mediated by the state vector. In brief, quantum mechanics provides the following as a definition of the quantum entity: A quantum entity is constituted by a pair, namely the degree(s) of freedom F and the state vector ψ(F) that encodes all of its properties. This inseparable pair, namely the degree of freedom and the state vector, embodies the condition in which the quantum entity exists.
2.12 Summary: quantum mathematics Classical physics is based on explaining the behavior of Nature based on attributing mathematical properties directly to the observed phenomenon; for example, a tangible force acts on a particle and changes its position. The logic of quantum mechanics is quite unlike classical physics. An elaborate mathematical superstructure connects the experimentally observed behavior of the particle’s degree of freedom – enigmatically enough the degree of freedom can never in principle ever be empirically observed – with its mathematical description [Baaquie (2013e)]. All our understanding of a quantum entity is based on theoretical and mathematical concepts that, in turn, have to explain a plethora of experimental data. In the case of quantum mechanics, the mathematical construction has led us to infer the existence of the quantum degree of freedom. The theoretical constructions of quantum mechanics are far from being arbitrary and ambiguous; on the contrary, given the maze of links from the quantum entity to its empirical properties, it is highly unlikely that there are any major gaps or redundancies in the theoretical superstructure of quantum mechanics. Quantum mechanics and quantum field theory – bedrocks of theoretical physics and of modern technology – synthesize a vast range of mathematical disciplines that constitutes its mathematical foundations and has given rise to the discipline of quantum mathematics. Quantum mathematics includes such diverse mathematical fields as calculus, linear algebra, functional analysis and functional integration, probability and information theory, dynamical systems, logic, combinatorics and graph theory, Lie groups and representation theory, differential and algebraic geometry, topology, knot theory, and number theory, to name a few. The relation of quantum mathematics to quantum mechanics is analogous to the connection of calculus to Newtonian mechanics: although calculus was discovered by Newton for explaining classical mechanics, calculus as a discipline goes far beyond Newtonian mechanics – having applications in almost every branch of science. Similarly, it is worth noting that quantum mathematics is a discipline that
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is far greater than quantum mechanics – with possible applications in all fields of science as well as the social sciences that are based on uncertainty and randomness. Quantum mathematics describes random, uncertain and indeterminate systems using the concept of the degree of freedom, which in turn defines a linear vector state space; the dynamics of the degrees of freedom is determined by the analog of the Hamiltonian or the Lagrangian, which are defined on the state space. The expectation values of random quantities – which are functions of the degrees of freedom – can be obtained by using either the techniques of operators and state space or by employing the Feynman path integral (functional integration) that entails summing over all possible configurations of the degrees of freedom. A leading example of quantum mathematics is the explanation of critical phenomena. Classical random systems undergoing phase transitions – such as a piece of iron becoming a magnet when it is cooled – are examples of critical phenomena and are described by classical statistical mechanics. Wilson (1983) solved the problem of classical phase transitions by describing it as a system that has infinitely many degrees of freedom and which is mathematically identical to a (Euclidean) quantum field theory. Experiments later validated the explanation of critical classical systems by quantum mathematics, and in particular by the mathematics of quantum field theory. In fact, based on the common ground of quantum mathematics, there is a two way relation between classical random systems and quantum mechanics. For example, the work of Wilson (1983) showed that all renormalizable quantum field theories, in turn, are mathematically equivalent to classical systems that undergo second order phase transitions. Phase transitions are mathematically described by quantum field theories in Euclidean time. If one restricts quantum mathematics to quantum mechanics, then one may ask questions such as “is probability conserved in phase transitions?” – questions that are clearly meaningless since systems undergoing phase transitions are in equilibrium and hence there is no concept of time evolution in phase transitions. Instead, using quantum mathematics, Wilson (1983) computed classical quantities such as critical exponents that characterize phase transitions, exponents that can be experimentally measured [Papon et al. (2002)]. From the example of phase transitions it can be seen that the symbols of quantum mathematics, when applied to other fields such as finance [Baaquie (2004), Baaquie (2010)], the human psyche [Baaquie and Martin (2005)], the social sciences [Haven and Khrennikov (2013)] and so on, have interpretations that are quite different from quantum mechanics The interpretations of quantum mathematics in these diverse fields have no fixed prescription but, instead, have to be arrived at from first principles [Baaquie (2013a)].
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The main thrust of the remaining chapters is on the mathematics of quantum mechanics, leaving aside questions of how these mathematical results are applied to physics, finance, and other disciplines. Various models are analyzed to develop the myriad and multifaceted principles and methods of quantum mathematics.
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3 Operators
Operators represent physically observable quantities, as discussed in Section 2.5. The structure and property of operators depend on the nature of the degree of freedom; operators act on the state space and in particular on the state vector of a given degree of freedom. The significance of operators in the interpretation of quantum mechanics has been discussed in Baaquie (2013e). The operators discussed in this chapter are mostly based on the continuous degree of freedom, which is analyzed in Section 3.1. Hermitian operators represent physically observable properties of a degree of freedom and their mathematical properties are defined in Section 3.3. The coordinate and momentum operators are the leading exemplar of a pair of noncommuting Hermitian operators and these are studied in some detail in Section 3.4. The Weyl operators yield, as in Section 3.5, a finite dimensional example of the shift and scaling operators; Section 3.8 provides a unitary representation of the coordinate and momentum operators. The term selfadjoint operator is used for Hermitian operators when there is a need to emphasize the importance of the domain of the Hilbert space on which the operators act – a topic not usually discussed in most books on quantum mechanics. Sections 3.10 and 3.11 discuss the concept of selfadjoint operators, in particular the crucial role played by the domain for realizing the property of selfadjointness. It is shown in Section 3.12 how the requirement of selfadjointness yields a nontrivial extension of Hamiltonians that include singular interactions.
3.1 Continuous degree of freedom Continuous and discrete degrees of freedom occur widely in quantum mechanics. An indepth analysis of a discrete degree of freedom is presented in Chapter 8. In this chapter, the focus is on analysis of a continuous degree of freedom and its state space and operators. The structure of the continuous degree of freedom is seen to
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3.1 Continuous degree of freedom a −2a
−a
0
31 +
8
8
–
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a
2a
Figure 3.1 Discretization of a continuous degree of freedom space F = .
emerge naturally by taking the continuum limit of an underlying system consisting of a discrete degree of freedom. Consider a quantum particle that can be detected by the position projection operators at any point of space; to simplify the discussion suppose the particle can move in only one dimension and hence can be found at any point x ∈ [−∞, +∞] = . Hence, the degree of freedom is F = and the specific values of the degree of freedom x constitute a real continuous variable. Since there are infinitely many points on the real line, the quantum particle’s degree of freedom has infinitely many possible outcomes. As shown in Figure 3.1, let the continuous degree of freedom x, −∞ ≤ x ≤ +∞, take only discrete values at points x = na with lattice spacing a and with n = 0, ±1, ±2, . . .; in other words, the lattice is embedded in the continuous line and the lattice point n identified with the point na in . To obtain the continuous position degree of freedom F, let a → 0 and the allowed values of the particle’s position x can take any real value, that is, x ∈ , and hence F → . The discrete basis vectors of the quantum particle’s state space V are represented by infinite column vectors with the only nonzero entry being unity in the nth position. Hence n : n = 0, ±1, ±2, . . . ± ∞, where, more explicitly
⎡ ⎤ ... ⎢0 ⎥ ⎢ ⎥ ⎥ n = ⎢ ⎢1 ⎥ : nth position. ⎣0 ⎦ ...
The basis vectors for the dual state space VD are given by
n = · · · 0 1 0 · · · ⇒ nm = δn−m .
(3.1)
The completeness of the basis states yields the following: +∞
nn = diagonal(. . . , 1, 1, . . .) = I : completeness equation,
n=−∞
where I above is the infinite dimensional unit matrix.
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The limit of a → 0 needs to be taken to obtain a continuous x; in terms of the underlying lattice, the continuous point x is related to the discrete lattice point n by −∞ ≤ x ≤ +∞ : x = lim [na], n = 0, ±1, ± . . . . ± ∞. a→0
The state vector for the particle is given by the ket vector x, with its dual vector given by the bra vector x. The basis state n is dimensionless; the ket vector √ x has a dimension of 1/ a since, from Eq. 3.14, the Dirac delta function has dimension of 1/a. Hence, due to dimensional consistency 1 1 x = lim √ n, x = lim √ n. a→0 a→0 a a
(3.2)
The position projection operator is given by the outer product of the position ket vector with the bra vector and is given by 1 nn. a→0 a
xx = lim
(3.3)
The scalar product, for x = na and x = ma, in the limit of a → 0, is given, from Eqs. 3.1, 3.2, and 3.14, by the Dirac delta function 1 δm−n ⇒ xx = δ(x − x ). a The completeness equation above has the following continuum limit: xx = lim
a→0
I= ⇒
+∞ n=−∞
∞
−∞
nn = lim a a→0
+∞
xx
(3.4)
(3.5)
n=−∞
dxxx = I : completeness equation.
(3.6)
Equation 3.5 shows that the projection operators given in Eq. 3.3 are complete and span the entire state space V. I is the identity operator on state space V; namely for any state vector ψ ∈ V, it follows from the completeness equation that Iψ = ψ. The completeness equation given by Eq. 3.6 is a key equation that is central to the analysis of state space, and yields ∞ ∞ dzxzzx = dzδ(x − z)δ(z − x ) = δ(x − x ), xIx = −∞
−∞
that follows from the definition of the Dirac delta function δ(x − x ). The above equation shows that δ(x − x ) is the matrix element of the identity operator I for the continuous degree of freedom F = in the x basis.
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The state space V(F) of a continuous degree of freedom F is a function space and it is for this reason that the subject of functional analysis studies the mathematical properties of quantum mechanics. For the case of F = , the state vector f is an element of V() and yields a state function f (x) given by f (x) = xf ; hence all functions of x, namely f (x), can be thought of as elements of a state space V(). Being an element of a state space endows the function f (x) with the additional property of linearity that needs to be consistent with all the other properties of f (x). It should be noted that not all functions are elements of a (quantum mechanical) state space.
Noteworthy 3.1 Dirac delta function The Dirac delta function is useful in the study of continuum spaces, and some of its essential properties are reviewed. Dirac delta functions are not ordinary Lebesgue measureable functions since they have support set with measure zero; rather they are generalized functions also called distributions. In essence, the Dirac delta function is the continuum generalization of the discrete Kronecker delta function. Consider a continuous line labelled by coordinate x such that −∞ ≤ x ≤ +∞, and let f (x) be an infinitely differentiable function. The Dirac delta function, denoted by δ(x − a), is defined by the following:
+∞
−∞
+∞
−∞
δ(x − a) = δ(a − x) : even function, 1 δ(c(x − a)) = δ(x − a), c dxf (x)δ(x − a) = f (a),
dxf (x)
n dn n d δ(x − a) = (−1) f (x)x=a . dx n dx n
The Heaviside step function (t) is defined by ⎧ ⎨1 t > 0 (t) = 12 t = 0 . ⎩ 0 t λ2 , the state 1 has the lower energy. Hence, for N → ∞, and labeling 1 =  = the vacuum state, yields λ ! 2 N N N L = λ1  + O . (8.40) λ1 Hence, for N → ∞, the periodic correlator given in Eq. 8.17 reduces to 1 N λ tr(μˆ r μˆ 0 ) Z 1 = μˆ r μˆ 0 .
Cr =
(8.41)
Recall from Eq. 6.40 that time ordering of operators is given by 6 Oˆ t Oˆ t t > t ˆ ˆ . T (Ot Ot ) = ˆ ˆ Ot Ot t < t Hence, it follows, in general, that Cr = T (μˆ k+r μˆ k ).
(8.42)
8.5 Ising model’s path integral The derivation given in Section 8.4 is for the partition function and correlator of the periodic chain L. Another derivation is now given (for a periodic lattice of size N ) of the partition function and the correlator by summing over all the possible 2N configurations. This derivation is an example of evaluating a discrete path integral.
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8.5.1 Ising partition function The partition function is given by Z = tr(LN ); one can evaluate the trace for Z by inserting, N times, the complete set of states given in Eq. 8.1, yielding μμ = I, μ , μ = −1, +1. μ=±1
Introducing an index μi on μ to distinguish the different completeness equations that are inserted into the trace to evaluate Z, one can write LN as an N fold matrix product and the trace is achieved by having periodic boundary conditions on the spin variables, namely μi = μi+N ; in particular μ1 = μN+1 . Hence the partition function is given by Z = tr(LN ) = μN  LLL · · · LLL ) *+ · · · LLL, μN μN
=
N−times
N &
i=1 μi =±1
=
e
N &
μi+1 Lμi =
N & N & i=1 μi =±1
i=1
eKμi+1 μi
i=1
: periodic boundary conditions μ1 = μN .
S
{μ}
The action S is given by S=K
N
(8.43)
μi+1 μi .
i=1
Note μ1 , μ2 , . . . μN are discrete random variables taking values of ±1. There are 2N possible spin configurations, and eS[μ1 ,...μN ] /Z is the (normalized) probability distribution for the occurrence of a particular spin configuration (also called a fluctuation). The action can be rewritten using the special property of the spin variables that (μn μn+1 )2 = 1; this fact yields eS = eK
(N
n=1 μn+1 μn
=
N &
eKμn μn+1
n=1
=
N &
[cosh K + sinh Kμn μn+1 ]
n=1
= cosh K N
N &
[1 + tanh Kμn μn+1 ] .
(8.44)
n=1
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Representing μn μn+1 by a bond connecting the lattice site n with n + 1, one can see that eS has an expansion that consists of the sum of all possible combinations C of bonds, with the term 1 arising from no bond to the term N n=1 μn that has one contribution from every bond. Consider the simple product (1 + a)(1 + b) = 1 + a + b + ab (1 + a)(1 + b)(1 + c) = 1 + a + b + ab + c + ac + bc + abc (
...... = ........
(8.45)
Since μ μ = 0, we conclude that, at each lattice site n, only even or no spin variables can contribute to the sum for Z. There are only two terms in the expansion for S that satisfy this condition, namely the term with no bonds and the term with ( all the bonds. Every sum contributes a factor of 2, since ui =±1 = 2, and hence ( N {μ} = 2 . The partition function is Z = tr LN = 2N coshN K[1 + tanhN K] N = λN 1 + λ2
and we obtain the expected result given in Eq. 8.26.
8.5.2 Path integral calculation of Cr The correlator, given by Eq. 8.17, can be written as 1 tr μˆ k+r μˆ k LN Z 1 ˆ r μˆ = tr LN−r μL Z 1 μk LN−r μL ˆ r μμ ˆ k . = Z μ =±1
Cr =
k
Inserting a complete set of states yields 1 Cr = μk LN−r μμ ˆ k+r μk+r Lr μk μk Z μ μ k k+r 1 = μk+r μk μk LN−r μk+r μk+r Lr μk Z μ μ k k+r 1 K (Nn=1 μn μn+1 e μk+r μk , = Z {μ} ( where {μ} is a sum over all possible configurations. Hence
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μk+r
μk+r + μk
μk
(tanh K)Nr
Figure 8.7 The two bond configurations that contribute to the correlation function ( S {μ} e μk+r μk /Z. Note that the correlation function is independent of k.
Cr =
1 S e μr μ0 Z {μ}
: path integral,
where the action is given in Eq. 8.43. Since the lattice is periodic, the correlator Cr does not depend on the index k and hence Cr =
1 S e μr μ0 . Z {μ}
Using the expression Eq. 8.44 for the Ising action yields Cr =
coshN K & (1 + tanh Kμn μn+1 )μr μ0 . Z n {μ}
(8.46)
The μr and μ0 terms must be canceled by bonds. Only two terms survive from the product, namely a product of bonds clockwise going from μr and μ0 and another term going counterclockwise, as shown in Figure 8.7. Hence performing the sum yields, as expected 2N coshN K tanhr K + tanhN−r K Z
1 tanhr K + tanhN−r K . = N 1 + tanh K
Cr =
(8.47)
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Figure 8.8 The spins are decimated (summed over) from the boundary.
8.6 Spin decimation Consider an open chain one dimensional Ising model with coupling constants Kn and the action given by
S=
N−1
Kn μn μn+1 ,
(8.48)
eS.
(8.49)
n=1
Z=
{μ}
The partition function Z can be evaluated by summing over the spin μN at the boundary, called spin decimation. Then decimate spin μN−1 , μN−2 ,. . . all the way to spin μ1 . In symbols one has S=
N−1
(8.50)
Kn μn μn+1 ,
n=1
Z=
N &
eKn μn μn+1
n=1 μn =±1
=
eKN−1 μN−1 μN
μN
eKN−2 μN−2 μN−1 . . .
μN−1
= eKN−1 μN−1 + e−KN−1 μN−1 = 2 cosh(KN−1 )
eKN−2 μN−2 μN−1 . . .
μN−1
e
KN−2 μN−2 μN−1
...
μN−1
Recursively decimating the spins from the boundary inwards, as shown in Figure 8.8, yields
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Z=
.N−1 &
/ 2 cosh(Kn )
μ1
n=1
=2
.N−1 &
/
2 cosh(Kn ) .
n=1
To evaluate μn μn+r note that the correlation function can be rewritten by differentiating the action with respect to the coupling constants Km ; one starts with the coupling constants Kn corresponding to the term in the action Kn μn μn+1 and keeps differentiating on the next lattice site until one reaches site n + r − 1. In symbols one has 1 μn μn+r eS Z ∂ 1 ∂ ∂ eS , = ··· Z ∂Kn ∂Kn+1 ∂Kn+r−1 n+r−1 n+r−1 & sinh(K ) & = tanh(K ). = cosh(K ) =n =n
E[μn μn+r ] =
(8.51) (8.52)
As expected, the limit K = K = constant yields E[μn μn+r ] → tanhr K
for K = K = const.
8.7 Ising model on 2 × N lattice Consider the Ising model on a two site by N steps in the time direction. At each step there are two spin variables μn and μn . Suppose the time lattice is a periodic lattice, that is μN+1 = μ1 and μN+1 = μ1 . The action is given by S=K
N n=1
μn μn
+K
N
μn μn+1 + K
n=1
N
μn μn+1
n=1
=
N
L(n).
n=1
The action is defined on the Ising “ladder”, as shown in Figure 8.9, and yields the Lagrangian L(n) =
K μn μn + μn+1 μn+1 + Kμn μn+1 + Kμn μn+1 . 2
(8.53)
To simplify the notation, let us make the simplification μn+1 = μ, μn+1 = μ μn = λ,
μn = λ .
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Figure 8.9 Ising “ladder” for 2 × N lattice.
This yields K λ λ + μμ + K λμ + λ μ . L μ, μ ; λλ = 2
(8.54)
The partition function is given by Z= eS[μ;μ ] = eS[μ,λ] . {μ;μ }
{μ,λ}
One has to choose a set of basis vectors to write out the matrix elements of the transfer matrix L. Since there are two spins at each time n, the natural basis states for each n are given by the tensor product of the basis states of the two lattice sites, namely μn ; μn ≡ μn ⊗ μn . In the simplified notation introduced above, the matrix elements of L are given by μ; μ Lλ; λ . The explicit construction of the basis vectors λ ⊗ λ , using the rules of tensor product of state vectors [Baaquie (2013e)], yields ⎛ ⎞ 1 ⎜0⎟ 1 1 ⎟ ⊗ =⎜ ⎝0⎠, 0 0 0 ⎛ ⎞ 0 ! ! ⎜ 1 0 1⎟ ⎟ ⊗ =⎜ ⎝0⎠, 0 1 0 ⎛ ⎞ 0 ! ! ⎜ 0 1 0⎟ ⎟ ⊗ =⎜ ⎝1⎠, 1 0 0 ⎛ ⎞ 0 ! ! ⎜ 0 0 0⎟ ⎟ =⎜ ⊗ ⎝0⎠. 1 1 1 !
1 = + ⊗ + =
2 = + ⊗ − =
3 = − ⊗ + =
4 = − ⊗ − =
!
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The matrix elements of L are defined with respect to the four basis vectors, and the matrix can be written as Lij ≡ iLj , where i, i = 1, 2, 3, 4 as given above and j  is the transpose of the i vectors. Hence ⎞ L14 L24 ⎟ ⎟, Lij = Lj i , L34 ⎠
⎛
L11 L12 L13 4 ⎜ .. L22 L23 L= Lij ij  = ⎜ ⎝ .. .. L33 i,j =1 .. .. ..
(8.55)
L44
with the other elements being fixed by the fact that L is a symmetric matrix, for example L11 = 1L1 and so forth. Thus the transfer matrix is given by
Lij = eL(μ,μ ;λλ ) , where the matrix elements are given by L11 = eL(1,1;1,1) = e3K ; L12 = L13 = eL(1,1;−1,1) = 1, L14 = eL(1,1;−1,−1) = e−K , L22 = eL(1,−1;1,−1) = eK , L23 = eL(1,−1;−1,1) = e−3K , L24 = eL(1,−1;−1,−1) = 1, L33 = eL(−1,1;−1,1) = eK , L34 = eL(−1,1;−1,−1) = 1, L44 = eL(−1,−1;−1,−1) = e3K . Collecting all the results yields the symmetric transfer matrix L given by ⎛
e3K ⎜ 1 L=⎜ ⎝ 1 e−K
1 eK e−3K 1
1 e−3K eK 1
⎞ e−K 1 ⎟ ⎟, 1 ⎠ e3K
with eigenvalues λ1 > λ2 > λ3 > λ4 (decreasing in magnitude) given by λ1 = e−3k (−1 + e4k ), λ2 = e−k (−1 + e4k ), 9 1 λ3 = e−4k ek + e3k + e5k + e7k − ek (1 + 22k ) 1 − 4e2k + 10e4k − 4e6k + e8k , 2 9 1 −4k k e + e3k + e5k + e7k + ek (1 + 22k ) 1 − 4e2k + 10e4k − 4e6k + e8k . λ4 = e 2
The partition function is given by Z ==
{μ,λ}
eS[μ,λ] = tr(LN ) =
4
λN i .
i=1
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8.8 Summary
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The Ising model can be extended to a square lattice by adding lattice sites to extend the Ising “ladder” and create an N ×N two dimensional lattice. The transfer matrix is 2N × 2N and has 2N eigenvalues λI with the partition function given by 2 N
Z=
[λI ]N .
I =1
The limit of N → ∞ of the two dimensional Ising model was exactly solved by Onsager in a landmark derivation and exhibits a second order phase transition [Papon et al. (2002)]. Similarly, one can go to higher dimensions by defining Ising spins on an N d lattice given by N × N × N . . . × N . The transfer matrix rapidly becomes more and more complicated. 8.8 Summary The Ising model is based on the simplest possible degree of freedom, namely one having only two possible values. The onedimensional Ising model is a toy model that has all the mathematical structures of quantum mechanics, from the state space to operators and onto path integrals and correlation functions. The periodic lattice was studied for explicitly deriving the partition function and the correlation function, and these derivations illustrate the general features of the computations that are carried out for more complex systems. One significant feature of the Ising model is the expansion of the action in a power series. This is possible since the Ising spin is a compact variable, taking values in a finite range. This property does not hold for Gaussian degrees of freedom and hence is a special feature of the Ising model. The generalization of the onedimensional Ising model to the 2 × N Ising ladder can be extended to defining the Ising model on an N × N lattice and shows the procedure that is required for defining the model in higher dimensions.
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9 Ising model: magnetic field
The Ising model in a magnetic field is a continuation of the discussions of Chapter 8. The equations have to be generalized to include the presence of a magnetic field. A number of new features are studied, in particular, the effect of the Ising model’s boundary conditions on the partition and correlation functions. The Ising model in a magnetic field is introduced in Section 9.1 and the evolution for this system is obtained. The magnetization of the Ising models is an important manifestation of the magnetic field and is discussed for a periodic lattice in Section 9.3. The concept of correlation function is discussed in Section 9.4 using the concept of linear regression. Magnetization for an open chain is discussed in Section 9.5, and block spin renormalization is discussed in Section 9.6.
9.1 Periodic Ising model in a magnetic field The earlier calculation of the Ising model for a periodic lattice is generalized to the case with a nonzero magnetic field. The calculation is more complex than the earlier case showing new features of the transfer matrix. The partition function for an N size periodic lattice is defined by ZN =
eS ,
{μ}
{μ}
≡
N &
, μN+1 = μ1 ,
n=1 μn =±1
where S≡
N
Kμj μj +1 + hμj .
(9.1)
j =1
Writing the partition function as ZN = tr LN
(9.2)
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9.1 Periodic Ising model in a magnetic field
yields the Hermitian (symmetric) transfer matrix ' 1 L(μ, μ ) = exp Kμμ + h μ + μ . 2 The eigenvalues and eigenfunctions of the transfer matrix are ! eK+h e−K , Lφi = λi φi , i = 1, 2, L= e−K eK−h 1/2 , λ1 = eK cosh h + e2K sinh2 (h) + e−2K 1/2 λ2 = eK cosh h − e2K sinh2 (h) + e−2K , ! 1 1 φ1 = √ , 2 a 1+a ! 1 a , φ2 = √ 1 + a 2 −1 a = eK λ1 − eK+h .
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(9.3)
(9.4)
(9.5) (9.6) (9.7)
The completeness equation is given by φi φi  = 1. i=1,2
From Eqs. 9.6 and 9.7 ˆ 1 = −φ2 μφ ˆ 2 = φ1 μφ
1 − a2 2a , φ1 μφ ˆ 2 = φ2 μφ ˆ 1 = . 2 1+a 1 + a2
(9.8)
The eigenvalues yield N L = λ1 φ1 φ1  + λ2 φ2 φ2  ⇒ LN = λN 1 φ1 φ1  + λ2 φ2 φ2 .
Hence, the partition function is given by N ZN = tr LN = λN 1 + λ2 .
(9.9)
(9.10)
The periodic Ising model with negative coupling, namely −K, is given by S ≡
N
−Kμj μj +1 + hμj .
j =1
By a change of variables for only the odd spins, that is, μ2n+1 → −μ2n+1 , one can change the sign of K for all the bonds. This yields S ≡
N
Kμj μj +1 + h(−1)n μj .
j =1
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Hence, one can see that −K corresponds to antiferromagnetic coupling, with the nearest neighbor spins tending to be oppositely aligned. The alternating sign of the magnetic field drives this antialignment. All the formulas derived for +K can be applied to the case for −K by a simple change of sign. 9.2 Ising model’s evolution kernel From Eq. 9.4, the transfer matrix of the Ising model in a magnetic field is given by ! eK+h e−K . L= e−K eK−h The eigenvalues and eigenvectors of L, given in Eqs. 9.5, 9.6, and 9.7, yield ! N ! ! 1 1 a λ1 1 a 0 N L = a −1 0 λN 1 + a 2 a −1 2 ! N N 2 N 1 λN 1 + a λ2 a λ1 − λ2 . (9.11) = N N a 2 λN 1 + a 2 a λN 1 − λ2 1 + λ2 The evolution kernel for the Ising model is given by 2 3 γ μ + μ . K μ , μ; N = μ LN μ = exp β + αμμ + 2 In matrix notation ! α+γ e−α β e . K=e e−α eα−γ
(9.12)
From Eqs. 9.11 and 9.12, after some algebra N 2 N N 2 N N 2 N λ a + a λ λ + λ + a λ λ 1 2 1 2 2 e2γ = 21 N , e4α = , N 2 a λ1 + λN a 2 λN 2 1 − λ2 √ N 2 N a N N 1/2 2 N 1/4 N 1/4 λ λ a eβ = − λ + a λ λ + λ . 1 2 1 2 1 2 1 + a2 In the limit of h → 0 a → 1, λ1 → κ1 , λ2 → κ2 , with κ1N = 2N coshN (K), κ2N = 2N sinhN (K). The evolution kernel for zero magnetic field h → 0 is hence given by the parameters ! 1 κ N + κ2N 1 α → ln 1N ; β → ln κ12N − κ22N − ln 2; γ → 0. N 2 2 κ1 − κ2
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9.3 Magnetization
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9.3 Magnetization The magnetization is defined by MN = E[μn ] =
1 μn eS . Z {μ}
Due to the periodicity of the lattice, μn = μn+N and hence MN is independent of n. Hence MN =
N 1 1 1 ∂ S E[μn ] = e · N n=1 N Z ∂h {μ}
1 ∂ ln(ZN ) N ∂h # 1 ∂ ∂ = ln(λ1 ) + ln 1 + ∂h N ∂h
=
λ2 λ1
!N $ .
Consider the limit of N → ∞, called the thermodynamic limit. Since λ1 > λ2 , the partition function is given by
−N . ZN = λN 1 1+O e Hence, the magnetization is given by ∂ M = lim MN = ln(λ1 ) + O(e−N ) N→∞ ∂h . / 1 e2K sinh h cosh h K = e sinh h + 1/2 . λ1 e2K sinh2 (h) + e−2K
(9.13)
The magnetization can also be derived using the transfer matrix. For a periodic lattice 1 μn e S MN = μn = ZN {μ} =
1 tr LN μˆ . ZN
(9.14)
From Eqs. 9.8, 9.9, and 9.10 2 1 φi LN μφ ˆ i MN = ZN i=1
=
1 N ˆ 1 + λN ˆ 2 λ1 φ1 μφ 2 φ2 μφ ZN
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Figure 9.1 Ising magnetization for the case of +K and −K.
1 (1 − a 2 ) N · · (λN 1 − λ2 ) ZN (1 + a 2 ) N 1 − a 2 λN 1 − λ2 = · . N 1 + a 2 λN 1 + λ2 =
(9.15)
Taking the limit of N → ∞ yields the magnetization 1 − a2 , (9.16) N→∞ 1 + a2 and it can be shown to be equal to the result given in Eq. 9.13. Figure 9.1 plots the magnetization given in Eq. 9.15 as a function of h with K = ±1 and for two cases, namely N = 2 and N = 100. It can be seen that the magnetization converges very fast to its large lattice value. For −K, the magnetization has a smaller value since, due to the antiferromagnetic coupling, the spins tend to antialign. M = lim MN =
9.3.1 Correlator The periodic chain correlation function is given by 1 μk μk+r eS , E[μk μk+r ] = ZN {μ} where the partition function is given by Eq. 9.10. From Eq. 8.17, in terms of the transfer matrix, the correlator is given by 1 tr LN−r μL ˆ r μˆ . Cr = ZN
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9.4 Linear regression
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The expression given above for the nonzero magnetic field looks the same as for the case of zero magnetic field, the only difference between the two cases being the values of the eigenvalues and eigenvectors of the transfer matrix L. The correlation function can then be written, similarly to Eq. 8.31, as 2 2 1 N−r r λi λj φi μφ ˆ j . Cr = ZN i,j =1
A straightforward but long derivation, using the result above and Eqs. 9.5 and 9.10, yields ⎡ r N−r ⎤ λ2 + λλ21 1 ⎢ ⎥ 2 2 2 λ1 (9.17) Cr = ⎣ 1 − a + 4a ⎦. N (1 + a 2 )2 λ2 1 + λ1 As expected, C0 = 1. Recall from Eq. 9.16 M=
1 − a2 1 + a2
and hence
N−r + λλ21 . N λ2 1 + λ1
r 4a
Cr = M2 +
2
1 + a2
2
λ2 λ1
(9.18)
Note that since λ1 > λ2 , we can define correlation length ξ by ! ! ! λ2 r λ2 N λ2 N−r ≡ exp(−r/ξ ), lim → 0. N→∞ λ1 λ1 λ1 Hence, for an infinitely large lattice, N → ∞, from Eq. 9.18 4a 2
Cr = M2 +
1 + a2
2 e
−r/ξ
+ O e−N .
(9.19)
The limit of h = 0 leads to a → 0 and results in M = 0; the value of Cr converges to the value of the correlator for zero magnetic field given in Eq. 8.36.
9.4 Linear regression For two random variables to be correlated means that the variables take their random values in tandem, that is, the value of one of them takes a predictable range of values if the other takes certain values and vice versa.
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The concept of correlation is very different from causation, in which the cause determines the effect: heat and fire are correlated as well as causally linked, with heat causing fires. Consider the height and weight of a person in a given age group; although both are random variables they are nevertheless related (correlated) since the taller the person is, the greater is the likelihood that the person is heavier and vice versa. However, although they are correlated, height is not taken to be the cause of weight and vice versa. Another example is intelligence and say height – which have no correlation – again without one being the cause of the other. Consider random variables X and Y . Suppose they are linearly related and let Y = p + qX.
(9.20)
The equation above is written so that the random variable X is considered as the independent random variable and Y the dependent random variable. For the case of height and weight, the height can be taken to be the independent random variable with the weight being the dependent random variable. One uses the average value and the correlation of X, Y and variance of X to fix the parameters p, q in the following manner. Let E[XY ]c = E[XY ] − E[X]E[Y ], σ 2 (X) = E[X2 ] − E[X]2 . Then q=
E[XY ]c , p = E[Y ] − qE[X]. σ 2 (X)
(9.21)
The estimate of the variance of Y from above is given by 2 σEst (Y ) = q 2 σ 2 (X) =
XY 2c . σ 2 (X)
(9.22)
The exact variance of Y is σ 2 (Y ). Hence, one measure of the accuracy of the linear regression between X and Y is given by the fractional error 7 2 σ 2 (Y ) − σEst (Y ) FE = σ 2 (Y ) 7 E[XY ]2c = 1− 2 . (9.23) σ (X)σ 2 (Y ) From the point of view of probability theory, the spins μ0 and μr are two random variables; let X = μr , Y = μ0 .
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The average value of the random variable is given by the magnetization, namely E[μr ] =
1 S e μr . ZN {μ}
(9.24)
For the periodic lattice, the magnetization is given by E[μr ] = E[μ0 ] = MN .
(9.25)
The expectation value of the product of two spin variables, namely μ0 and μr , is given by E[μr μ0 ] =
1 S e μr μ0 . ZN {μ}
(9.26)
The connected two spin correlation function is given by E[XY ]c = Gr = E[μr μ0 ] − E[μr ]E[μ0 ].
(9.27)
For the case of the Ising spin on a finite periodic lattice, from Eqs. 9.15 and 9.18, the correlator is Gr = Cr − M2N =
(1 − a ) (1 + a 2 )2 2 2
2λN 2 N λ 1 + λN 2
⎡ λ2 N−r ⎤ λ2 r + 4a λ1 ⎣ λ1 + λ2 N ⎦ . 2 2 (1 + a ) 1 + λ1 2
Furthermore, the variance of μr is given by
σ 2 (μr ) = E μ2r − (E [μr ])2 = 1 − M2N = σ 2 (μ0 ),
(9.28)
(9.29)
where the last equation follows due to the periodic lattice. From the point of view of probability theory all the spins in the lattice, namely μn with n = 1, 2, . . . N are random variables. The behavior of all the spins can be fully described by specifying how the different random variables are correlated. One can start with the simplest relation between the random variables’ as given by the linear regression. From Eqs. 9.20 and 9.21 μr p + qμ0 , Gr Gr q= 2 , = σ (μr ) 1 − M2N p = μr − qμ0 = [1 − q]MN .
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From Eq. 9.23 the fractional error for the Ising spins is given by 7 1−
FE = =
Gr 2 1 − M2N
0 r=0 . 1 r >> 1
(9.30)
For simplicity, consider an infinitely large lattice N → ∞ that yields, from Eq. 9.19,
μr p + e−r/ξ μ0 , p = 1 − e−r/ξ M, 9 F E = 1 − e−2r/ξ . For r ξ we have to leading order that the random variables μr and μ0 are decorrelated and take values independently of each other; that is r ξ ⇒ μr M
: independent of μ0 .
The fractional error FE is small when r, the distance of the two spins μ0 and μr , is less than ξ and hence the linear regression is a good fit; when r ξ the error FE ∼ 1, which is equal to the magnitude of the spin, and hence the linear regression is no longer a good fit. In summary, the correlation length ξ is the crucial quantity in the validity of linear regression. The correlation length ξ specifies the distance within which the spins are correlated. In fact, in statistical mechanics and quantum field theory, the correlation length ξ plays a central role in describing physical phenomena. To fully specify the theory one needs to have a generalized nonlinear regression that expresses all the product spins, namely all possible combination of bonds, in terms of the other remaining spins. For instance, one needs to specify μ0 μn is terms of products of other spins excluding the spins μ0 and μn . Note that one cannot arbitrarily assign values to the regression of various products of spins since these must satisfy consistency conditions. At most, one can try and define a theory perturbatively by specifying the regression coefficients for the product of a few random variables. The assignment of the probability distribution – in statistical mechanics and quantum theory – is given by the expression eS[μ0 ,...μN−1 ] /Z for the (normalized) probability distribution for a particular spin configuration. This completely and uniquely defines the theory, as well as determining the regression and correlation of all the random spin variables in a selfconsistent manner.
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μN
μ1
Figure 9.2 Open lattice with boundary spins given by μ1 , μN .
9.5 Open chain Ising model in a magnetic field Consider the one dimensional Ising model for an open chain of length N with arbitrary boundary values for μ1 , μN as shown in Figure 9.2. The open chain partition function is defined by eS ZN = {μ}
and the action is given by S≡K
N−1
μj μj +1 + h
j =1
≡
{μ}
N &
N
μj ,
j =1
.
n=1 μn =±1
The partition function can be written in terms of the transfer matrix as ⎡ ⎤ N−1 N−1 h h ⎣ ⎦ ZN = exp K μj μj +1 + (μj + μj +1 ) exp (μN + μ1 ) . 2 j =1 2 {μ} j =1 (9.31) The sum for the magnetic field runs from 1 to N − 1 since it extends to the boundaries. The term in the action ' h exp (μN + μ1 ) 2 is left over due to the open chain condition. The transfer matrix is given by ' 1 (9.32) L(μ, μ ) = exp Kμμ + h μ + μ , 2 which is the same as the one given in Eq. 9.3 for the periodic lattice. From Eq. 9.31, the partition function ZN can be written as ⎤ ⎡ N−1 & h ⎦ ⎣ ... μj Lμj +1 exp (μN + μ1 ) ZN = 2 μ μ j =1 1
N
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=
⎡ ⎤ N−2 & h ... μ1  ⎣ Lμj μj ⎦ LμN e 2 (μN +μ1 )
μ1
=
μ1
j =2
μN
μ1 L
N−2
μN
#
= tr
h
· LμN e 2 (μN +μ1 )
μ1
$ LN−1 μN μ1 e
h 2 (μN +μ1 )
.
μN
In matrix notation, the partition function is given by ZN = tr(LN−1 B),
(9.33)
where the boundary condition B is given by h μN μ1 e 2 (μN +μ1 ) . B= μ1
μN
In components h
h
μN Bμ1 = e 2 μN e 2 μ1 ⇒ B =
! eh 1 . 1 e−h
(9.34)
Hence, from Eqs. 9.6 and 9.7 ⇒ ZN = λN−1 φ1 Bφ1 + λN−1 φ2 Bφ2 1 2 h h h 2 h 2 e 2 + ae− 2 e− 2 − ae 2 + λN−1 . = λN−1 1 2 1 + a2 1 + a2
(9.35)
The case of h = 0 yields the partition function ZN = 2(2 cosh K)N−1 .
(9.36)
9.5.1 Open chain magnetization Consider the expectation value μk of single spin variable μk ; note that for an open chain μk depends on how far the lattice site k is from the boundaries. In particular ⎛ ⎞ N−1 & 1 ⎝ μj Lμj +1 ⎠ μk e h2 (μ1 +μN ) ··· μk = ZN μ μ j =1 1
=
N
1 tr Lk−1 μL ˆ N−k M , ZN
(9.37)
where, from Eq. 8.14
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9.6 Block spin renormalization
!
μˆ =
i
191
1 0 . 0 −1
A long calculation yields
h h h 2 h 2 1 1 N−1 N−1 2 2 2 + ae − 2 2 − e− 2 λ e 1 − a − λ ae 1 − a μk = 1 2 ZN (1 + a 2 )2 3 2 k−1 k−1 N−k h 2 −h ae . λ + λ λ − 1 + a − ae + 2a λN−k 1 2 1 2
Note that for h → 0, one recovers the expected result
(9.38)
E[μk ] = 0. 9.6 Block spin renormalization Wilson’s (1983) concept of renormalization plays a central role in quantum field theory. The essential idea is that at each length scale, there is an effective theory that completely describes the physics at that length scale. The onedimensional Ising model provides a toy model to examine the essential features of renormalization theory; the concept of length scale in the case of the Ising model is the distance between two spins in a correlation function. The large distance action is related to the short distance action by the procedure of renormalization. In Wilson’s (1983) approach, the short distance degrees of freedom are summed over (integrated out) to generate the action appropriate for describing the longer distance physics. Let the lattice spacing for the Ising model be denoted by a, described by action S0 . Consider the odd and even spins of the onedimensional Ising model. The distance between the even spins is twice the distance between adjacent spins. Hence if all the odd spins are summed over, the new lattice will have a spacing of a1 = 2a and describe a new effective action S1 ; the transformation relating the action for the original lattice S0 to the action of the new lattice S0 is the renormalization transformation R and S1 = R[S0 ] [Kardar (2007)]. Successive transformations form a group, and hence the name renormalization group for this procedure for studying the distinct scales of a system. If one repeats the renormalization transformation l times, then the lattice spacing of the final lattice is al = 2l a and the action Sl is given by Sl = Rl [S0 ]. The correlation on the original lattice is say ξ0 = La; hence, the correlation on the lattice with spacing al is given by ξl = 2−l ξ0 , where ξl is the correlation length measured by the scale of the final lattice, namely al = 2l a. Divide the lattice into odd and even sites and we sum over all the spins residing at the even sites, thus generating a lattice of double the original lattice. The partition function remains invariant. The renormalization transformation is given by
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λ a
μ’
x
λ’
μ’’ x
x
μ
μ‘ 2a
λ’’
μ’’
x
μ
μ’ 4a
Figure 9.3 Blockspin renormalization.
Z=
e S0 =
{μ:all}
S0 = g0 + K0
e S1 ,
(9.39)
μn μn+1 .
(9.40)
{μ:odd}
n
Let the even spins be labeled by λn as shown in Figure 9.3; the renormalization transformation is given by eS0 = e S0 . e S1 = {μ:even}
{λ}
One can think of integrating over the even spins as combining two spins together, as shown in Figure 9.3, to generate a new blockspin that is defined on the new lattice and described by its own effective action. For a lattice of infinite size, this process can be repeated indefinitely. Each λn spin is coupled only to its nearest neighbor; hence the sum over the λn spins can be done separately over each λn spin. As shown in Figure 9.3, the Ising spin at the even site labeled by λ = ±1 couples to its nearest neighbor spins μ, μ , the two bonds λμ and μ λ that contain the spin variable λ; summing over λ yields the fundamental renormalization group transformation eK0 λ(μ+μ ) = eg1 +K1 μμ , (9.41) e2g0 λ=±1
where it is postulated that the effective bonds coupling the lattice at double the lattice spacing are given by the right hand side of Eq. 9.41. The new coupling constants g1 , K1 are now determined in terms of the original coupling constants g0 , K0 . Consider the following cases for Eq. 9.41: • μ = 1 = μ and μ = −1 = μ both yield the same equation, namely
2e2g0 e2K0 + e−2K0 = eg1 +K1 .
(9.42)
• μ = 1 = −μ and μ = −1 = −μ both yield the same equation 2e2g0 = eg1 −K1 .
(9.43)
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Equations. 9.42 and 9.43 yield eg1 = 2e2g0 cosh1/2 2K0 , eK1 = cosh1/2 2K0 , and we obtain the renormalization group transformation K1 =
1 1 ln cosh 2K0 , g1 = 2g0 + ln cosh 2K0 + ln 2. 2 2
(9.44)
The lattice spacing for S1 is twice the lattice spacing of S0 ; hence, the correlation lengths ξ as computed from the two actions should scale accordingly, namely 1 1 ξ1 = ξ0 ⇒ ξ(K1 ) = ξ(K0 ). 2 2
(9.45)
To prove Eq. 9.45, recall that Eq. 8.38 yields the correlation lengths ξ(K0 ) = −
1 1 and ξ(K1 ) = − . ln(tanh K0 ) ln(tanh K1 )
The expression for K1 in Eq. 9.44 yields 2 ⇒ tanh(K1 ) = [tanh(K0 )]2 . 2 e2K1 ± 1 = eK0 ± e−K0 Hence, the result given in Eq. 9.45 is realized as ξ(K1 ) = −
1 1 1 1 =− = ξ(K0 ). ln(tanh K1 ) 2 ln(tanh K0 ) 2
(9.46)
Note the remarkable result that the new coupling constant K1 on the larger lattice “remembers” that it is the result of summing over the smaller lattice with coupling constant K0 . One of the fundamental insights that the renormalization group provides about the different length scales in a problem is the following: that each scale 2 of the system has its own corresponding coupling constant K . The recursion equation connecting coupling constants gl , Kl defined for a lattice of size 2l a, with coupling constants gl+1 , Kl+1 defined for a lattice of size 2l+1 a, is given by gl+1 = 2gl +
1 1 ln cosh 2Kl + ln 2 = R(gl ), Kl+1 = ln cosh(2Kl ) = R(Kl ). 2 2
One can write the renormalization group recursion equations more generally as (Kl , gl ) = R(Kl−1 , gl−1 ) = R (K0 , g0 ),
(9.47)
where R is a 2 × 2 matrix. The coupling constant Kl is the renormalized coupling constant, describing the physics at length scale 2l a and is obtained from the initial (bare) coupling constant
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K0 by the repeated application of the renormalization group transformation R. The correlation length is given by 1 1 ξ = ξ−1 = ξ0 . 2 2 The fixed point of the renormalization group transformation is given by 1 K ∗ = R(K ∗ ) ⇔ ξ ∗ = ξ ∗ 2 and hence leads to the result 1 ξ∗ = ξ∗ 2
⇒ ξ∗ = 0
or
ξ ∗ = ∞.
The case of ξ ∗ = 0 corresponds to a decoupled system. A system is said to be critical – undergoing a phase transition – if its correlation length is infinite, namley ξ ∗ = ∞; hence the fixed point of the renormalization transformation corresponds to the system being critical. Note the remarkable fact that the fixed point of the renormalization group transformation R depends only on R and not on the details of the (Ising) system we start from; this feature of phase transitions leads to the property of universality in that many different types of phase transitions are described by the same critical system defined by R. The fixed point is analyzed as follows: • To find the fixed point of Kl , consider the limit liml→∞ Kl → K ∗ . • Fixed point of K0 = ∞ = K ∗ result from the fact that Kl = K ∗ = ∞. The initial Ising system (fixed by K0 ) is strongly correlated with the correlation length being ξ ∗ = ∞. Hence, under renormalization, the correlation length remains infinite no matter how large the effective lattice spacing. • For any Kl 0, ! 4Kl2 1 ≈ Kl2 < Kl . Kl+1 ln 1 + 2 2 Hence liml→∞ Kl → K ∗ = 0. The system is decoupled at large enough distances for any initial value K0 < 0. In fact, for any K0 < ∞, liml→∞ Kl → 0 = K ∗ . For any initial finite correlation length, the large distance correlation is always zero, namely ξ ∗ = 0. The flow of the coupling constant Kl is shown in Figure 9.4; if one starts with any finite value for K0 < ∞, then the effective coupling Kl for larger and larger lattice size flows towards zero, with the onedimensional Ising model becoming decoupled at large distances.
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9.6 Block spin renormalization
0=tanh K*
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tanh K*=1
K
Figure 9.4 Flow of coupling constant under renormalization.
Near K ∗ ∞, Kl+1
1 e2Kl ln 2 2
! Kl −
1 ln 2 < Kl . 2
Hence, for any Kl < ∞ Kl+1 < Kl ,
lim Kl → 0.
l→∞
9.6.1 Block spin renormalization: magnetic field One can introduce a magnetic field h, as in Eq. 9.1, and repeat the block spin calculation starting with the initial Ising model given by S0 ≡
+∞
Kμj μj +1 + hμj .
j =−∞
The renormalization group transformation is given by h exp K μμ + (μ + μ ) + g 2 h exp Kλ μ + μ + μ + μ + hλ + 2g . = 2 λ=±1
(9.48)
To solve for the renormalized interaction it is convenient to set ⎧ ⎨x = e K , y = e h , z = e g ⎩
(9.49)
x = e K , y = e h , z = e g .
Similarly to the case for zero magnetic field, the four possible configurations of the bond this time yield only three equations in three unknowns. The solution is given by Kardar (2007) ⎧ 4 z = z8 x 2 y + x −2 y −1 [x −2 y + x 2 y −1 (y + y −1 )2 ] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 2 2 −2 y −1 y = y 2 xx −2y+x (9.50) y+x 2 y −1 ⎪ ⎪ ⎪ ⎪ 2 −2 −1 −2 2 −1 ⎪ ⎪ ⎩x 4 = (x y+x y )(x 2y+x y ) . −1 (y+y )
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Ising model: magnetic field
tanh(h)
+1
1 tanh K*=0
tanh(K)
tanh K*=1
Figure 9.5 Flow of coupling constant K and magnetic field h under renormalization, from an initial value of K0 , h0 .
The renormalization of coupling constants with three equations is a generalization of the renormalization group equations given in Eq. 9.47; taking the logarithm of Eq. 9.50 yields (Kl+1 , gl+1 , hl+1 ) = R(Kl , gl , hl ),
(9.51)
where R is now a 3 × 3 matrix. Similarly to the case of hl = 0, the flow of gl is irrelevant for the critical properties of the Ising system in the presence of a magnetic field. The coupling constants Kl , hl have flows as shown in Figure 9.5. The system is critical for K ∗ = ∞ and for any value of the magnetic field [Kardar (2007)]. For K0 < ∞, as one recurses and goes to long distance, the system decouples with the coupling constant going to zero, namely Kl → 0; the magnetic field flows to hl → h∗ , namely a system with a fixed magnetic field h∗ , the value of h∗ being fixed by the initial values K0 , h0 . Figure 9.5 shows the renormalization flow for K0 = K ∗ = ∞ and h0 = 0. 9.7 Summary The Ising model with a magnetic field has many new features. An exact solution of the partition function, of the magnetization, and of the correlation function can
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be obtained using the transfer matrix. Unlike the zero magnetic field case, the expansion eS in a power series does not lead to a tractable method for calculation due to the difficulty in doing the combinatorics in the presence of a magnetic field. Linear regression in the presence of a magnetic field provides an intuitive and physical interpretation of the concept of the correlation function. An important feature of all lattice systems is that the lattice spacing a does not appear explicitly. The consequence of this plays a central role in the concept of renormalization and of the renormalization group. Let the physical correlation length of a system described by the Ising model be L, in units of say meters; let the lattice spacing be a meters; then on the lattice, since all quantities are dimensionless, the dimensionless correlation length is given by ξ(K0 ) = L/a. The correlation for the lattice of spacing 2a with coupling constant K1 is ξ(K1 ) and hence the dimensionless correlation is given by ξ(K1 ) = L/(2a). The dimensionless numerical value of the correlation ξ(K1 ) is seen to be half the value of the correlation for lattice a with coupling constant K0 given by ξ(K0 ), namely ξ(K1 ) = L/(2a) = ξ(K0 )/2. The result of the renormalization transformation leads to the nontrivial result that the effective lattice spacing, instead of appearing explicitly, appears through the value of the coupling constant Kl which corresponds to a lattice of spacing 2l a.
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10 Fermions
The degrees of freedom studied so far have been either real or complex variables. These variables commute under multiplication, in the sense that two numbers a, b satisfy ab = ba; commuting variables are generically called bosonic variables, or bosonic degrees of freedom. Typical of the bosonic case are the degrees of freedom for a collection of quantum mechanical particles. Interactions of fundamental particles are generally mediated by bosonic fields such as the Maxwell electromagnetic field, whereas mass is usually carried by particles that are fermions, the most familiar being the electron. Unlike bosonic variables, fermionic variables, also called fermionic degrees of freedom, anticommute; namely, if ζ, η are two fermionic variables, then ζ η = −ηζ . Two key features distinguish fermionic from bosonic variables: • Fermions obey the Pauli exclusion principle, which states that no two fermions can occupy the same quantum state. This is the reason the concept of intensity does not apply to a fermion. A high intensity electric field is a reflection of the presence of a large number of photons, which are bosons, in the same quantum state; for photons, any number of photons can be in the same quantum state. In contrast, an electron is either in a quantum state or it is not; in particular, a fermion exists at a point or there is no electron there. • The state function of a multibosonic system is totally symmetric in that the exchange of any two bosonic degrees of freedom yields the same state function. In contrast, a multifermion system is totally antisymmetric: the exchange of any two fermion degrees of freedom gives the same state – but with a negative sign. The Pauli exclusion principle implies a discrete nature for fermions since a fermion degree of freedom has only two possibilities, either occupying a state or not occupying a state. In discussing the fermion Hilbert space in Section 10.3,
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10.1 Fermionic variables
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it will be seen that the Hilbert space of a single fermion is identical (isomorphic) to the Hilbert space of a discrete degree of freedom that takes only two values. Both the key features of fermions, namely obeying the Pauli exclusion principle and the state function being antisymmetric, can be mathematically realized by introducing a new type of variable, namely fermionic variables; similarly to the bosonic case, fermionic degrees of freedom can be described by either real or complex fermionic variables. Fermionic variables are defined in Section 10.1 and fermion integration is discussed in Section 10.2. The fermionic Hilbert space as well as its dual space are defined in Section 10.3. The concept of the antifermionic Hilbert space is discussed in Section 10.4. Gaussian integration for real and complex fermions is discussed in Section 10.6. The fermionic path integral is defined for the particle and antiparticle system in Section 10.8, and the transition probability amplitude is obtained in Section 10.11. A simple onedimensional toy model is constructed in Section 10.12 to show how quark confinement can arise by coupling the fermions to a gauge field. 10.1 Fermionic variables The defining property of fermions is the Pauli exclusion principle, namely that at most only one fermion can occupy a quantum state. In other words, for fermions, a state either has no fermions, or at most one fermion. Let0F be the fermion vacuum state, and let aF† be the fermion creation operator. Then 0F : ground state; no fermions, aF† 0F : one fermion, (a † )2F 0 = 0 : null state.
(10.1)
The second defining property of fermions is that two different fermions must give an antisymmetric state function on being exchanged; hence two distinct fermionic creation operators, represented by say a1† ,a2† , must satisfy the antisymmetry a1† a2† 0 = −a2† a1† 0, which is realized by imposing the anticommutation relation a1† a2† = −a2† a1† ⇒ {a1† , a2† } = 0, where the anticommutator is defined for any two quantities A, B by {A, B} ≡ AB + BA.
(10.2)
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Instead of working with fermionic creation and annihilation operators acting on the ground state 0, one can instead describe fermions using a calculus distinct from the calculus based on real numbers that is used for describing bosons. An independent and selfcontained formalism for realizing all the defining properties of fermions is provided by a set of anticommuting fermionic variables ψ1 , ψ2 , . . . ψN and its conjugate ψ¯ 1 , ψ¯ 2 , . . . ψ¯ N , defined by the properties {ψi , ψj } = −{ψi , ψj }, {ψ¯ i , ψj } = −{ψj , ψ¯ i }, {ψ¯ i , ψ¯ j } = −{ψ¯ j , ψ¯ i }. Hence, it follows that ψi2 = 0 = ψ¯ i2 . Fermionic differentiation is defined by δ δ ψj = δi−j , ψ¯ j = 0 δψi δψi and δ2 δ2 δ2 δ2 =− ⇒ = 0 = . δψi δψj δψj δψi δψi2 δ ψ¯ i2 Similarly, all the fermionic derivative operators δ/δψi ,δ/δ ψ¯ i anticommute.
10.2 Fermion integration " +∞ Similarly to the case of −∞ dxf (x) which is invariant under x → x + a, that is " +∞ " +∞ −∞ dxf (x) = −∞ dxf (x + a), we define fermion integration by ¯ ¯ ¯ (ψ¯ + η). d ψf (ψ) = d ψf ¯ (10.3) Since ψ¯ 2 = 0, Taylors expansion shows that the most general function of the variable ψ¯ is given by ¯ f = a + bψ. It follows that rules of fermion integration that yield Eq. 10.3 are given by ¯ d ψ = 0 = dψ, ¯ ¯ d ψ ψ = 1 = dψψ,
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10.3 Fermion Hilbert space
¯ d ψdψψ ψ¯ = 1 = −
¯ ¯ d ψdψ ψψ.
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(10.4)
For N fermionic variables ψi , with i = 1, 2, . . . N, one has the generalization / .N & dψn ψi1 ψi2 . . . ψin = i1 ,i2 ,...in , (10.5) n=1
where i1 ,i2 ,...in is the completely antisymmetric epsilon tensor. Consider a change variable for a single variable, namely ψ = aχ + ζ, where a is a constant and ζ is a constant fermion. From Eq. 10.4, the nonzero fermion integral yields 1 1 = dψψ = dψ(aχ + ζ ) = dψaχ ⇒ dψ = dχ. (10.6) a Note that this is the inverse for the case of real variables, since x = ay yields dx = ady. For the case of N fermions, the antisymmetric matrix Mij ) = −Mj i yields the change of variables ψi =
N
Mij χj ⇒ ψ = Mχ .
j =1
Similarly to Eq. 10.6, it follows that N & i=1
where Dψ =
CN i=1
dψi =
N 1 & 1 dχj ⇒ Dψ = Dχ, det M j =1 det M
(10.7)
dψi and so on. 10.3 Fermion Hilbert space
The discussion in Section 2.4 on state space and its dual is valid for fermions as well. There are two distinct fermionic variables, namely the variable ψ and its dual ¯ Comparing with the bosonic case, if one takes the fermionic variable ψ¯ to be ψ. the analog of the coordinate x, then ψ is the analog of the conjugate momentum variable p. The two different state spaces Vψ¯ and Vψ are based on the coordinate variable ψ¯ and its conjugate ψ, respectively. There are consequently two Hilbert in Figure 2.3. spaces, namely Vψ¯ and Vψ , that are dual to each other, as shown " ∗ In analogy with bosonic variables x for which φφ = dxφ (x)φ(x) ≥ 0, the norm for the fermionic Hilbert space needs to be defined so as to yield a positive norm fermionic Hilbert space.
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¯ A fermion Choose Vψ¯ to be the state space of the fermionic degree of freedom ψ. state function is then given by the scalar product of the dual coordinate state vector ¯ ∈ Vψ with f ∈ Vψ¯ and yields ψ ¯ = ψf ¯ = a + bψ. ¯ f (ψ) The dual state of f , denoted by f , is defined such that f f = a2 + b2 > 0.
(10.8)
Physical (normalizable) state functions have f f = 1 and yield the interpretation f f = 1 ⇒ a2 + b2 = 1, a2 = probability, there is no fermion, b2 = probability, there is one fermion. Since a, b are complex numbers, the space of all physical state functions is equal to a threedimensional sphere S 3 . Note that a fermion state function, similarly to the boson case, is equivalent to all state functions related to it by a global phase eiφ . Factoring out the phase from the physically distinct state functions yields, as discussed in Section 8.1.1, the Hilbert space Vψ¯ ≡ S 3 /S 1 = S 2 : Bloch sphere. Hence, similarly to a spin 1/2 system, the distinct physical states of a single fermion Hilbert space are parameterized by the points of a twodimensional sphere. Or more formally, each state vector of the single fermion space of states corresponds to one point on a twodimensional sphere. The single fermion state space is seen to be isomorphic to the state space of the Ising spin discussed in Section 8.1.1, and shows that in essence a fermion is a discrete degree of freedom. Define the dual state by ψ¯ → ψ ⇒ f D (ψ) = f ψ = a ∗ + b∗ ψ. To achieve the required scalar product, note that ¯ D ¯ (ψ)e−ψψ f (ψ) f f = d ψdψf ¯ ∗ ¯ ¯ + b∗ ψ)e−ψψ (a + bψ) = d ψdψ(a ¯ ¯ = d ψdψ a2 + b2 ψ ψ¯ + a ∗ bψ¯ + ab∗ ψ (1 − ψψ).
(10.9)
(10.10)
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Using the rules for fermion integration given in Eq. 10.4, the positive definite scalar product is given by f f = a2 + b2 . 10.3.1 Fermionic completeness equation ¯ ψ) being analThe fermionic variables have a phase space representation with (ψ, ogous to the coherent state representation of the creation and destruction operators ¯ be the dual fermionic eigen(a † , a). Let ψ be the fermionic eigenstate and ψ state. The completeness equation for the fermion degree of freedom is given by ¯ −ψψ ¯ ¯ I = d ψdψψe ψ. (10.11) The fermion completeness equation is similar to the completeness equation for coherent states discussed in Section 5.12, and to Eq. 5.80 in particular. The fermion basis states are overcomplete – as is the case for the bosonic coherent basis states – and the metric on the fermion state space that accounts for the overcompleteness ¯ is given by exp{−ψψ}. The inner product of the basis with its dual being is given by the selfconsistency equation that follows from the completeness equation and yields ¯
¯ ψψ = eψψ .
(10.12)
To verify that Eq. 10.12 is indeed consistent with the completeness equation given in Eq. 10.11, let ζ¯ , ζ be fermionic variables; then ¯ ¯ ¯ = d ψdψ ¯ ¯ ¯ ζ¯ ζ = d ψdψ ζ¯ ψe−ψψ ψζ 1 + ζ¯ ψ 1 − ψψ 1 + ψζ ¯ ¯ ¯ − ψψ ¯ ¯ = d ψdψ ζ¯ ψ ψζ = d ψdψ 1 + ζ¯ ζ ψ ψ¯ = eζ ζ as expected. The inner product of the basis states and completeness is selfconsistently determined since one requires the other: to prove completeness one needs the inner product. In particular, a proof of the resolution of the identity as given in Eq. 10.11 is the following: ¯ ¯ 2 ¯ ¯ ζ¯ dζ ψe−ζ ψ ζ¯ ζ e−ψζ ψ I = d ψdψd ¯ ¯ ¯ ¯ ¯ = d ψdψd ζ¯ dζ ψe−ζ ψ eζ ζ e−ψζ ψ ¯ −ψψ ¯ = I, ¯ ψ = d ψdψψe
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where the result follows from the inner product given in Eq. 10.12 and performing the ζ¯ , ζ integrations. 10.3.2 Fermionic momentum operator ¯ so we need to determine the representation of the The state space depends on ψ, dual coordinate ψ on the state space. The fermion coordinate operator ψˆ has the fermionic coordinate eigenstate given by ˆ ψψ = ψψ, where the coordinate eigenvalue ψ is fermionic. The scalar product yields δ ψψ ¯ (e ), δ ψ¯ δ ⇒ ψ= . δ ψ¯ ¯
¯ ψψ ˆ ¯ ψ = ψψψ = ψeψψ = ¯ ⇒ ψψψ =
δ ψψ ¯ e ¯ δψ
Hence, on state space Vψ¯ the dual coordinate ψ yields ¯ = ψf (ψ)
δ ¯ f (ψ). δ ψ¯
(10.13)
Hence, as mentioned in Section 10.3, the variable ψ is the analog of the momentum operator p and is evidenced by its action on state space, as given in Eq. 10.13. In terms of the fermionic variables the creation and annihilation operators have the realization δ ¯ with {a † , a} = 1. a= a † = ψ, ¯ δψ The identification of the fermion variables with the fermion creation destruction operators is consistent with the identification made earlier, in the discssion on Eq. 10.11, of the fermion completeness equation with the completeness equation for the coherent state space. 10.4 Antifermion state space Let χ, χ¯ be a set of fermionic variables; the dual coordinate eigenstate is defined by χ  ∈ Vχ ; hence χ f = a + bχ. This change of definition for the coordinate of the state space will lead to the conclusion that Vχ is the state space for antifermions. The completeness equation continues to be
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205
d χ¯ dχ χ¯ e−χ¯ χ χ .
(10.14)
I=
Consistency with the completeness equation for the antifermionic variables requires that the inner product continues to be the same as the particle case, namely χχ ¯ = exp{χ¯ χ }.
(10.15)
To following derivation provides a consistency check for the above completeness equation: ¯ ηη ¯ = d χ¯ dχ eχ¯ η e−χ¯ χ eηχ = d χ¯ dχ (1 + χ¯ η) (1 − χ¯ χ) (1 + ηχ) ¯ = d χ¯ dχ (−χ¯ χ + χ¯ ηηχ) ¯ = d χ¯ dχ (−χ¯ χ) (1 + ηη) ¯ = exp{ηη}. ¯ In the completeness equation given in Eq. 10.14, the variables χ , χ¯ have been interchanged for the state space vectors – as compared to the completeness equation ¯ as in Eq. 10.11 – but with the metric being unchanged, namely for variables ψ, ψ, ¯ exp{−χ¯ χ } and exp{−ψψ}. To compensate for the difference in these two cases, an extra minus sign needs to introduce the conjugation of the state space vector f (χ ). Conjugation is defined by χ → −χ¯ . 1 Hence f D (χ¯ ) = f ∗ (−χ¯ ) ⇒ a ∗ − b∗ χ¯ . It is verified that the rule for conjugation yields a positive definite norm for the state space Vχ , f f = d χ¯ dχf D (χ¯ )e−χ¯ χ f (χ ) = d χ¯ dχ (a ∗ − b∗ χ)(a ¯ + bχ)(1 − χ¯ χ ) = d χ¯ dχ (a2 + b2 )χ χ¯ = a2 + b2 . The fermion momentum operator is defined by χ χ ¯ χ¯ = χ¯ eχ¯ χ = −
δ χ¯ χ e δχ
and yields 1 This is because the order of integration in the scalar product is reversed compared to the fermion case, for
¯ which, under conjugation ψ → ψ.
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χ¯ f (χ ) = −
δ f (χ ). δχ
The fact that anticommuting variables χ , χ¯ represent antifermions becomes clear when the antifermions are combined with fermions, as is done in the following section.
10.5 Fermion and antifermion Hilbert space One can choose either of the Hilbert spaces Vψ¯ or Vχ to be fermionic state space; once a choice is made for the fermion state space, it automatically allows for the introduction of the concept of antifermions. The representation of antifermions is fixed by the choice that is made for the fermion state space. The normal convention is to choose Vψ¯ to be the fermionic state space and ψ¯ the fermionic degree of freedom and Vχ to be the antifermionic state space and χ to be the anitfermions degree of freedom. The state space and path integral for the fermion and antifermion system is discussed below. A system containing both particle and antiparticle has a state space given by the tensor product of the fermion and antifermion state spaces, namely Vψ¯ ⊗ Vχ . The most general state vector is given by ¯ χ ) = ψ, ¯ χf = a + bψ¯ + cχ + d ψχ, ¯ f (ψ, a2 + b2 + c2 + d2 = 1.
(10.16)
The interpretation of the state vector is the following: • • • •
a2 = probability of the system having no fermion or antifermion, b2 = probability of the system having one fermion, c2 = probability of the system having one antifermion, d2 = probability of the system having one fermion and one antifermion.
Hermitian conjugation for the fermion and antifermion state space is defined by the operations: 1. Complex conjugate all the coefficients; 2. Reverse the order of all the fermion variables; 3. Make the substitution ! ! ! ψ 1 0 ψ¯ → . χ 0 −1 χ¯
(10.17)
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The completeness for the fermion–antifermion degrees of freedom is ¯ ¯ χ . ¯ I = d ψdψd χ¯ dχ ψ, χ¯ e−ψψ−χ¯ χ ψ, Consider a state vector f given in Eq. 10.16; its dual state vector, using the rules for fermion and antifermion conjugation, is given by f D (ψ, χ¯ ) = f ψ, χ ¯ = a ∗ + b∗ ψ¯ − c∗ χ¯ − d ∗ χ¯ ψ. The rule for conjugation yields the positive definite inner product 0 1 ¯ ¯ ¯ χ f f f = d ψdψd χ¯ dχ f ψ, χ ¯ e−ψψ−χ¯ χ ψ, ¯ − χ¯ χ + ψψ ¯ χ¯ χ ¯ = d ψdψd χ¯ dχ a ∗ + b∗ − c∗ χ¯ − d ∗ χ¯ ψ 1 − ψψ ¯ × a + b + cχ + d ψχ ¯ χ¯ χ ¯ = d ψdψd χ¯ dχ a2 + b2 + c2 + d2 ψψ = a2 + b2 + c2 + d2 .
10.6 Real and complex fermions: Gaussian integration The rules of fermion integration are used for evaluating real and complex fermion Gaussian integration. Consider N real fermion variables χn with n = 1, 2, . . . , N. Define the partition function Z [J ] =
N & n=1
' 1 dχn exp − χn Mnm χm + Jn χn . 2
(10.18)
The external source Jn is fermionic. For N odd, Z [J ] is always zero. To see this, consider the case of J = 0; on expanding the exponential, one always has powers of the even product of fermions; hence, term by term, the partition function is given by terms that are all zero, namely N &
dχi even products of χj = 0.
i=1
The term M is a real antisymmetric matrix. One way of evaluating the partition function is to try and diagonalize M. However, one can only use a real transformation since all the fermions are real. Matrix algebra yields the result that every
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real antisymmetric matrix M can be brought into a blockdiagonal form by an orthogonal transformation O in the following manner: ⎞ ⎛ 0 λ1 ⎟ ⎜−λ1 0 ⎟ ⎜ ⎟ ⎜ 0 λ 2 ⎟ ⎜ ⎟ ⎜ T ⎜ −λ 0 ⎟ O = O T O, 2 (10.19) M=O ⎜ ⎟ .. ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ 0 λN ⎠ ⎝ 2 −λ N 0 2
where O T O = 1 : orthogonal. Let us perform a change of variables, and from Eq. 10.7 Oχ = ξ : real ⇒
N &
dχi = det(O)
i=1
N &
dξi =
i=1
N &
dξi .
i=1
The partition function factors into a product of N/2 terms and yields
' 1 dξi exp − ξn nm ξm Z [0] = 2 i=1 ! !' N/2 & 0 λi 1 χ1 dξ1 dξ2 exp − ξ1 ξ2 = −λi 0 χ2 2 i=1 N/2 N/2 & & dξ1 dξ2 e+λi ξ1 ξ2 = λi . (10.20) = N &
i=1
i=1
From Eq. 10.19 det M =
N/2 & i=1
det
0 −λi
λi 0
! =
N/2 &
λ2i .
i=1
Hence, from Eq. 10.20, the partition function is given by √ Z [0] = det M
(10.21)
The square root of an antisymmetric matrix is known as a Pfaffian and has many remarkable mathematical properties.
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The general case of the partition function in the presence of an external fermionic source Ji can be derived from the results obtained. Rewriting the action given in Eq. 10.18 yields ' 1 −1 1 −1 −1 M χ + M J + J MJ . (10.22) Z [J ] = Dχ exp − χ + J M 2 2 Using the fundamental invariance of fermion integration under the shift of the fermion integration variable given in Eq. 10.3 allows us to shift the integration variable, χ → χ − J M −1 , and yields, from Eqs. 10.22 and 10.21 1 1 T 1 T −1 −1 Z [J ] = Dχe− 2 χ Mχ e 2 J M J = Z [0] e 2 J M J √ 1 T −1 ⇒ Z [J ] = det Me 2 J M J . The propagator can be obtained by fermionic differentiations on Ji , 6 N D% % % 2 % 0 1 1 1 δ δ2 % −1 DχeS+J χ %% χi χj = · = exp JmT Mmn Jn % % δJi δJj Z δJi δJj 2 mn=1 J =0 J =0 ! % % 1 −1 δ 1 1 1 −1 = = Mj−1 M Jk − Jk Mkj−1 eS %% i − Mij δJi 2 jk 2 2 2 J =0 0 1 ⇒ χi χj = Mj−1 . i
10.6.1 Complex Gaussian fermion Consider the N dimensional Gaussian integral for fermions ψn and ψ¯ n , Z[J ] =
N &
d ψ¯ n dψn exp −ψ¯ n Mnm ψm + J¯n ψn + ψ¯ n Jn ,
n=1
where Mnm = −Mmn is an antisymmetric matrix. For real fermions ψ = ψ ∗ . For complex fermions ψ = ψ1 + iψ2 . An antisymmetric matrix M = −M T can be diagonalized by a unitary transformation ⎛ ⎞ λ1 ⎜ ⎟ † .. M = U† ⎝ ⎠ U, U U = 1. . λN
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In matrix notation M = U † U,
U U † = 1,
det(U U † ) = 1,
(10.23)
where = diag(λ1 , . . . λN ). Since the fermions ψ, ψ¯ are complex, let us define the change of variables using the unitary matrix U , and from Eq. 10.72 ¯ † = η, ¯ ψU DηD η¯ =
η = U ψ,
1 1 ¯ Dψ × D ψ¯ = DψD ψ. det(U ) det(U † )
(10.24)
Hence, the fermion integrations completely factorize and yield N & Z[0] = d η¯ n dηn exp{− λn η¯ n ηn } n
=
&
n=1
d η¯ n dηn e
n
−λn η¯ n ηn
=
&
λn = det M.
(10.25)
n
The partition function with an external source is given as before by a shift of fermion integration variables. Write the partition function as ¯ Z[J ] = D ψDψ exp − ψ¯ − J¯M −1 M ψ − M −1 J + J¯M −1 J . Using the fundamental property of fermion integration that it is invariant, a constant shift of fermion variables as given in Eq. 10.3 yields ψ¯ → ψ¯ + J¯M −1 , ψ → ψ + M −1 J and hence Z[J ] =
¯ ¯ D ψDψ exp −ψMψ + J¯M −1 J = (det M) exp J¯M −1 J .
The fermion Gaussian integration obtained in Eq. 10.25 can be directly done using ¯ the rules of fermion integration. On expanding the exponential term exp{ψMψ}, N ¯ only one term – ψMψ /N! containing the product of all the fermion variables – is nonzero inside the integrand. Using the notation of summing over repeated indexes, Eq. 10.5 yields N 1 ¯ ψMψ ¯ ¯ ¯ = D ψDψ ψMψ D ψDψe N! 1 ¯ ψ¯ i1 ψj1 ψ¯ i2 ψj2 · · · ψ¯ iN ψjN = Mi1 j1 Mi2 j2 · · · MiN jN D ψDψ N! 2 Note that for real fermions one could not use a unitary transformation for a change of variables as this would
lead to the transformed fermions being complex.
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1 Mi j Mi j · · · MiN jN i1 i2 ···iN j1 j2 ···jN N! 1 1 2 2 = det M. =
10.7 Fermionic operators Operators for fermions, such as the Hamiltonian and momentum have a representation in terms of fermionic variables. Given that the dual of the fermionic coordinate ψ¯ is given by ψ, all operators for fermions are expressed in terms of both the fermion coordinate and its dual. In this sense, the operators of the fermion degree of freedom are defined analogously to bosonic operators that are defined on phase space, as in Section 5.4. Consider a fermionic operator O; the operator is a mapping from Vψ¯ to itself and hence, similar to the bosonic case given in Eq. 2.5, O is an element of the tensor product space Vψ¯ ⊗ Vψ , O ∈ V ⊗ VD ≡ Vψ¯ ⊗ Vψ . The matrix elements of O are given by ¯
¯ ¯ ψ)ψψ ¯ ¯ ψ)eψψ . ψOψ = O(ψ, = O(ψ, In particular, the Hamiltonian operator H is defined by ¯
¯ ψ = H (ψ, ¯ ψ)ψψ ¯ ¯ ψ)eψψ ψH = H (ψ,
(10.26)
and yields, from Eq. 10.26, ¯ ¯ ¯ ψ)]eψψ ¯ −H ψ = ψ[1 − H ]ψ = [1 − H (ψ, + O( 2 ) ψe ¯
¯
e−H (ψ,ψ) eψψ + O( 2 ).
(10.27)
10.8 Fermionic path integral One way of understanding the difference between a fermion and an antifermion is to examine the evolution of fermions in (Euclidean) time. For clarity, consider a time lattice with spacing and let lattice time be denoted by n. The fermion degrees of freedom are defined on the lattice and denoted by ψ¯ n ; ψn . Consider a typical action for the lattice fermions given by SP = − ψ¯ n ψn + 2K ψ¯ n+1 ψn =
n
n
L(n),
n
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where the fermion Lagrangian is given by L(n) = −ψ¯ n ψn + 2K ψ¯ n+1 ψn
= 2K ψ¯ n+1 − ψ¯ n ψn + (2K − 1)ψ¯ n ψn . The path integral is given by ZP =
&
d ψ¯ n dψn eS .
(10.28)
(10.29)
n
The path integral is written in the fermion coherent state basis, similarly to the coherent basis path integral for bosons (real and complex degrees of freedom) as discussed in Section 5.14. Note that the fermion variable ψn at time n propagates to variable ψ¯ n+1 at time n + 1. The partition function is written by repeatedly using the completeness equation given in Eq. 10.11, and yields ¯ ¯ ¯ ψ¯ n+1 e−H ψn e−ψn ψn ψ¯ n e−H ψn−1 e−ψn−1 ψn−1 . . . ZP = D ψDψ Hence, from the definition of fermionic operators given in Eq. 10.27 ¯
e2K ψn+1 ψn = ψ¯ n+1 e−H ψn ¯
¯
¯
e−H (ψn+1 ,ψn ) ψ¯ n+1 ψn = e−H (ψn+1 ,ψn ) eψn+1 ψn .
(10.30)
Dropping the index of time on the fermion variables yields the particle Hamiltonian ¯ ψ) 2K ψψ ¯ − ψψ ¯ −HP (ψ, ! ¯ ψ) 1 − 2K ψψ ¯ = ψH ¯ ψ. ⇒ HP (ψ, Note that, as is the case for coherent states (Section 5.13), the Hamiltonian is automatically normal ordered. The limit of continuous time is taken by defining the following continuum ¯ fermions ψ(t), ψ(t) and Hamiltonian: √ √ ψ¯ t = 2K ψ¯ n , ψt = 2Kψn , t = n, ! δ 1 − 2K 1 − 2K ¯ ψ = HP ψ, , ω= ψ¯ t ψt = ωψ¯ t . (10.31) 2K δψt 2K The continuum Lagrangian L(t) and action, from Eqs. 10.28 and 10.31, are given by 2K − 1 ∂ ψ¯ t ¯ L(n) = − ψt + ψt ψt ≡ L(t), ∂t 2K
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SP =
dtL(t), L(t) = −
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∂ ψ¯ t ψt + ωψ¯ t ψt . ∂t
The term SP is the action for a particle propagating forward in time. Consider another action that describes the time evolution of antiparticles, namely χ¯ n χn + 2K χ¯ n χn+1 SA = − n
with
n
ZA =
D χ¯ Dχ eSA .
(10.32)
The antifermion variable χ¯ n propagates from time n to the variable χn+1 at time n + 1. If one thinks of the variable χn+1 as representing a particle, then one can think of the antiparticle as being equivalent to a particle propagating backwards in time; although this way of thinking is not required, it does help to develop some physical intuition about the peculiarities of antiparticles. Given the nature of the state space of the antiparticles, ZA can be decomposed as ZA = D χ¯ Dχ . . . χn+1 e−HA χ¯ n e−χ¯n χn χn e−HA χ¯ n−1 . . . (10.33) and by inspection χn+1 e−HA χ¯ n = e2K χ¯n χn+1 = e−HA (χ¯n χn+1 ) eχ¯n χn+1 ! 1 − 2K χ¯ χ = χH χ. ¯ ⇒ HA = Note that the order of the matrix element of the antifermions, namely χH χ, ¯ is the reverse of the fermion case. We define continuum fermion variables by ! √ √ 1 − 2K χ¯ t = 2K χ¯ n , χt = 2Kχn , ω = , t = n. 2K Anticommuting the fermionic variables and ignoring an irrelevant constant yields, similarly to Eq. 10.31, the continuum antifermion Hamiltonian HA = −ωχt χ¯ t .
(10.34)
After normal ordering, interpreting HA as an operator yields, using χ¯ = −δ/δχ , HA = ωχt
δ . δχt
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10.9 Fermion–antifermion Hamiltonian From Eq. 10.31, a Hamiltonian for fermions is given by ¯ = ωψ¯ HP = ωψψ
δ . δ ψ¯
(10.35)
The eigenstates and eigenenergies of the Hamiltonian are given by HP n = En n 0 = 1 1 = ψ¯
E0 = 0 E1 = ω.
From Eq. 10.34, a typical antifermion Hamiltonian is given by HA = ωχ ˜
δ = −ωχ ˜ χ¯ δχ
(10.36)
with eigenstates and eigenenergies A 0 = 1 A 1
=χ
E0 = 0 E1 = ω.
¯ For a single fermion degree of freedom, the Hamiltonian H has the form ωψψ and this is all that one can construct. Hence, it is necessary to look at more complicated systems, such as considering systems coupling fermion and antifermion as well as coupling of the fermions to bosons. Consider a fermion and antifermion system. The Hilbert state space is four dimensional since there are four possible states for the system, as enumerated in Eq. 10.16. A simple Hamiltonian for the fermion and antifermion system is a sum of the fermion and antifermion system with a coupling term, namely δ δ ¯ + ω χ + λψχ H = ωψ¯ ¯ δχ δψ ¯ ¯ − ω χ χ¯ + λψχ ≡ ωψψ
(10.37)
with real values for ω, ω , λ. The normalized eigenfunctions and eigenvalues are given by H n = En n . One can directly verify that 1 = χ 2 = ψ¯ ¯ 3 = ψχ
E1 = ω E2 = ω E3 = ω + ω .
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Note that the eigenstates do not form a complete set since the Hilbert space is four dimensional, the reason being that this Hamiltonian is not Hermitian. Using the rules of conjugation given in Eq. 10.17, ψ¯ → ψ, χ → −χ¯ ¯ χ¯ → −χ ψ → ψ, and reversing the order of the fermion variables shows that ¯ − ω χ χ¯ − λχ¯ ψ = H. H † = ωψψ
(10.38)
A Hermitian Hamiltonian for the fermion–antifermion system is given by δ δ δ2 ¯ + ω χ + λψχ ¯ δχ δ ψ¯ δ ψδχ ¯ + ω χ¯ χ − λψχψ ¯ = ωψψ χ¯ .
H = ωψ¯
Hermitian conjugation, explicitly shown below, shows that the Hamiltonian is Hermitian, ¯ + ω χ¯ χ − λχ¯ c ψ c χ c ψ¯ c H † = ωψψ ¯ + ω χ¯ χ − λ (−χ) ψ¯ (−χ¯ ) ψ = ωψψ ¯ + ω χ¯ χ − λχ ψ¯ χ¯ ψ = ωψψ ¯ ¯ + ω χ¯ χ − λψχψ χ¯ = ωψψ = H. The eigenfunctions can be read off by inspection and are 0 = 1 E0 = 0 1 = ψ¯ E1 = ω 2 = χ E2 = ω ¯ 3 = ψχ E3 = ω + ω − λ. The Hamiltonian given in Eq. 10.37 can be made Hermitian by adding the term required, and yields δ δ ¯ + χ¯ ψ). + ω χ H = ωψ¯ + iλ(ψχ ¯ δχ δψ
(10.39)
Two of the four orthogonal eigenstates can be obtained by inspection, and this yields 1 = χ 2 = ψ¯
E1 = ω E2 = ω.
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For the other two eigenfunctions consider the ansatz ¯ + ic. = ψχ Applying the Hamiltonian given in Eq. 10.39 on yields
λ ¯ −i . H = ω + ω − cλ ψχ ω + ω − cλ One obtains the eigenvalue condition c=−
λ , ω + ω − cλ
which has two solutions, namely 9 1 c± = ω + ω ± (ω + ω )2 + 4λ2 , c+ c− = −1. 2λ Hence, the remaining two eigenfunctions and eigenvalues are given by
1 ¯ + ic+ , E3 = ω + ω − λc+ 3 = 8 ψχ 2 1 + c+
(10.40)
1 ¯ + ic− , E4 = ω + ω − λc− . 4 = 8 (10.41) ψχ 2 1 + c− 9 2 2 The interpretation of the state 9 3 is that c+  / 1 + c+  is the likelihood that the system has no particles and 1/ 1 + c+ 2 is the likelihood of having a fermion– antifermion pair, with a similar interpretation for 4 . 10.9.1 Orthogonality and completeness To illustrate the workings of fermion calculus, the orthogonality of the states 3 , 4 is explicitly computed. Using the rules for forming the conjugate state function yields ¯ ¯ ¯ χ 4 3 4 = d ψdψd χ¯ dχ 3 ψ; χe ¯ −ψψ−χ¯ χ ψ; ¯ χ¯ χ ¯ χ]e−ψψ− ¯ = d ψdψd χ¯ dχ †3 [ψ; χ¯ ]4 [ψ; ¯ χ¯ χ ¯ ¯ + ic− e−ψψ− = d ψdψd χ¯ dχ − χψ ¯ − ic+ ψχ ¯ ¯ χ¯ χ = (1 + c+ c− ) d ψdψd χ¯ dχ ψψ = (1 + c+ c− ) = 0, since c+ c− = −1.
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The completeness equation can be expressed in terms of the eigenfunctions by I=
4
i i .
(10.42)
i=1
To verify Eq. 10.42, one needs to prove that 4 ¯ χi i ψ, χ¯ = exp{ψψ ¯ + χ¯ χ }. ψ, i=1
Using the explicit form of the state functions derived above yields 4
¯ χi i ψ, χ¯ = χ¯ χ + ψψ ¯ + N+ ψχ ¯ + ic+ [−χ¯ ψ − ic+ ] ψ,
i=1
¯ + ic− [−χ¯ ψ − ic− ] + N− ψχ
2 2 ¯ + c+ ¯ χ¯ χ = χ¯ χ + ψψ N+ + c− N− + [N+ + N− ] ψψ
¯ + χ¯ ψ] − i[c+ N+ + c− N− ][ψχ 1 1 where N+ = , N− = . 2 2 1 + c+ 1 + c− From Eq. 10.40 it follows that 2 2 N+ + c− N− = 1 = N+ + N− , c+
c+ N+ + c− N− = 0. Hence 4
¯ χi i ψ, χ¯ = 1 + χ¯ χ + ψψ ¯ + ψψ ¯ χ¯ χ = exp ψψ ¯ + χ¯ χ , ψ,
i=1
thus verifying Eq. 10.42. 10.10 Fermion–antifermion Lagrangian Consider the propagation of a fermion–antifermion system given by the Lagrangian (10.43) Ln = −ψ¯ n ψn − χ¯ n χn + 2K ψ¯ n+1 ψn + χ¯ n χn+1 . The term Ln consists of a fermion propagating forward in time and its antifermion (since K is the same for both) propagating “backward” in time. Define a twocomponent spinor ! ! ψu ψ ψ= = (10.44) ψ χ
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and ψ¯ = (ψ¯ and let
χ¯ )
! 1 0 γ0 = . 0 −1
(10.45)
(10.46)
Then Ln = −ψ¯ n ψn + K ψ¯ n+1 (1 + γ0 )ψn + ψ¯ n (1 − γ0 )ψn+1 . The fermion action is S=
Ln .
(10.47)
(10.48)
n
Consider the continuum limit by defining t = n and take the limit of → 0, yielding K ψ¯ n+1 ψ = K(ψ¯ n+1 − ψ¯ n )ψn + K ψ¯ n ψn ∼ = K∂0 ψ¯ n ψn + K ψ¯ n ψn .
(10.49)
K χ¯ n χn+1 = K χ¯ n ∂0 χ + K χ¯ n χn .
(10.50)
Similarly
Hence
Ln = (−1 + 2K) ψ¯ n ψn + χ¯ n χn + 2K ∂0 ψ¯ n ψn + χ¯ n ∂0 χn .
(10.51)
Let us define continuum fermionic variables by ψ¯ t = (2K)1/2 ψ¯ n , ψt = (2K)1/2 ψn , χ¯ t = (2K)1/2 χ¯ n , χt = (2K)1/2 χn . The continuum Lagrangian is given by !
2K − 1 L(t) = ψ¯ t ψt + χ¯ t χt + ψ¯ t (−∂0 )ψt + χ¯ t ∂0 χt . 2K We define
1 2K − 1 ⇒ 2K = m = − 2K 1 + m ! ! ψ 1 0 ¯ = ψ¯ χ¯ = . γ0 = χ 0 −1
(10.52)
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10.11 Fermionic transition probability amplitude
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Hence, the continuum action and Lagrangian are given by +∞ dtL, S= −∞
¯ 0 ∂0 + m) L(t) = −(γ : onedimensional Dirac Lagrangian.
10.11 Fermionic transition probability amplitude The derivation done earlier for finding the eigenstates and eigenenergies of the fermionic Hamiltonian can be obtained directly working with the fermion transition probability amplitude. Recall for a fermion particle ¯
¯ −H ψ = e2K ψψ . ψe
(10.53)
The eigenvalue and eigenstates that are given by 0 ∼ 1 and 1 ∼ ψ¯ can be directly obtained by performing fermion integration in the following manner: −H ¯ −H ςe−ς¯ ς ς ¯ 0 = dςd ς ¯ ψe ¯ 0 ψe ¯ = dςd ςe ¯ 2K ψς e−ς¯ ς ¯ = dςd ς(1 ¯ + 2K ψς)(1 − ς¯ ς ) = 1, ¯ = ψ ¯ 0 = 1, E0 = 0. ⇒ 0 (ψ)
(10.54)
For the eigenstate 1 ∼ ψ¯ consider the calculation ¯ −H ¯ ¯ ψe 1 = dςd ς¯ e2K ψς e−ςς ς¯ ¯ = 2K ψ¯ dςd ς¯ ς ς¯ = 2K ψ¯ = e−E1 ψ, ¯ ⇒ ψ1 = ψ, which yields the expected answer. The eigenenergy is given by 1 ⇒ E1 = − ln(2K) ! 1 1 = − ln 1 + m m + O().
(10.55)
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Similarly for the antifermion, recall χ e−H χ ¯ = e2K χ¯ χ .
(10.56)
Hence, for the antifermion ground state, e−H 0 = 0 since 0 ∼ 1. For χ1 = χ one has ¯ −ς¯ ς ς1 χ e−H 1 = χ e−H ςe e2K ς¯ χ e−ς¯ ς ς = ¯ ςς (1 + 2K ςχ)(1 ¯ − ςς)ς ¯ = ςς ¯ = 2Kχ (−ς¯ ς ) = 2Kχ . (10.57) Note that the transition amplitude automatically yields a normal ordered Hamiltonian with the energy given by 1 E1 = − ln(2K) m.
(10.58)
Until now, fermions and antifermions have equal mass but have not been coupled, and hence their contrasting properties have not come into play. One can cou¯ χ¯ χ , as well as by their ple them by nonlinear terms in the Lagrangian such as λψψ coupling to gauge fields, and this is briefly explored below.
10.12 Quark confinement Consider a onedimensional toy model of quarks (fermions) and antiquarks (antifermions) given by Ln = −ψ¯ n ψn + 2K ψ¯ n+1 ψn + χ¯ n χn+1 − χ¯ n χn . A gauge transformation on the fermions is defined by ψn → eiφn ψn
ψ¯ n → ψ¯ n e−iφn
χn → eiφn χn
χ¯ n → χ¯ n e−iφn .
To leave Ln invariant we need fix the nearest neighbor term. Let us introduce gauge field eiBn and modify Ln to Ln = −ψ¯ n ψn − χ¯ n χn + 2K ψ¯ n+1 e−iBn ψn + χ¯ n eiBn χn+1 .
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Under a gauge transformation, let Bn have the transformation eiBn → eiφn eiBn e−iφn+1 , and hence the combined transformations on the fermions and gauge field leave the Lagrangian L invariant. Note that Bn is an angular variable taking values in [−π, +π]. The partition function is & π dBn Z= d ψ¯ n dψn d χ¯ n dχn eS . 2π −π n Since eiBn couples nearest neighboring instants of time, one can derive as before the transition amplitude for quark–antiquark system, namely ψ¯ n+1 , χn+1 e−H ψn , χ¯ n , by generalizing Eq. 10.30. One can choose to include the integration over the gauge field in the definition of the transition probability amplitude for the fermions, and this yields π dBn Ln ψ¯ n+1 , χn+1 e−H ψn , χ¯ n = e . −π 2π Dropping the subscripts from Ln yields π dB ¯ −iB ψ 1 + 2K χe ¯ χe−H ψ, χ¯ = ¯ iB χ 1 + 2K ψe ψ, −π 2π ¯ χ¯ χ . = 1 + (2K)2 ψψ The transition probability amplitude acting on a general state function  yields ¯ ¯ χe−H ζ, ξ¯ e−ζ¯ ζ −ξ¯ ξ ζ¯ , ξ  ψ, χ = d ζ¯ dςd ξ¯ dξ ψ,
¯ ξ¯ χ e−ζ¯ ζ −ξ¯ ξ (ζ¯ , ξ ). = d ζ¯ dζ d ξ¯ dξ 1 + (2K)2 ψζ (10.59) The only nonzero integral is
d ζ¯ dζ d ξ¯ dξ ζ ζ¯ ξ ξ¯ = 1.
Solving the eigenvalue equation of the quark–antiquark Hamiltonian e−H n = e−En n using Eq. 10.59 yields the following eigenfunctions and eigenvalues. Note that single quark ψ¯ and antiquark χ states are confined since they have infinite energy and cannot propagate in time; they are fixed at whatever moment in
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n (ζ¯ , ξ )
En
State
1 ψ¯ χ ¯ ψχ
0 ∞ ∞ 2 − ln(2K)
vacuum one quark one antiquark quark+antiquark
¯ , are time they are created. Only the paired states of quark–antiquark, namely ψχ the finite energy eigenstates and hence can propagate in time. 10.13 Summary The fermion variable takes only two values and is fundamentally different from a real variable. The fermion degree of freedom describes a system that is essentially discrete and at the same time quite distinct from the Ising variable, that belongs to the category of real variables. A differential and integral calculus was developed for the fermion variable and the concepts applicable to a real variable were generalized to the fermion case. Gaussian integration was defined for both real and complex fermions and the results are similar to the real variable case, but with a few significant differences. Fermion and antifermion variables emerge naturally, based on the manner in which conjugation of the state vector is defined. The state space and Hamiltonian for fermions and antifermions were derived and the state space was shown to behave in the manner that one intuitively expects for a discrete system. A few simple models of the fermion and antifermion path integral, Hamiltonian, and state space were discussed. A onedimensional toy model based on fermion and antifermion degrees of freedom coupled to a gauge field was shown to exhibit quark confinement.
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Part four Quadratic path integrals
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11 Simple harmonic oscillator
A simple harmonic oscillator, or oscillator in short, is one of the most important systems in physics as well as in quantum mechanics. The path integral and Hamiltonian of a simple harmonic oscillator are leading exemplars for the description of a wide class of physical systems. An oscillator is also a theoretical model of great utility, primarily due to its simplicity that allows for the exact derivation of many results. The simple harmonic oscillator can be generalized to the case of infinitely many oscillators, and is also the starting point for analysis of many nonlinear quantum systems. Path integrals for the simple harmonic oscillator are all based on Gaussian integration, which has been discussed in Section 7.2. The harmonic oscillator is described by Gaussian path integrals, and is also a bedrock of the general theory of path integration. Moreover, all perturbation expansions of nonlinear path integrals about the oscillator path integral, which includes the semiclassical expansion, are based on results of Gaussian integration The properties of the harmonic oscillator are analyzed here in coordinate space representation, and its path integral as well as its correlation functions are analyzed. All formulas and derivations are for Euclidean time. In Sections 11.1 and 11.2 the Hamiltonian and state space of the oscillator are studied and the correlator is derived using state space methods. In Section 11.2 the infinite time oscillator is introduced and in Sections 11.3–11.5 the path integral for the oscillator is evaluated. In Sections 11.6–11.11 the oscillator path integral is studied on a finite lattice and the path integral is reduced to a finite dimensional ordinary multiple integral. In Section 11.12 the finite lattice is derived using the technique of the transfer matrix.
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11.1 Oscillator Hamiltonian The oscillator Hamiltonian H has a kinetic term equal to the free particle case – as in Eq. 5.69 – and which has an operator form valid both in Minkowski and Euclidean time; the oscillator Hamiltonian has a quadratic potential and hence is given by H =−
1 1 ∂2 + mω2 x 2 . 2m ∂x 2 2
Let us define the creation and annihilation operators as follows: mω ∂ 1 † = a − a† , x=√ a+a , ∂x 2 2mω ! ! 1 ∂ ∂ 1 † , a =√ , mωx + mωx − a=√ ∂x ∂x 2mω 2mω which yield the commutation equations
∂ x, = −I ⇒ a, a † = I. ∂x We define the oscillator ground state 0 by a0 = 0. The Hamiltonian and its eigenfunctions and eigenvectors are given by 1 H = ωa † a + ω, 2 (a † )n n = √ 0, mn = δn−m , a0 = 0, n! 1 H n = En n, En = nω + ω. 2 Furthermore, the oscillator algebra [a, a † ] = I yields the basis states √ √ a † n = n + 1 n + 1, a n = n n − 1, which are complete, namely
nn = I.
(11.1)
n
11.2 The propagator The Heisenberg operator is given by ˆ −tH ≡ xˆt . xH (t) = etH xe
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11.2 The propagator
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x^ t
Time
Figure 11.1 The correlator of two coordinate operators acting on the vacuum state.
Consider the correlation function, shown in Figure 11.1, D(t, t ) = 0T [xH (t)xH (t )]0,
(11.2)
which is a measure of the time interval over which disturbances on the vacuum are correlated. Given the special role of D(t, t ) – the correlator of two coordinate operators – it is called the propagator. Taking t = 0 and t > 0, the correlator is given by ωτ
ˆ −tH x0, ˆ D(t) = e 2 0xe 1 + = 0(a + a † )e−tωa a (a + a † )0 2mω 1 + = 1e−tωa a 1 2mω 1 −ωt = e . 2mω
(11.3)
(11.4)
For the general case given in Eq. 11.2, it can readily be shown that D(t, t ) =
1 −ωt−t  . e 2mω
(11.5)
11.2.1 Finite time propagator The propagator for finite time can be evaluated exactly for the case of the harmonic oscillator. The oscillator basis is used for the derivation and another derivation will be given later using the path integral. Figure 11.2 shows two coordinate operators acting at two different times, with a finite time interval with periodic time given by τ . To simplify the notation let xH (t) ≡ xˆt ; the finite time correlator is D t, t , τ =
1 ˆ ˆ tr T xˆt xˆt e−τ H , Z(τ ) = tr e−τ H . Z(τ )
(11.6)
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Figure 11.2 The correlator of two coordinate operators for periodic time.
Taking t = 0 and t > 0 yields, for D (t, 0, τ ) = D (t, τ ), 1 1 ˆ ˆ ˆ ˆ −t H xˆ tr xˆt xˆ0 e−τ H = tr e−(τ −t)H xe D (t, τ ) ≡ Z(τ ) Z(τ ) N = , Z(τ ) ˆ ˆ ˆ −t H xˆ . where N = tr e−(τ −t)H xe The numerator N is given by using the complete oscillator basis states n given by Eq. 11.1; hence ∞ % % ˆ ˆ % % n %e−(τ −t)H xe ˆ −t H xˆ % n N= n=0
=
∞ %% 1 −(τ −t)(n+ 1 )ω %% ˆ 2 e n % a + a † e−t H a + a † % n. 2mω n=0
Using the oscillator algebra yields √ 1 −(τ −t)(n+ 1 )ω 2√ ˆ 2 e n + 1 n + 1 + n n − 1 e−t H 2mω n=0 √ 3 √ × n + 1 n + 1 + n n − 1 ∞
N=
∞
1 −τ (n+ 1 )ω 2 e = (n + 1) e−ωt + neωt 2mω n=0 ∞
e− 2 −nωτ −ωt + eωt + e−nωτ e−ωt . ne e = 2mω n=0 ωτ
Note the two summations ∞ ∞ ∞ 1 ∂ −nωτ e−ωτ −nωτ −nωτ e = , ne = − e = . 1 − e−ωτ n=0 ∂(ωτ ) n=0 (1 − e−ωτ )2 n=0
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Hence, the partition function is given by Z(τ ) = tr(e
−τ Hˆ
)=e
− ωτ 2
∞
e−nωτ =
n=0
1 , 2 sinh ωτ 2
and the numerator simplifies to . / ωτ e−ωt + eωt e−ωτ e−ωt e− 2 + N= 2mω 1 − e−ωτ (1 − e−ωτ )2 ωτ e− 2 e−ωt e−ωτ + eωt e−ωτ + e−ωt − e−ωτ e−ωt = 2mω (1 − e−ωτ )2 . / ωτ ωτ eωt e− 2 + e−ωt e 2 1 = ωτ ωτ 2 2mω e 2 − e− 2 − ωt 1 cosh ωτ 2 . = 2mω 2 sinh2 ωτ 2 Hence, the propagator is given by
N 1 cosh ωt − ωτ 2 . D(t, τ ) = = Z(τ ) 2mω sinh ωτ 2
(11.7)
In general, for two arbitrary times t and t , the correlator is given by time ordering the operators and yields 1 cosh ωt − t  − ωτ 2 . (11.8) D t, t ; τ = 2mω sinh ωτ 2 Consider the limiting case of τ → ∞, ωτ
lim e−τ H e− 2 00
τ →∞
and hence, from Eq. 11.6,
lim D(t, t ; τ ) →
τ →∞
ωτ tr T xˆt xˆt e− 2 00 ωτ
tre− 2 00 = 0T xˆt xˆt 0.
Equation 11.2 shows that the finite time correlator reduces to the one for infinite time. Directly taking the limit of τ → ∞ in Eq. 11.8 yields the limiting value lim D(t, t ; τ ) →
τ →∞
1 −ωt−t  , e 2mω
as was obtained earlier in Eq. 11.5.
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11.3 Infinite time oscillator The action for the infinite time oscillator is given by m +∞ 2 S=− dt x˙ + ω2 x 2 2 −∞ ! d2 m 2 dtx − 2 + ω x =− 2 dt ! m d2 2 dtdt xt − 2 + ω δ(t − t )xt , =− 2 dt and yields Att
! d2 2 = − 2 + ω δ t − t . dt
The inverse of Att is given by $ # dp ip(t−t ) 1 1 −1 δ(t − t ) = 2 Att = e d2 2 d 2π − dt 2 + ω − dt 2 + ω2 dp eip(t−t ) 1 −ωt−t  = = . e 2 2 2π p + ω 2ω Hence
1 Dx xτ x0 eS E[xτ x0 ] = Z 1 1 −ωτ  = A−1 , e τ0 = m 2mω
(11.9)
(11.10)
as expected from Eq. 11.4. To interpret the meaning of the correlator one can repeat the earlier linear regression analysis discussed in Section 9.4. Since E[xτ ] = 0 and E[x02 ] = 1/(2mω), the random variables are related by the linear regression xτ
A−1 τ0 x0 e−ωτ  x0 . E[x02 ]
11.4 Harmonic oscillator’s evolution kernel The simple harmonic oscillator’s action and Lagrangian is given by L=−
m 2 1 x˙ − mω2 x 2 . 2 2
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The Euclidean time path integral is given by
K(xf , xi ; τ ) = x e
−τ H
x =
Dx eS .
The continuum action for finite time is given by S=
τ
0
m Ldt = − 2
τ
dt x˙ 2 + ω2 x 2 .
0
Consider the classical field equation δS = mx¨c (t) − mω2 xc (t) = 0, δx(t) ⇒ xc (t) = Ae−ωt + Beωt ,
(11.11) (11.12)
xc (0) = x, xc (τ ) = x .
boundary conditions:
(11.13)
Define new quantum variables ξ(t) by x(t) = ξ(t) + xc (t) that obey the boundary conditions ξ(0) = 0 = ξ(τ ). Hence, the path integral yields
Dx e = S
Dξ eS[ξ +xc ] ,
where Dx = Dξ . The action, for ξ˙ = dξ/dt, is 3 2 m τ 2 dt ξ˙ + x˙c + ω2 (ξ + xc )2 S[ξ + xc ] = − 2 0 m τ 2 =− dt ξ˙ + 2ξ˙ x˙c + x˙c2 + ω2 ξ 2 + 2ξ xc + xc2 2 0 m m dt x˙c2 + ω2 xc2 − m dt ξ˙ 2 + ω2 ξ 2 . =− ξ˙ x˙c + ω2 ξ xc − 2 2
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Note, since ξ(0) = ξ(τ ) = 0, ' τ τ d ˙ξ x˙c = (ξ x˙c ) − ξ x¨c = ξ(0)x˙c (0) − ξ(τ )x˙c (τ ) − ξ x¨c dt dt 0 0 = − ξ x¨c dt. The crossterms of xc and ξ cancel using the classical equation, Eq. 11.11, and S[xc + ξ ] = S[xc ] + S[ξ ].
(11.14)
Hence, the evolution kernel factorizes and yields K(x , x; τ ) = N eSc , where using Dx = Dξ , we have N = Dξ eS[ξ ] : independent of x, x . It is shown in Eq. 11.24 that S[ξ ] =N = Dξ e yielding the result
K(x , x; τ ) =
(11.15)
mω , 2π sinh ωτ
mω e Sc . 2π sinh ωτ
(11.16)
The classical action Sc is given by m τ 2 Sc = − dt x˙c + ω2 xc2 2 0 m τ m τ dtxc −x¨c + ω2 x¨c = − (xc x˙c ) 0 − 2 2 0 m = − [xc (τ )x˙c (τ ) − xc (0)x˙c (0)] . 2 Equation 11.12 yields x˙c (t) = ω −Ae−ωt + Beωt . The classical action is hence mω 2 2ωτ B e − 1 + A2 1 − e−2ωτ . Sc [xc ] = − 2 The boundary conditions, from Eq. 11.13, are
(11.17)
xc (0) = A + B = x, xc (τ ) = Ae−ωτ + Beωτ = x ,
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and yield ωτ 1 1 xe − x , B = x − xe−ωτ , (11.18) 2 sinh ωτ 2 sinh ωτ and hence the classical solution for S0 , from Eqs. 11.12 and 11.18, is given by A=
ωτ 1 xe − x e−ωt + x − xe−ωτ eωt . (11.19) 2 sinh ωτ From Eq. 11.17, the classical action is mω 2 Sc = − (11.20) x + x 2 cosh ωτ − 2xx , 2 sinh ωτ yielding the final result for the evolution kernel 3 2 mω mω 2 K(x , x; τ ) = exp − x + x 2 cosh ωτ − 2xx . 2π sinh ωτ 2 sinh ωτ (11.21) xc (t) =
11.5 Normalization
−τ Hˆ
x. Consider the periodic trace of K, namely Recall K(x , x; τ ) = x e −τ H trK = dxxe x = dxK(x, x; τ ). Using K = N exp{Sc } yields +∞ dxeSc (x,x;τ ) trK = N −∞ 2 3 mω = N dx exp − [cosh τ − 1]x 2 sinh ωτ 7 sinh ωτ 2π =N . mω 2(cosh ωτ − 1)
(11.22)
In the oscillator basis H = ωa † a +
ω 2
the trace yields, using this basis, tr(K) = tr e−τ H τω τω † † ne−τ ωa a n = e− 2 tr e−τ ωa a = e− 2 n
= e−
τω 2
∞
e
n=0
−nτ ω
= e−
τω 2
1 . 1 − e−τ ω
(11.23)
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Hence, from Eqs. 11.22 and 11.23 N =
mω . 2π sinh(ωτ )
(11.24)
11.6 Generating functional for the oscillator Consider the simple harmonic oscillator in the presence of an external source j (t) given by 1 1 L = − mx˙ 2 (t) − mω2 x 2 (t) + j (t)x(t) 2 2 and with the finite time action τ τ τ 2 1 2 2 S=− m x˙ + w x + j (t)x(t) = S0 + j (t)x(t). 2 0 0 0
(11.25)
The path integral for the normalized generating functional is given by " 1 1 S Dxe = DxeS0 + j (t)x(t) , Z[j ] = Z Z Z = DxeS . The generating functional contains the full content of the quantum system; all the correlation functions for the action S0 , in the presence of the boundary conditions, can be evaluated by functional differentiation. In particular, the first few correlators are given by 1 1 δZ[j ] %% Dx x(t)eS0 , = % Z δj (t) j =0 Z 1 δ 2 Z[j ] %% 1 Dx x(t)x(t )eS0 , = % Z δj (t)δj (t ) j =0 Z % δ 3 Z[j ] 1 1 % Dx x(t), x(t )x(t )eS0 , = % Z δj (t)δj (t )δj (t ) j =0 Z and so on.
11.6.1 Classical solution with source Similarly to the derivation in Section 11.4, one can use the classical equations of motion to evaluate the Gaussian path integral. The classical solution obeys the equation
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!
−
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235
d2 J (t) + w 2 xc = , 2 dt m xc = xH + xI ,
(11.26)
where xH , xI are the homogeneous and inhomogeneous solutions respectively. Note that the homogeneous solution depends on the boundary conditions. In particular, ! d2 2 − 2 + w xH = 0 ⇒ xH = A e−wt + B ewt . dt The inhomogeneous solution is given by τ 1 1 1 J = dt e−wt−t  J (t ) xI (t) = t d2 m 2mw 0 − dt 2 + w 2 t τ 1 −w(t−t ) −w(t −t) dt e J (t ) + dt e J (t ) , = 2mw 0 t
(11.27)
where d2 − 2 + w2 dt
!−1
e−wt−t  δ(t − t ) = . 2w
(11.28)
We fix boundary conditions by xc (0) = x = xH (0) + xI (0), xc (τ ) = x = xH (τ ) + xI (τ ). Hence 1 x(0) = x = A + B − 2mω
x(τ ) = x = A e−ωτ + B eωτ
τ
dte−ωt J (t), 0 τ 1 − dte−ω(τ −t) J (t), 2mω 0
which yields 1 x − e−ωτ x − A = 2 sinh(ωτ ) 1 e−ωτ x − x − B = 2 sinh(ωτ )
1 τ dt sinh ω(τ − t)J (t) , m 0 e−ωτ τ dt sinh(ωt)J (t) . mω 0
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After considerable simplifications, mω 2 S[xc ; j ] = − (x + x 2 ) cosh ωτ − 2xx 2 sinh ωτ τ τ x x + dtJ (t) sinh w (τ − t) + dtJ (t) sinh(wt) sinh wτ 0 sinh wτ 0 τ t 1 + dt J (t) sinh w (τ − t) sinh wt J t . mw sinh wτ 0 0 (11.29) The generating functional is hence given by the decomposition x(t) = xc (t) + ξ(t), ξ(0) = 0 = ξ(τ ), that yields
S[xc ;j ] ˜ Dxe = Ae Dξ eS0 [ξ ] , ˜ ⇒ A Dξ eS0 [ξ ] = N : independent of x, x , and j (t), S
with normalization constant N given by Eq. 11.24. The generating functional, from Eq. 11.29, is given by 1 DxeS = eS[ξ ;j ] , Z[j ] = Z with the normalization constant N canceling out.
11.6.2 Source free classical solution The classical solution yields the inhomogeneous term xI , as in Eq. 11.26, arising from the source j (t). To obtain the final answer given in Eq. 11.29 requires a fair amount of algebra. Given the importance of the simple harmonic oscillator, another derivation of the same result is given here. This derivation turns out to be useful in situations that are more complicated than the simple one that we are considering. For this derivation, consider the classical solution for the source free action, that is only S0 [x] – the external source j (t) is not included in S0 [x] – and hence the classical solution does not have an inhomogeneous term xI . The change of variables x = xc + ξ now yields a path integral for which the quantum variable is coupled to the source j (t) – and one needs to perform a path integral over the quantum degree of freedom ξ(t) to obtain the result.
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Recall, from Eq. 11.25, that the simple harmonic action for the generating functional is given by τ τ 2 1 x˙ + w2 x 2 . j (t)x(t), S0 = − m S = S0 + 2 0 0 The classical solution for S0 , from Eq. 11.19, is given by xc (t) =
ωτ 1 xe − x e−ωt + x − xe−ωτ eωt . 2 sinh ωτ
Define the quantum variables ξ by x(t) = xc (t) + ξ(t), ξ(0) = 0 = ξ(τ ). The classical solution yields, from Eq. 11.14, the factorization S0 [xc + ξ ] = S0 [xc ] + S0 [ξ ]. The path integral for the generating functional, up to an irrelevant normalization, is hence given by DxeS = eF0 [xc ;j ] Dξ eF [ξ ;j ] , τ F0 [xc ; j ] = S0 [xc ] + j (t)xc (t), 0τ F [ξ ; j ] = S0 [ξ ] + dtj (t)ξ(t). 0
The following path integral over the quantum variables ξ(t) needs to be performed: Dξ exp{F [ξ ; j ]}, ξ(0) = 0 = ξ(τ ). The boundary conditions ξ(0) = 0 = ξ(τ ) are satisfied by the Fourier sine expansion for ξ(t), nπt ξn , ξ(t) = sin τ n=1 ∞
where, for each n, ξn is an independent and real (indeterminate ) integration variable. Hence, using the orthogonality of the sine functions, τ mπt τ nπt sin = δn−m dt sin τ τ 2 0
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yields ∞ ∞ 1 2 F [ξ ; j ] = − λn ξn + jn ξn , 2 n=1 n=1 ' τ 1 nπt nπ 2 2 . λn = mτ + w , jn = j (t) sin 2 τ τ 0
All the ξn variables have decoupled and we obtain, up to a normalization, D 6 +∞ +∞ ∞ ∞ & 1 2 Dξ exp {F [ξ ; j ]} = dξn exp − λn ξn + jn ξn 2 −∞ n=1 n=1 n=1 7 6 ∞ D 1 1 1 = exp jn jn . 2πλn 2 n=1 λn The result further simplifies, ∞ n=1
jn
1 jn = λn
D(t, t ) =
∞ n=1
τ
dt 0
τ
dt j (t)D(t, t )j (t ),
0
nπt 1 nπt sin . sin λn τ τ
Using the identity +∞ cos(θ) π cosh[a(π − θ)] 1 = − 2 2 2 a +n 2a sinh πa 2a n=1
(11.30)
yields D t, t =
1 sinh w (τ − t) sinh wt , t > t mw sinh wτ
and hence τ 1 τ dt dt j (t)D t, t j t 2 0 0 τ t 1 = dt dt J (t) sinh w(τ − t) sinh wt J t . mw sinh wτ 0 0 Note, from Eq. 11.19, that τ j (t)xc (t) = 0
τ x dtJ (t) sinh w(τ − t) sinh wτ 0 τ x dtJ (t) sinh(wt). + sinh wτ 0
(11.31)
(11.32)
(11.33)
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The term S0 [xc ] is given in Eq. 11.20; hence, from Eqs. 11.32 and 11.33 DxeS = exp {S[xc ; j ]} , τ τ 1 τ S[xc ; j ] = S0 [xc ] + j (t)xc (t) + dt dt j (t)D(t, t )j (t ), 2 0 0 0 and, as expected, S[xc ; j ] is given by Eq. 11.29. Furthermore, the generating functional is given by 1 Z[j ] = DxeS Z τ 21 τ τ 3 = exp dt dt j (t)D(t, t )j (t ) + j (t)xc (t) , 2 0 0 0 with the source free classical solution S0 [xc ] canceling out due to the normalization by Z. 11.7 Harmonic oscillator’s conditional probability Consider the Lagrangian given by
dx 2 1 2 2 L=− m ( ) +ω x . 2 dt
The evolution kernel is the probability amplitude that the harmonic oscillator’s degree of freedom will reach a final point xf from initial point xi in time τ . It is given, up to a normalization constant that cancels out, by K(xf , xi ; τ ) = xf e−τ H xi 2 3 mω 2 (xi + xf 2 ) cosh ωτ − 2xi xf . = N exp − 2 sinh ωτ The conditional probability quantifies the likelihood that the outcome of an experiment will yield the value of the coordinate to be xi , given that xi has occurred; it is given by1 P (xf xi ; τ ) = "
K(xf , xi ; τ )2 . dxf K(xf , xi ; τ )2
The denominator can be simplified as follows: x 2 +x 2 cosh ωτ −2xi xf − mω dxf K(xf , xi ; τ )2 = dxf e 2 sinh ωτ i f 2π sinh ωτ = exp −mωxi2 tanh ωτ , 2mω 1 The normalization constant N cancels out and is set to unity in this section.
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yielding the conditional probability given by mω P (xf xi ; τ ) = exp mω tanh ωτ xi2 K(xf , xi ; τ )2 . π sinh ωτ Note that the normalization of the evolution kernel drops out of P (xi , xf ; τ ). As expected, one obtains the normalization +∞ dxf P (xf xi ; τ ) = 1. −∞
The conditional probability is different for Euclidean time compared to Minkowski time. 11.8 Free particle path integral A path integral derivation is given of the evolution kernel for a free particle degree of freedom moving in one dimension (d = 1), which was obtained earlier in Eq. 5.72 using the eigenfunctions of the free particle Hamiltonian. Let = t/N ; from Eq. 4.30 the path integral for finite is given by a multiple integral. Using the infinitesimal form of Eq. 5.72, which also directly follows from the Dirac–Feynman formula, yields (set = 1) K(x, x ; τ ) =
N−1 &
K(xn+1 , xn ; )
n=0

= ≡
m 2π
! N−1 & n=1 m
m 2π
+∞ −∞
m
dxn e− 2
m
(N−1 n=0
m
(xn+1 −xn )2
2
Dx e− 2 (x−xN−1 ) · · · e− 2 (x2 −x1 ) e− 2 (x1 −x ) , 2
2
boundary conditions : x = xN , x = x0 , where

Dx ≡
m 2π
! N−1 & n=1
m 2π
(11.34)
+∞ −∞
dxn .
(11.35)
Note the identity +∞ m 1 − m · 1 (x2 −x )2 m m 2 2 dx1 e− 2 (x2 −x1 ) − 2 (x1 −x ) = . e 2 2 2π −∞ 2 One can evaluate the path integral exactly by performing the Dxintegrations recursively, starting from the end with x1 . The successive integrations over the variables x1 → x2 → x3 · · · → xN−1 yield
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K(x, x ; τ ) =
11.9 Finite lattice path integral
m − m · 1 (x−x )2 = e 2 N 2πN

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m − m (x−x )2 , e 2τ 2πτ
(11.36)
which is the result obtained in Eq. 5.72. The case for a free particle in arbitrary ddimensional space follows from Eq. 11.36, since the ddimensional transition amplitude factorizes into separate onedimensional components. To obtain the evolution kernel in Minkowski time, recall from Eq. 5.16 that τ = it. Hence, from Eq. 11.36, the Minkowski time evolution kernel, denoted by subscript M, and restoring the in the formula, yields m − m (x−x )2 m i m (x−x )2 = , (11.37) e 2τ e 2t KM (x, x ; t) = 2πτ 2πit which was obtained in Eq. 5.72 using the free particle Hamiltonian.
11.9 Finite lattice path integral A rigorous definition of the path integral can be given by approximating it by a finite dimensional multiple integral. One can then take the continuum limit and obtain the functional integration required for defining a path integral. There is another reason for considering the finite approximation of this integral. For numerical simulation one always approximates the path integral by a finite dimensional multiple integral. Hence it is important to have some exact analytical results for a finite system so as to compare the numerical simulations with it. One of the few exact path integral results that one can derive is for the finite approximation of the harmonic oscillator path integral, and in this section the oscillator is analyzed in some detail. Consider an open lattice with boundary values of xN and x0 ; the action is given by N−1 N−1 m 1 2 2 S=− xn2 , (xn+1 − xn ) − mω 2 n=0 2 n=1 N−1 & dxn eS. K(xN , x0 ; τ ) = xN e−τ H x0 = N i=1
To simplify the analysis, consider time to be periodic with boundary condition xN = x0 . Furthermore, for a symmetric labeling of the lattice sites, consider a periodic lattice of size 2N + 1 as shown in Figure 11.3.
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–N
0
1
N
Figure 11.3 Finite periodic lattice, with symmetric numbering of the lattice sites.
The degrees of freedom are periodic in the discrete time index n, where t = n, = τ/(2N + 1), xn = xn+2N +1
+N 1 = e2π ipn/(2N +1) xp . 2N + 1 p=−N
For simplicity, use the notation 2πp/(2N + 1) = k; in this simplified notation one writes xn =
eikn xk ≡
k
+N 1 e2π ipn/(2N +1) xp , 2N + 1 p=−N
and one has the identity N n=−N
e
i(k+k )n
=
+N
2π i
e 2N+1 n(p+p ) = (2N + 1)δ(p + p ) ≡ δk+k .
n=−N
The action is hence given by +N N m mω2 2 2 S=− x (xn+1 − xn ) − 2 n=−N 2 n=−N n N 3 m 2 ik e − 1 xk (eik − 1)xk + 2 ω2 xk xk ei(k+k )n =− 2 n=−N k,k m ik e − 12 + 2 ω2 xk x−k , =− 2 k
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where note that, due to the simplified notation being used, one has ! πp ik 2 2 2 2 + 2 ω2 . 1 − e  + ω = 4 sin 2N + 1 In the momentum basis, both the gradient and quadratic term are diagonalized, and hence the path integral can be performed exactly.
11.9.1 Coordinate and momentum basis The change of integration variables from real space variables xn to momentum space variables xk is analyzed below. There are 2N + 1 real integration variables xn , and these are transformed to 2N + 1 variables xk that are complex, and hence there seem to be too many xk variables. However, note that the xn coordinates are all real and hence in the Fourier ( expansion given by xn = k eikn xk , Eq. 11.38 yields xn∗ = xn ⇒ xk∗ = x−k , with k = 0, ±1, ±2, . . . ± N. Hence, although there are 2N + 1 complex variables xk , they are not all independent, and one can choose only the set of complex variables xk – with k = 0, 1, 2, . . . N – to be the independent complex variables. Note the special case of the zero momentum mode, which is real since x0 = ∗ : real. xk=0 = xk=0 In summary, there are 2N + 1 independent momentum space variables; writing the momentum modes in real and imaginary components yields the 2N + 1 independent momentum space variables to be xk = xkR + ixkI : k = 1, 2, . . . N, x˜0 ≡ xk=0 .
(11.38)
The change of 2N + 1 variables from xn to xk has a Jacobian equal to one, and yields +N & n=−N
+∞ −∞
dxn =
+∞ −∞
d x˜0
N & k=1
+∞
−∞
dxkR
+∞ −∞
dxkI
≡
Dx.
11.10 Lattice free energy To illustrate the workings of the momentum space variables, the partition function is evaluated as follows. Let the lattice simple harmonic Hamiltonian be denoted by HLat ; the partition function for the periodic lattice is given by
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Z = tr(exp {−(2N + 1)HLat } +N m (2N +1)/2 & dxn eS = 2π n=−N N & m (N ik 2 2 2 dxkR dxkI e− 2 k=−N {1−e  + ω }x−k xk , = N d x˜0 k=1
where HLat is the lattice Hamiltonian. Note, since k ≡ 2πp/(2N + 1), ! πp ik 2 2 2 2 + 2 ω2 ≡ dk . 1 − e  + ω = 4 sin 2N + 1 From Eq. 11.38 one has the simplification 2 2 x−k xk = xkR − ixkI xkR + ixkI = xkR + xkI . Hence the action completely factorizes into independent Gaussian integrations for each variable xk , k = 0, 1, 2, . . . N. This factorization is the reason in the first place why the variables were transformed from real space to momentum space, since in the momentum basis both the quadratic potential and the kinetic energy are diagonal. The partition function is given by ∞ N m 2N+1 & m m 2 2 2 2 R I − 2 dk [(xkR )2 +(xkI )2 ] dxk dxk e dx0 e− 2 · ω x0 Z= 2π −∞ k=1 .N / ! ! m N 2π N & 1 m 1/2 2π 1 = · 2 2 2π m dk 2π m ω k=1 7 $ # ! N 1 1 & 1 = exp − = ln (dk ) d0 k=1 dk 2 k = e−(2N+1)F ,
(11.39)
where d0 = 2 ω2 has been used in obtaining Eq. 11.39. The free energy for the simple harmonic partition function is given by F = free energy per lattice site ! ' 1 1 2 k 2 2 + ω ln(dk ) = ln 4 sin = 2 k 2 k 2 ! ' N 1 1 πp 2 2 2 = + ω . ln 4 sin 2 2N + 1 p=−N 2N + 1
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Taking the limit of N → ∞ yields ! ' 1 −π dk 2 k 2 + ω0 , ω02 = lim 2 ω2 . F = ln 4 sin →0 2 −π 2π 2 11.11 Lattice propagator The propagator (correlator) for the lattice simple harmonic oscillator on a periodic lattice is given by 1 Dx xn+m xm eS . Dn = E[xn+m xm ] = Z The propagator does not depend on m due to periodicity of the lattice. Transforming to the momentum space basis yields 1 DXk xp xk eS Dn = eik(n+m) eik m Z k,k +,)* = eikn eim(k+k ) xk xk k,k
=
1 ikn im(k+k ) e e δk+k ik m 1 − e 2 + 2 ω2 k,k
eikn = m k 1 − eik 2 + 2 ω2 +N 1 cos(2πpn/(2N + 1)) ≡ . 8m 2N + 1 p=−N sin2 π pn + 2 ω2 2N+1 4
(11.40)
It can be shown by a direct and tedious calculation that the Fourier sum given in Eq. 11.40 yields Dn = where 2 ω2 ω = mω 1 + 4
r n + r 2N+1−n , 2ω (1 − r 2N+1 )
(11.41)
!1/2 ,
2 ω2 2 ω2 r =1+ − ω 1 + 2 4
#
!1/2 =
ω − 2

$2
ω 2 +1 2
.
(11.42)
Note the important fact that Eq. 11.41 is convergent for large N because r obeys the bounds 0 < r < 1.
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To define the continuum limit, the lattice parameters are defined in terms of continuum time t and total time τ as τ t n = (2N + 1), = , τ 2N + 1 ! 1 ωτ , +O r =1− 2N + 1 (2N + 1)2 ! 1 . ω = mω + O (2N + 1)2 For limit N → ∞, the propagator is given by 2N+1
2N+1
r n− 2 + r 2 −n , 2N+1 2N+1 2mω r − 2 − r 2 !2N+1 ωτ = 1− → e−ωτ , 2N + 1
Dn = r 2N+1
r n−
2N+1 2
t
1
1
t
ωτ
= r ( τ − 2 )(2N +1) → e−ωτ ( τ − 2 ) = e−ωt+ 2 , ωτ
1 e−ωt+ 2 + eωt− ∴ Dn → ωτ ωτ 2mω e 2 − e− 2
ωτ 2
.
Hence, one obtains the continuum limit given by 1 cosh ωt − ωτ 2 , Dn = 2mω sinh ωτ 2 which agrees with the result obtained earlier in Eq. 11.7. 11.12 Lattice transfer matrix and propagator The propagator can be found exactly for the lattice simple harmonic oscillator using the exact transfer matrix [Creutz and Freedman (1981)]. The path integral can be expressed as a trace due to the periodic boundary conditions. Hence Z= DXeS (periodic) 2N+1
= tr(T
).
To obtain the transfer matrix T, consider two adjacent sites; then the integrand of the lattice path integral for Z is given by the product · · · xn+2 T xn+1 xn+1 T xn xn T xn−1 · · ·
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The terms linking variables xn and xn+1 are entirely due to matrix element xn+1 T xn , whereas terms like xn2 in S can be symmetrically divided between adjacent matrix elements of T . A Hermitian transfer matrix is given by x T x = e− Recall, for [x, p] = i, one has
x e
2 − 2m pˆ
mω2 2 4 x

x = 
⇒ Tˆ =
m
2
e− 2 (x−x ) e−
mω2 2 4 x
.
m − m (x −x)2 , e 2 2π 2π − mω2 x 2 − pˆ 2 − mω2 x 2 e 4 e 2m e 2 . m
If one is working to only lowest order in , then using the CBHformula one can ˆ easily show that T ∼ e− H , where H = p2 /2m + mω2 x 2 /2. For the finite size lattice, we need the operator expression for T that is exact in , that is, correct to all powers of . Note, for α = /m and u2 = m2 ω2 , it can be shown that [Creutz and Freedman (1981)] α 2 u2 T x − iαT p, 2 ! α 2 u2 α 2 u2 2 T x. T p + iαu 1 + [p, T ] = 2 4 [x, T ] =
Using the commutators given above it can be shown that [T , H ] = 0, where 1 1 H = p2 + β 2 x 2 , 22 α 2 u2 2 ω2 β =u 1+ = mω 1 + = ω , 4 4 where the last relation follows from Eq. 11.42. The eigenstates of Hˆ also diagonalize T . Let the creation and annihilation operators be defined as 1 ω x + ip , a=√ 2ω 1 a† = √ ω x − ip , 2ω † [a, a ] = 1.
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One can show that [a, T ] = (r − 1)T a with α 2 u2 α 2 u2 r =1+ − αu 1 + 2 4
!1/2 .
Since n are eigenstates of T , T n = λn n, and the commutator [a, T ] yields (r − 1)T an = (aT − T a)n or
(r − 1)λn−1 n − 1 = (λn − λn−1 )n − 1, (r − 1)λn−1 = λn − λn−1 , λn = r ⇒ λn = Kr n λn−1
(K = constant).
Hence T = AKr (H /W ) . To determine the constant K, consider the trace of T ∞
1 Kr 1/2 trT = K . r n+1/2 = A 1−r n=0 Note that from Eq. 11.43 it follows that ./2 α 2 u2 αu r= 1+ . − 4 2 Hence
8 2 2 1 + α 4u − αu 1 r 2 = 8 = . 1−r αu α 2 u2 αu αu 1+ 4 − 2 1/2
The definition of T yields 1 trT = A
dxxe−
αu2 2 4 x
α
e− 2 p e− 2
αu2 2 4 x
x
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11.13 Eigenfunctions from evolution kernel
=
−∞
=
+∞
dxe− 
2π · αu2
which yields K = 1. Consequently
αu2 2 2 x
α
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xe− 2 p x 2
1 1 = , 2πα αu
T =
2π Hˆ /w . r m
The path integral is hence given by Z = tr(T 2N+1 ) ! 2N+1 2 2πr 1 = . m 1 − r 2N+1 Furthermore, the propagator is given by 1 n 2N+1−n tr xT xT Z n 1 2N+1−n r + r , = 2ω (1 − r 2N+1 )
Dn = E [xn+m xm ] =
which is seen to agree with the earlier result given by Eq. 11.41 11.13 Eigenfunctions from evolution kernel Recall K(x, x ; τ ) = xe−τ H x = e−τ En xψn ψn x n
=
e−τ En ψn (x)ψn∗ (x ),
n
where ψn , En are the eigenfunctions and eigenvalues of H . Expanding the sum yields K x, x ; τ = eτ E0 ψ0 (x)ψ0∗ x + e−τ E1 ψ1 (x)ψ1∗ x + · · · The evolution kernel can be expanded in a power series of e−τ En term by term, which yields the eigenfunctions. Consider the harmonic oscillator with 3 2 mω mω 2 K x, x ; τ = exp − (x + x 2 ) cosh ωτ − 2xx . 2π sinh(ωτ ) 2 sinh ωτ
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For τ → ∞, one has to leading order

mω − mω x 2 +x 2 , K(x, x ; τ ) e e 2 2π mω 1/4 mω 2 ω ⇒ E0 = e− 2 x . ψ0 (x) = 2 2π To next leading order ωτ 3 K x, x ; τ ψ0 (x)ψ0 (x ) e− 2 + 2mωxx e− 2 ωτ ' 1 5 + + 2m2 ω2 x 2 x 2 − mω(x 2 + x 2 ) e− 2 ωτ + · · · 2
− ωτ 2
and hence by inspection 3 E1 = ω 2 5 E2 = ω 2
, ,
mω 1/4 √ mω 2 · 2mω · xe− 2 x , 2π mω 1/4 1 mω 2 ψ2 (x) = √ 2mωx 2 − 1 e− 2 x . 2π 2 ψ1 (x) =
11.14 Summary The simple harmonic oscillator is a leading exemplar in the study of path integrals since many of the key ideas can be fully worked out for the oscillator. Starting from the Hamiltonian, the propagator of the oscillator was derived using state space methods. The oscillator path integral was then defined and the evolution kernel was evaluated, both in the presence and absence of an external current. The path integral for discrete time and for a finite interval was shown to reduce to an ordinary multiple integral. The propagator for the finite and discrete time was exactly evaluated using both the multiple integral formulation and the transfer matrix formulation, showing the consistency of the integral and differential formulations. The evolution kernel was lastly used to derive the ground state and the first few excited states of the simple harmonic oscillator.
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12 Gaussian path integrals
Several path integrals are exactly evaluated here using Gaussian path integration. A few general ideas are illustrated using the advantage of being able to exactly evaluate Gaussian path integrals. Path integrals defined over a particular collection of allowed indeterminate paths can sometimes be represented by a Fourier expansion of the paths. This leads to two important techniques for performing path integrations: 1. Expanding the action about the classical solution of the Lagrangian; 2. Expanding the degree of freedom in a Fourier expansion of the allowed paths. Various cases are considered to illustrate the usage of classical solutions and Fourier expansions, and these also provide a set of relatively simple examples to familiarize oneself with the nuts and bolts of the path integral. The Lagrangian of the simple harmonic oscillator is used for all of the following examples; all the computations are carried out explicitly and exactly. The following different cases are considered: • Correlators of exponential functions of the degree of freedom are discussed in Section 12.1. • The generating functional for periodic paths is evaluated in Section 12.2. • The path integral required for evaluating the normalization constant for the oscillator evolution kernel is discussed in Section 12.3. The path integral entails summing over all paths that start from and return to the same fixed position. • Section 12.4 discusses the evolution kernel for a particle starting at an initial position xi and, after time τ , having a final position that is indeterminate. • The path integral of a free particle in the presence of a fixed external current j is discussed in Section 12.5. • Section 12.6 discusses the evolution kernel for a system with indeterminate initial and indeterminate final positions.
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• The evolution kernel of the harmonic oscillator was derived using the classical solutions. Given its importance, the evolution kernel of the oscillator is derived using the Fourier expansion in Section 12.7. In Section 12.8 the path integral for a free particle in the presence of a magnetic field is exactly evaluated; this example has a new feature in that the magnetic field couples to the velocity of the particle. The calculation is carried out in Minkowski time.
12.1 Exponential operators Gaussian integration has the important property, as seen in Eq. 7.8, that the generating function can be exactly evaluated. This result allows the exact study of exponential operators of the coordinate operators, namely eix(t)/a , where a is an arbitrary length scale required for making the exponent dimensional. For the case of the simple harmonic oscillator, the degree of freedom x(t) is sometimes called log normal since, in the path integral formulation, x(t) can be considered a Gaussian (normal) random variable. The correlators of the exponential operator can be exactly evaluated. From Eqs. 7.8 and 11.9 the infinite time case yields " 1 DXei dtj (t)x(t) eS Z[j ] = Z ' 1 −ωt−t  dtdt j (t)e = exp − j (t ) . (12.1) 2mω Consider the correlator of two operators " 1 1 DXeS eix(t)/a e−ix(t )/a = DXei dτj (τ )x(τ ) eS , G(t, t ) = Z Z
1 δ(τ − t) − δ(τ − t ) . ⇒ j (τ ) = a Hence 1 dτ dτ e−ωτ −τ  G(t, t ) = exp − 2mωa 2 '
× δ(τ − t) − δ(τ − t ) δ τ − t − δ(τ − t ) ' 1 −ωt−t  1−e . = exp − mωa 2
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Taking the limit of ω → 0 yields ' t−t  1 G(t, t ) exp − 2 t − t  = e− ξ , ma where correlation time is given by ξ = ma 2 . Note that for ω = 0, namely, the case of a free particle, it can be shown that the correlation of a product of operators eix1 (t)/a1 eix2 (t)/a2 . . . eix1 (t)/an is nonzero only if 1 1 1 + ... + = 0. a1 a1 an 12.2 Periodic path integral The finite time action for a simple harmonic oscillator is m τ S=− dt (x˙ 2 + ω2 x 2 ). 2 0
(12.2)
The correlator for finite time is given in Eq. 11.6, 1 ˆ ˆ tr T xˆt xˆt e−τ H , Z(τ ) = tr(e−τ H ). D t, t , τ = Z(τ ) Due to the trace, the propagator is given by the Feynman path integral over all periodic paths, namely for x(t + τ ) = x(t). More precisely, the propagator is given by 1 DXx(t)x(t )eS , D t, t , τ ≡ Dt−t = Z x(t + τ ) = x(t). To calculate the propagator, consider the generating function given by ' " 1 1 S+ j (t)x(t) DXe j (t)Dt−t j (t ) . = exp Z[j ] = Z 2
(12.3)
All the periodic path x(t) are given by the Fourier expansion x(t) =
+∞
e2π int/τ xn .
(12.4)
n=−∞
Since
τ
dte2π int/τ e2π imt/τ = τ δn−m ,
(12.5)
0
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the action is given by mτ S=− 2 n
6
2πn τ
D
!2 +ω
2
xn x−n .
(12.6)
The Fourier modes xn have all decoupled and one can exactly evaluate the path integral. From Eq. 12.3, using Gaussian integration yields ' 1 j (t)Dt−t j (t ) , Z[j ] = exp 2 where Dt−t =
+∞ 1 1 2π in(t−t )/τ e 2π n 2 mτ n=−∞ + ω2 } τ
+∞ 1 τ 2 e2π in(t−t )/τ . = mτ 2π n=−∞ n2 + ( ωτ )2 2π
(12.7)
Using the identity given in Eq. 11.30, namely +∞
einθ π cosh[(π − θ)a] = , 2 2 a +n a sinh πa n=−∞ yields Dt−t
! cosh ωτ − ωτ t − t  1 τ 2 2π 2 π = mτ 2π ωτ sinh( ωτ ) 2 ωτ 1 cosh 2 − ωτ t − t  = 2ωm sinh( ωτ ) 2
and reproduces the result obtained earlier in Eq. 11.8 using the oscillator algebra.
12.3 Oscillator normalization The normalization of the simple harmonic oscillator is given by the path integral mω S[z] = , Dze 2π sinh ωτ boundary conditions : z(0) = 0 = z(τ ). As can be read from the boundary conditions, the path integral corresponds to the degree of freedom starting and ending at the same point after time τ .
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A Fourier expansion for z(t) yields ! ! ∞ ∞ nπt z(t) nπ nπt zn , zn . sin z(t) = = cos τ dt τ τ n=1 n=1 Note that all the Fourier coefficients zn∗ = zn are real since z(t) is real. Using the orthogonality relation, ! ! τ ! ! τ mπt mπt τ nπt nπt sin = cos = δn−m dt sin dt cos τ τ τ τ 2 0 0 yields the simplification for the action ' ∞ τ m nπ 2 2 S[z] = − + ω zn2 . 2 2 n=1 τ The " action has completely factorized into a sum over the Fourier modes and the DZ path integral factorizes into infinitely many discrete integrations, one for each zn . We obtain (N is a normalization constant) ∞ +∞ & τ 2 Dz eS[z] = N dzn e− 2 λn zn n=1
=N
∞ &
−∞
7
n=1
=N
∞ &
τ
1
n=1
Since
∞ &
1+
1+
n=1
the path integral yields
# $1/2 ∞ & 2π 1 =N nπ 2 τ λn + ω2 n=1 (ωτ )2 n2 π 2
1/2 .
θ2 sinh θ = , n2 π 2 θ 
Dz e
S[z]
=N
ωτ . sinh ωτ
Recall from Eq. 11.16, for the simple harmonic oscillator, ωτ ˜ K(x , x; τ ) = AN e Sc . sinh ωτ For ω → 0, the simple harmonic oscillator reduces to the free particle. The m (x − x )2 . The evolution kernel classical action is given by limω→0 Sc (ω) → − 2τ for a free particle, from Eq. 11.36, is given by
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lim K(x, x ; τ ) =
ω→0
Hence
AN = yielding the result
K(x , x; τ ) =

m − m (x−x )2 . e 2τ 2πτ m , 2πτ
mω eSc. 2π sinh ωτ
12.4 Evolution kernel for indeterminate final position The simple harmonic oscillator’s action is given by m τ 2 (x˙ + ω2 x 2 )dt. S=− 2 0 The transition amplitude of the degree of freedom from the initial to the final position is denoted by K(x, x ; τ ). The transition amplitude K(x; τ ) from x to all possible positions at the final time is +∞ −τ H x = dx K(x, x ; τ ). K(x; τ ) = dx xe −∞
There are three ways of obtaining K(x; τ ): • One can first evaluate the full evolution kernel K(x, x ; τ ) and then integrate over x . • Since the boundary condition [dx(t)/dt]t=τ = 0 is equivalent to integrating over all possible values of the final position xf , one finds the classical solution for x(t) with boundary conditions given by dx(t) %% = 0. (12.8) x(0) = x, % dt t=τ The classical solution provides the classical action that contains the dependence on the final position x. • Another method for evaluating K(xi ; τ ) is by directly expanding the paths using a Fourier expansion that respects the boundary conditions given in Eq. 12.8, namely ! ∞ 2n + 1 πt . x(t) = x + xn sin 2 τ n=0
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Evolution kernel of harmonic oscillator The simple harmonic oscillator is given by Eq. 11.21. This yields 3 2 mω mω 2 2 K(x, x ; τ ) = exp − x + x cosh ωτ − 2xx . 2π sinh ωτ 2 sinh ωτ Hence, the kernel for reaching an arbitrary point after time τ is given by K(x; τ ) = dx K(x, x ; τ ) ' 2π sinh ωτ mω cosh ωτ 2 x = N exp − 2 sinh ωτ mω cosh ωτ ' mω 2 1 1 1 sinh ωτ x eF , × exp = 2 mω cosh ωτ sinh ωτ cosh ωτ where
mωx 2 sinh2 ωτ 1 mω 2 =− F =− x cosh ωτ − 2 sinh ωτ cosh ωτ 2 sinh ωτ cosh ωτ mω 2 =− x tanh ωτ. 2 Classical solution
The evolution kernel is determined by using the classical solution with appropriate boundary conditions, δS = 0 = x¨ − ω2 x, xc (t) = Ae−ωt + Beωt . δx(t) We impose the following boundary conditions on the classical solution: xc (0) = x;
dxc (τ ) = 0, dt
which yields x = A + B,
dx(τ ) = 0 = −ω Ae−ωτ − Beωτ dt
and hence A = Be2ωτ =
x e2ωτ . 1 + e2ωτ
The classical solution is given by xc (t) =
2ωτ −ωt x e e + eωt , 2ωτ 1+e
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with the classical action m [xc (τ )x˙c (τ ) − xc (0)x˙c (0)] 2 mx xω 2ωτ = −e + 1 2ωτ 2 1+e mω 2 mωx 2 sinh ωτ =− x tanh ωτ. =− 2 cosh ωτ 2
Sc = −
The evolution kernel is hence K(x; τ ) = N eSc . Finding the classical solution does not allow for determination of the normalization N , since one cannot use the composition law to generate the equation for N . Fourier expansion The boundary conditions x(0) = x;
dx(τ ) =0 dt
are realized by the Fourier expansion x(t) = x +
∞ n=0
xn sin
2n + 1 2
!
πt . τ
All possible paths obeying the given boundary condition are obtained by varying the real coefficients xn . The orthogonality equations tf (t − ti ) (t − ti ) cos mπ dt cos nπ τ τ ti tf (t − ti ) τ (t − ti ) sin mπ = δm−n , m, n ≥ 1 (12.9) = dt sin nπ τ τ 2 ti yield the action ∞ m τ (2n + 1)2 π 2 2 + ω xn2 S=− 2 2 n=0 4τ 2 $ # ! ∞ πt mω2 τ 2n + 1 − dt x 2 + 2x sin[ ]xn . 2 0 2 τ n=0
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12.4 Evolution kernel for indeterminate final position
Note
τ
2n + 1 2
dt sin 0
Thus
!
πt τ
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259
! ' 2τ 2n + 1 =− cos π −1 (2n + 1)π 2 2τ = . (2n + 1)π
. ∞ m π2 S=− (2n + 1)2 + 2 8τ n=0
2ωτ π
!2 / xn2 −
mτ ω2 2 x 2
∞ 1 2τ mxω2 − xn . π (2n + 1) n=0
Performing the path integral yields K(x; τ ) = const
.
& n
2π (2n + 1)2 + ( 2ωτ )2 n
/1/2 eF = N eF .
Using the numerical identity ∞ n=0
π4 tanh ωτ = 1− 2ωτ 2 6 ω2 τ 2 4 2 2 ωτ (2n + 1) + ( π ) (2n + 1) 1
!
yields !2
∞
1 8τ × × 2 mπ (2n + 1)2 n=0 ! mτ ω2 2 32mτ 3 ω4 2 π 4 tanh ωτ x =− x + 1− 2 π4 64ω2 τ 2 ωτ ! 2 2 mτ ω 2 mω τ tanh ωτ =− 1− x2 x + 2 2 ωτ mωx 2 =− tanh ωτ, 2
mτ ω2 2 F =− x + 2
2τ mxω2 π
.
1
/
(2n + 1)2 + ( 2ωτ )2 π
and thence the expected solution K(x; τ ) = N eSc . A careful treatment of the normalization of the path integral, as discussed in detail in Section 12.3, can be used to evaluate N .
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12.5 Free degree of freedom: constant external source Consider a free degree of freedom with action with an external current j given by ! τ τ 1 dx 2 S=− m dt +j dtx 2 dt 0 0 and the path integral
K(xf , xi ; τ ) =
DxeS ,
boundary conditions : x(0) = xi ; x(τ ) = xf . The general solution for an arbitrary time dependent j (t) given in Eq. 11.29 can be used to perform the path integral. A different approach is to employ a change of variables to simplify the source free quadratic action and then evaluate the path integral. Consider change of variables, t dx(t) dt ξ(t ), = ξ(t) ⇒ x(t) = xi + dt 0 τ x(τ ) = xi + dt ξ(t ). 0
The requirement that x(τ ) = xf needs to be put in as a constraint in the path integral. Hence τ K = Dξ δ(xi + dtξ(t) − xf )eS 0 "τ dη Dξ eiη(xi −xf ) eiη 0 dtξ(t) eS = 2π dη ˜ Dξ eiη(xi −xf ) eS . = 2π The action is given by τ τ t τ 1 2 S˜ = iη dtξ(t) − m dtξ − j dt dt ξ(t ) − j xi τ 2 0 0 0 τ τ0 1 2 =− m dtξ + dt[iη − j (τ − t)]ξ − j xi τ. 2 0 0 Performing the ξ path integral yields1 dη iη(xi −xf ) F˜ −j xi τ K= e , e 2π
(12.10)
1 Using the results given in Section 11.8 for the free particle path integral one can obtain the normalization
given in Eq. 12.10.
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where 1 F˜ = 2m
τ
dt[iη − j (τ − t)]2 = −
0
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τ 2 i 1 2 3 η − j ητ 2 + j τ . 2m 2m 6m
Doing the ηintegration yields K(xf , xi ; τ ) = N eF , where2 m τ2 2 1 2 3 (xi − xf − j) + j τ − j τ xi 2τ 2m 6m m mτ 1 2 3 = − (xi − xf )2 − (xi + xf )j + j τ , 2 24m  2τ m N = . 2πτ F =−
Hence, the evolution kernel is given by

K(xf , xi , τ ) =
m F e . 2πτ
The normalization is the same as the case of j = 0. 12.6 Evolution kernel for indeterminate positions A particle with paths that have indeterminate initial and final positions can be studied using a Fourier expansion of its possible paths. Let the particle have mass m and spring constant ω, and be subject to an external force j ; the particle’s Lagrangian and action, from initial time and position ti , xi to final time and position tf , xf , are given by ! tf 1 dx 2 1 dtL, L = − m − mω2 x 2 + j x. (12.11) S= 2 dt 2 ti The transition amplitude is given by K(x; ti ; xf , tf ) = xf e
−(tf −ti )H
xi =
DxeS .
For the case when the initial and final positions xi , xf are indeterminate the transition amplitude is given by +∞ K(ti , tf ; j ) = dxi dxf K(xi , ti ; xf , tf ). (12.12) −∞
2 There are some typographical errors in the result given for this computation in Feynman and Hibbs (1965).
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The integration over the initial and final positions that results in Z(ti , tf ; j ) has a simple expression " in terms of the boundary conditions imposed on the path integration measure DX. Instead of the initial and final positions being fixed, the paths x(t) now have dx(tf ) dx(ti ) =0= : Neumann BCs. (12.13) dt dt The Neumann boundary conditions (BCs) allow one to do an integration by parts of the action given in Eq.(12.11), yielding the action tf tf d2 1 2 dtx(t) − 2 + ω x(t) + dtj (t)x(t). (12.14) S=− m 2 dt ti ti The generating functional is given by the path integral K(ti , tf ; j ) = DxeS .
(12.15)
" The path integral DxeS is performed over all paths (functions) x(t) that satisfy the Neumann boundary conditions given in Eq. (12.13). All such functions can be expanded in a Fourier cosine series as follows: ∞ (t − ti ) , τ ≡ tf − ti , x(t) = a0 + an cos nπ τ n=1 ∞ +∞ & dan : infinite multiple integral, DX = N n=0
−∞
(N is a normalization constant). The orthogonality equations tf (t − ti ) (t − ti ) cos mπ dt cos nπ τ τ ti tf (t − ti ) τ (t − ti ) = sin mπ = δm−n , m, n ≥ 1 dt sin nπ τ τ 2 ti
(12.16)
yield, for the action given in Eq. 12.14, 6 D ∞ 1 1 nπ S = − mω2 τ a02 + 1 + ( )2 an2 2 2 n=1 ωτ 6 D tf ∞ (t − ti ) + dtj (t) a0 + an cos nπ τ ti i=1 ∞ ∞ 1 2 =− κn an + jn an , 2 n=0 n=0
(12.17)
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where
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nπ 2 1 2 κ0 = mω τ, κn = mω τ 1 + , n ≥ 1, 2 ωτ tf (t − ti ) , n = 0, 1, . . . ∞. jn = dtj (t) cos nπ τ ti 2
All the Gaussian integrations over the variables an have decoupled in the action S given in Eq. 12.17. The path integral has been reduced to an infinite product of single Gaussian integrations, each of which can be performed using Eq. 7.7. Hence, from Eqs. 12.15 and 12.17 ∞ +∞ & 1 2 S dan e− 2 κn an +jn an K(ti , tf ; j ) = DXe = N 6 = exp
n=0 ∞
D
−∞
1 1 jn jn , 2 n=0 κn
(12.18)
yielding 1
K(ti , tf ; j ) = e 2
" tf ti
dtdt j (t)D(t,t ;ti ,tf )j (t )
,
(12.19)
where the function D(t, t ; ti , tf ) is the propagator for the simple harmonic oscillator. Using Eq. 12.19 to factor out the j (t)s from Eq. 12.18 yields D(t, t ; ti , tf ) 6 D ∞ 1 1 (t − ti ) (t − ti ) = 1+2 . cos nπ nπ 2 cos nπ mω2 τ τ 1 + ( ) τ ωτ n=1 (12.20) Let θ = t − ti > 0 and θ = t − ti > 0; then 2
∞ cos(nπθ/τ ) cos(nπθ /τ ) n=1
1 + ( nπ )2 ωτ
∞ ωτ 2 cos(nπ(θ + θ )/τ ) + cos(nπ(θ − θ )/τ ) = . π ( ωτ )2 + n 2 π n=1
(12.21)
The summation over integer n is performed using the identity3 ∞ cos(nθ) n=1
a2
+
n2
=
1 π cosh(π − θ )a − 2 2a sinh πa 2a
(12.22)
3 The formula given in Eq. 11.30 is valid for any complex number a, and will be applied in later discussions for a case where a is indeed a complex number. The branch of the square root of a 2 that is taken on the right
hand side need not be specified since the rhs is a function of a 2 .
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and yields the result
cosh ω τ − θ − θ  + cosh ω τ − (θ + θ ) D(t, t ; ti , tf ) = . (12.23) 2mω sinh ωτ Hence, from Eq. 12.23, and since τ = tf − ti , the propagator is given by
D(t, t ; ti , tf ) cosh ω (tf − ti ) − t − t  + cosh ω (tf − ti ) − (t + t − 2ti ) = . (12.24) 2mω sinh ω(tf − ti ) Note that the propagator (also called a Green’s function) satisfies the differential equation d2 2 m − 2 + ω D t, t ; ti , tf = δ t − t : Neumann BCs. dt The case of infinite time for Eq. 12.24 is obtained by taking the limit of ti → −∞ , tf → +∞, and yields 1 −ωt−t  , e 2mω which has been derived earlier using different methods. D(t, t ) =
(12.25)
12.7 Simple harmonic oscillator: Fourier expansion Given the importance of the simple harmonic oscillator’s evolution kernel, a derivation is given below using the technique of Fourier expansions. Consider a quantum particle with an initial position xi and, after time τ , with a final position xf . The evolution kernel for this case has been obtained earlier in Section 11.6 based on evaluating the solution of the classical equation of motion. The limitation of finding the classical solution is that for many nonlinear Lagrangians, it is usually not possible to obtain the classical trajectory. The Fourier expansion avoids this problem by directly enumerating all the paths with a given initial and final position and hence can be used for a great variety of problems. Consider the Fourier expansion for the possible paths of the quantum particle, ∞ (2n + 1)πt xn. cos (12.26) x(t) = xf + 2τ n=0 The boundary conditions are x(0) = xf +
∞
xn ,
n=0
x(τ ) = xf .
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At time t = τ , the final condition is automatically satisfied. Note the velocity of the particle at t = 0 is zero, since ∞
dx(0) (2n + 1)π = sin[0]xn = 0. dt 2τ n=0 As discussed for the two previous cases, zero velocity at t = 0 means that the position x(0) is indeterminate. Hence, to fix the initial position of the particle to be xi , one needs to put a deltafunction constraint into the path integral, namely xi = xf +
∞
xn : constraint.
n=0
Hence, the evolution kernel is given by K(xi , xf ; τ ) = xf e−τ H xi ∞ +∞ ∞ & dxn δ( xn + xf − xi )eS =N =
n=0
−∞
(12.27)
n=0
+∞
Dx −∞
dη iη((∞ n=0 xn +xf −xi ) e S , e 2π
(12.28)
where N is a normalization constant. Using the orthogonality relation ! τ (2m + 1)πt (2n + 1)πt sin dt sin 2τ 2τ 0 τ (2m + 1)πt τ (2n + 1)πt cos = δn−m = dt cos 2τ 2τ 2 0 yields a simplification for the action, dx(t) 2 m τ 2 2 dt + ω x (t) S=− 2 0 dt . !2 / ∞ m π2 2ωτ m (2n + 1)2 + xn2 − ω2 τ xf2 =− 2 8τ n=0 π 2 + since
∞ 2mω2 τ (−1)n xf xn , (2n + 1)π n=0
τ 0
(2n + 1)πt dt cos 2τ
=−
2τ (−1)n . (2n + 1)π
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All the Fourier modes xn have completely factorized in the action. Performing the Gaussian path integration (over the Fourier modes xn ) given in Eq. 12.28 yields +∞ dη F (η) K(xi , xf ; τ ) = N e , −∞ 2π 2 ∞ 2mω2 τ 1 1 8τ n F (η) = (−1) xf . iη + 2m π 2 n=0 (2n + 1)2 + 2ωτ 2 (2n + 1)π π Using the numerical identities ∞ n=0 ∞ n=0 ∞ n=0
1 (2n + 1)2 +
2ωτ 2 = π
π2 tanh (ωτ ), 8ωτ
sinh2 ωτ (−1)n π3 2 · , = 2ωτ 2 2 3 8(ωτ ) cosh(ωτ ) (2n + 1) + π (2n + 1)
! tanh ωτ π4 , = 5 2 2 1− 2 2 2ω τ ωτ (2n + 1)4 + 2ωτ (2n + 1) π 1
yields the result sinh2 ( ωτ ) 1 2 tanh(ωτ ) tanh(ωτ ) 2 2 η − 2ixf η + xf mω2 τ 1 − F =− 2mω cosh(ωτ ) 2 ωτ 1 2 − xf mω2 τ − iη(xi − xf ) 2 xf 1 tanh(ωτ ) 2 − xf2 mω tanh(ωτ ). =− η − iη xi − 2mω cosh(ωτ ) 2
!
Performing the η integration gives the final result 7 2πmω G K(xi , xf ; τ ) = N e , tanh(ωτ ) with
2 xf mω 1 xi − − xf2 mω tanh(ωτ ) 2 tanh(ωτ ) cosh(ωτ ) 2 mω xi xf 2 2 =− x + xf − 2 + ξ. 2 tanh(ωτ ) i cosh(ωτ )
G=−
The remainder ξ is zero since 1 1 1 2 ξ = − mωxf + tanh(ωτ ) = 0. − 2 tanh(ωτ ) cosh2 (ωτ ) tanh(ωτ )
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Once G has been determined, the overall normalization of the evolution kernel can be fixed by using the composition law, as in Section 11.5. Hence, the evolution kernel is given by mω eG , K(xi , xf ; τ ) = 2π sinh(ωτ ) which agrees with the result obtained earlier in Eq. 11.21.
12.8 Evolution kernel for a magnetic field A free charged particle moving in an electromagnetic field couples with the vector potential which, in vector notation, is given by A. The Lagrangian, for x = (x, y, z), is given in Minkowski time by dx 1 L = m˙x2 + eA · x˙ , x˙ ≡ . 2 dt
(12.29)
The calculation in this section is carried out for Minkowski time to get a flavor for such calculations; in particular, it will be seen that the computations in Minkowski time have a plethora of i factors which are quite unnecessary in obtaining the result for Euclidean time. Consider a free charged particle interacting with a constant external magnetic field of strength B acting along the zdirection. The vector potential A is chosen to be in the symmetric gauge, 1 A = B(−y, x, 0). 2
(12.30)
The Lagrangian is then given by 2 eB 1 ˙ − x y) ˙ , σ = , L(t) = m x˙ 2 + y˙ 2 − mσ (xy 2 2m
(12.31)
where σ is the cyclotron frequency. The particle moves only in the xyplane and hence the zaxis can be completely dropped from the problem. The evolution kernel is given by the path integral K(xb , yb , tb ; xa , ya , ta ) = xb , yb , e−i(tb −ta )H xa , ya = DxDyeiS , (12.32) where H is the Hamiltonian of the system; the action S is given by tb dtL(t). S= ta
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The action is quadratic in the degrees of freedom and hence the evolution kernel can be obtained using the classical solution. The equations of motion for the Lagrangian in Eq. 12.29 yield the classical solution xc , yc given by x¨c = 2σ y˙c , y¨c − 2σ x˙c .
(12.33)
Shifting the degree of freedom x, y by xc , yc x → x + xc ,
y → y + yc
and keeping the same notation for the sake of simplicity yields the action S [x + xc , y + yc ; T ] = Sc [xc , yc ; T ] + S0 [x, y; T ] + S, T = tb − ta , (12.34) tb ˙ c + y x˙c − x y˙c − yx S = dt {m (x˙ x˙c + y˙ y˙c ) − mσ (xy ˙ c )} . ta
The shifted quantum degrees of freedom x, y have new boundary conditions given by x(ta ) = 0 = x(tb ), y(ta ) = 0 = y(tb )
(12.35)
and lead to the result S = 0. The decoupling of the classical solution – since S = 0 – from the quantum variables is true for any quadratic Lagrangian. From Eqs. 12.32, 12.34, and 12.35, the Minkowski time evolution kernel is given by iSc [xc ,yc ;T ] DxDyeiS0 [x,y;T ] K(xb , yb ; xa , ya ; T ) = e = N (T )eiSc [xc ,yc ;T ] .
(12.36)
Classical action The classical action is evaluated using the Lagrangian given in Eq. 12.29, ' tb 1 2 dt (12.37) m x˙c + y˙c2 − mσ (x˙c yc − xc y˙c ) . Sc [xc , yc ; T ] = 2 ta Integrating by parts, Eq. 12.37 becomes4 1 Sc [xc , yc ; T ] = m (x˙b xb + y˙b yb − x˙a xa − y˙a ya ) . 2 " 4 Since tb dt (x x¨ + y y¨ + 2σ x˙ y − 2σ x y˙ ) = 0. c c c c c c c c ta
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Furthermore, integrating Eq. 12.33 gives x˙c = 2σ (yc + C), y˙c = −2σ (xc − D),
(12.38)
where C and D are constants of the integration. Equation 12.38 is the equation for a twodimensional harmonic oscillator, x¨c = −ω2 (xc − D), y¨c = −ω2 (yc + C), ω = 2σ.
(12.39)
The solutions of Eq. 12.39 are given by xc = A cos ωt + B sin ωt + D, yc = −A sin ωt + B cos ωt − C. The constants C and D are derived from the boundary conditions, xa = A cos ωta + B sin ωta + D, ya = −A sin ωta + B cos ωta − C, xb = A cos ωtb + B sin ωtb + D, yb = −A sin ωtb + B cos ωtb − C.
(12.40)
Solving Eq. 12.40 yields (xb − xa ) cos σ T − (yb − ya ) sin σ T , 2 sin σ T (xb − xa ) sin σ T − (yb − ya ) cos σ T , A=− 2 sin σ T (yb − ya ) cos σ T 1 D = (xb + xa ) + , 2 2 sin σ T (xb − xa ) cos σ T 1 C = (yb + ya ) + , 2 2 sin σ T where T = tb − ta , T = ta + tb . Furthermore, Eq. 12.38 yields B=
(12.41)
xc x˙c + yc y˙c = 2σ (Cxc + Dyc ). Equation 12.37, together with Eqs. 12.38 and 12.41, yields Sc [xc , yc ; T ] = mσ {C(xb − xa ) − D(yb − ya )}, and we obtain the final result cos σ T {(x − x )2 + (y − y )2 } b a b a Sc [xc , yc ; T ] = mσ + yb xa − xb ya . (12.42) 2 sin σ T The normalization of the evolution kernel N (T ) is given by the consistency equation as discussed in Section 4.8, K(xb , yb ; xa , ya ; 2T ) = dξ dηK(xb , yb ; ξ, η; T )K(ξ, η; xa , ya ; T ), (12.43) ⇒ N (2T ) = N 2 (T )(T ).
(12.44)
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The evolution kernel is given by Eq. 12.36 and 12.42; performing the Gaussian integration given on the right hand side of Eq. 12.43 yields, from Eq. 12.44, the following recursion equation: N (2T ) = N 2 (T )
iπ sin(σ T ) mσ ⇒ N (T ) = . mσ cos(σ T ) 2πi sin σ T
(12.45)
The final result for the Minkowski time path integral, from Eqs. 12.36, 12.42, and 12.45, is thus mσ K(xb , yb ; xa , ya ; T ) = N (T )eiSc [xc ,yc ;T ] = 2πi sin σ T i mσ cos σ T 2 3 2 × exp (xb − xa ) + (yb − ya )2 + imσ yb xa − xb ya . 2 sin σ T (12.46) The evolution kernel for Euclidean time τ > 0 is given by τ = iT and mσ KE (xb , yb ; xa , ya ; τ ) = 2π sinh σ τ 1 mσ cosh σ τ 2 3 × exp − (xb − xa )2 + (yb − ya )2 + imσ yb xa − xb ya . 2 sinh σ τ 12.9 Summary The techniques of using classical solutions and Fourier expansions for analyzing path integrals were illustrated using Gaussian path integration. The exact solutions obtained show the flexibility and power of these techniques, which have generalizations to many nonlinear systems. The evolution kernel for various quadratic Lagrangians, in particular of the simple harmonic oscillator, reveal the rich mathematical structure of these relatively simple theories. The evolution kernel of a charged particle in a magnetic field, evaluated for both Minkowski and Euclidean time, shows the equivalence of these approaches as well as illustrating the relative simplicity of the results in Euclidean time.
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Part five Action with acceleration
Introduction to part five The most widely used actions in quantum mechanics have a kinetic term that is the velocity squared of the degree of freedom and a potential term that depends on the degree of freedom.1 The kinetic term fixes the equal time commutation equation of the degree of freedom, as shown in Section 6.7. Furthermore, the velocity term in the Lagrangian entails that all the possible indeterminate paths obey two boundary conditions, which in turn yields a state space that depends on the degree of freedom only. In Section 11.13, this property of the indeterminate paths was used for deriving the state function of the harmonic oscillator. The action with acceleration has a kinetic term that is given by the acceleration squared of the degree of freedom, in addition to the usual velocity and potential terms. It is an example of higher derivative Lagrangians, discussed in Simon (1990). The higher derivative quantum systems have many remarkable properties not present for the usual cases studied so far. The action with acceleration arises in many diverse fields and has been widely studied; it describes the behaviour of “stiff” polymers, of cell walls, of the formation of microemulsions, the properties of chromoelectric flux lines in quantum chromodynamics, as well as the Big Bang singularity in cosmology. The Euclidean action and path integral were studied by Kleinert (1986), where a list of the applications of the model is given. Bender and Mannheim (2008b) have extensively studied the Lagrangian and Hamiltonian of the model, both in Euclidean and Minkowski time; Mannheim and Davidson (2005) showed that the model in Minkowski time was free of ghost states using the concept of a PTsymmetric Hamiltonian; in Bender and Mannheim (2010) they demonstrated the reality of the model’s eigenspectrum. The acceleration action has been applied to 1 Velocity dependent terms can be included in the potential.
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the study of cosmology, and quantum and conformal gravity by Mannheim (2011a) and Mannheim (2011b). The Euclidean model was studied by Hawking and Hertog (2002) for its role in quantum gravity and in string theory’s Dbrane dynamics; the Euclidean path integral was used by Fontanini and Trodden (2011) for analyzing the ghost states for Minkowski time. The acceleration action appears in the study of mathematical finance in describing the time dependence of financial derivative instruments and in the study of interest rates and equity [Baaquie (2010)]. The Euclidean path integral has been studied extensively by Baaquie et al. (2012) and Yang (2012) for describing the correlation function of equities. The model has been applied in the study of options by Baaquie and Yang (2013) and to the study of microeconomics by Baaquie (2013b). The acceleration Lagrangian yields a pseudoHermitian Hamiltonian that has been studied in Mostafazadeh (2002) and Wang et al. (2010). The analysis of the Euclidean Hamiltonian in this book is a continuation of the study carried out of the Minkowski time Hamiltonian by Bender and Mannheim (2008a), (2008b). The Euclidean case has some new features that are absent for Minkowski time, the most important being a transparent positivity of the Euclidean state space. The path integral for the pseudoHermitian Hamiltonian has been studied, starting from the propagator, by Jones and Rivers (2009) and Rivers (2011). For a special value of the parameters, the acceleration Hamiltonian becomes essentially nonHermitian. As shown by Mannheim and Davidson (2000), (2005), and elaborated further by Mannheim (2011b), at the special value of the parameters the Hamiltonian is described by a Jordan block matrix; the Jordan block Hamiltonian was shown to be pseudoHermitian by Bender and Mannheim (2011). The following three chapters analyze the acceleration Lagrangian and Hamiltonian and are largely based on papers by Baaquie (2013c), (2013d). The quantum mechanical system for the acceleration Lagrangian is defined in Euclidean time and using the path integral. Many of the unnecessary complications that appear in Minkowski time formulation – and which obscure key features of the system – are absent in the Euclidean time formulation. Furthermore, applications in biophysics and finance are directly based on the Euclidean formulation and consequently this system is interesting in its own right, regardless of its connection with the theory in Minkowski time.
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13 Acceleration Lagrangian
The Lagrangian and path integral for the action with acceleration are defined in Section 13.1 and the quadratic potential is chosen in Section 13.2 so that the path integral can be performed exactly using the classical solution. The propagator is derived from the infinite time action in Section 13.3. The Hamiltonian is obtained for the Euclidean time Lagrangian using the technique of Dirac brackets in Section 13.4, and it is shown in Section 13.5 that the Hamiltonian yields the expected path integral. The evolution kernel is carefully studied in Section 13.6 to prove that the Lagrangian based path integral is equivalent to the one derived from the quantum Hamiltonian. It is shown in Section 13.7 how the boundary conditions on the indeterminate paths (integration variables) of the path integral are transformed in going from the Hamiltonian to the action based formulation of the path integral. In Section 14.14, the acceleration Hamiltonian is extended to many degrees of freedom.
13.1 Lagrangian Euclidean time is defined by the analytic continuation of Minkowski time tM = −it, where t is Euclidean time. The degree of freedom x does not change in going from Minkowski to Euclidean time but Minkowski velocity vM picks up an extra factor of i since it is related to Euclidean velocity v, due to Eq. 5.17, by v=−
dx dx = ivM . =i dτ dtM
(13.1)
The minus in the definition of Euclidean velocity, v = ivM , is consistent with t = itM . The nonHermitian Euclidean Hamiltonian and path integral are well behaved and a complexification of the degree of freedom is required.
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The Euclidean time Lagrangian with acceleration is given by ! !2 γ d 2x α dx 2 L=− − − (x), 2 dt 2 2 dt with the acceleration action for finite Euclidean time τ given by τ S[x] = dtL.
(13.2)
(13.3)
0
The Feynman path integral for finite Euclidean time is given by % S [x] % K(xf , x˙f ; xi , x˙i ) = DX e % ,
DX = N˜
τ &
(xi ,x˙i ;xf ,x˙f )
(13.4)
+∞
(13.5)
dx(t), −∞
t=0
where N˜ is a normalization constant. The paths have four boundary conditions, dx(0) = x˙i initial position and velocity, (13.6) dt dx(τ ) (13.7) x(τ ) = xf ; = x˙f final position and velocity. dt To evaluate the path integral exactly using the classical solution, consider the quadratic potential x(0) = xi ;
(x) =
β 2 x , 2
(13.8)
which yields the Lagrangian γ L=− 2
d 2x dt 2
!2
α − 2
dx dt
!2 −
β 2 x . 2
(13.9)
The path integral given Eq. 13.4 is now quadratic and can be evaluated exactly using the classical solution. Let xc (t) be the classical solution given by δS[xc ] =0 δx(t)
(13.10)
that satisfies the boundary conditions, dxc (0) = x˙i initial position and velocity, dt dxc (τ ) xc (τ ) = xf ; = x˙f final position and velocity. dt Consider the change of integration variables, from x(t) to ξ(t), xc (0) = xi ;
x(t) = xc (t) + ξ(t),
(13.11)
(13.12)
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with boundary conditions given by dξ(0) = 0 initial position and velocity, dt dξ(τ ) ξ(τ ) = 0; = 0 final position and velocity. dt The change of variables yields ξ(0) = 0;
S[x] = S[xc + ξ ] = S[xc ] + S[ξ ], K(xf , x˙f ; xi , x˙i ) = DX eS = N (τ )eS [xc ] , N (τ ) = Dξ eS [ξ ] .
(13.13)
(13.14)
The evolution kernel given in Eq. 13.14 has been evaluated explicitly in Hawking and Hertog (2002) by solving for the classical solution xc (t) and then obtaining S[xc ] and N (τ ). As can be directly verified from the classical action S[xc ], the classical solution xc (t) yields another equally valid classical solution x˜c (t) given by x˜c (t) = xc (τ − t),
(13.15)
with boundary conditions, from Eq. 13.11, given by d x˜c (0) dxc (τ ) =− = −x˙f , dt dt d x˜c (τ ) dxc (0) =− = −x˙i , x˜c (τ ) = xc (0) = xi ; dt dt ⇒ S[xc ] = S[x˜c ],
x˜c (0) = xc (τ ) = xf ;
⇒
Sc [xf , x˙f ; xi , x˙i ] = S[xi , −x˙i ; xf , −x˙f ].
(13.16) (13.17)
(13.18)
The evolution kernel, from Eqs. 13.14 and 13.18, hence has the symmetry K(xf , x˙f ; xi , x˙i ) = K(xi , −x˙i ; xf , −x˙f ).
(13.19)
13.2 Quadratic potential: the classical solution The acceleration Lagrangian given by Eq. 13.9 is rewritten with a slight change of notation, more suitable for the classical solution, in the following manner: τ 1 ˙ 2 + cx 2 , S = dtL. (13.20) L = − a x¨ 2 + 2b(x) 2 0 The parameterization chosen in Eq. 13.20 is more suitable for studying the classical solutions than the one given in Eq. 13.9
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The Euler–Lagrangian equation in Eq. 13.10 yields the equation of motion; the classical solution xc (t) satisfies the equation of motion .... a x c (t) − 2bx¨c (t) + cxc (t) = 0. (13.21) Let us choose the boundary conditions, Initial values : x(0) = xf = x1 ,
x(0) ˙ = x˙f = x2 = −vf ,
Final values : x(τ ) = xi = x4 ,
x(τ ˙ ) = x˙i = x3 = −vi .
We define the parameters r and ω by 7 √ b + i ac − b2 . r + iω ≡ a
(13.22)
(13.23)
From Eq. 13.21 the classical solution of equations of motion is given by xc (t) = ert (a1 sin ωt + a2 cos ωt) + e−rt (a3 sin ωt + a4 cos ωt).
(13.24)
The parameters a1 , . . . , a4 are obtained from the boundary conditions and are given by Yang (2012) as a1 = r 2 xf e2rτ sin(2τ ω) + ωvf e2rτ − rvf e2rτ sin(2τ ω) + rωxf e2rτ cos(2τ ω) − rωxf − ωvf − 2r 2 xi erτ sin(τ ω) − 2rvi erτ sin(τ ω) − ωerτ e2rτ − 1 cos(τ ω) (vi + rxi ) − ω2 xi erτ sin(τ ω) + ω2 xi e3rτ sin(τ ω) , a2 = r 2 xf −e2rτ + rvf e2rτ + re2rτ cos(2τ ω) rxf − vf − ω2 xf e2rτ − rωxf e2rτ sin(2τ ω) + ω2 xf − ωvi erτ sin(τ ω) + ωvi e3rτ sin(τ ω) + ω2 xi erτ e2rτ − 1 cos(τ ω) + rωxi erτ sin(τ ω) + rωxi e3rτ sin(τ ω) , a3 = erτ r 2 xf erτ sin(2τ ω) + ωvf erτ − ωvf e3rτ + rvf erτ sin(2τ ω) + rωxf e3rτ − rωxf erτ cos(2τ ω) − 2r 2 xi e2rτ sin(τ ω) − 2rvi e2rτ sin(τ ω) − ω e2rτ − 1 cos(τ ω) (rxi − vi ) − ω2 xi e2rτ sin(τ ω) + ω2 xi sin(τ ω) , a4 = erτ r 2 xf −erτ − rvf erτ + rerτ cos(2τ ω) rxf + vf − ω2 xf erτ + ω2 xf e3rτ + rωxf erτ sin(2τ ω) − ωvi e2rτ sin(τ ω) − ω2 xi e2rτ − 1 cos(τ ω) − rωxi e2rτ sin(τ ω) − rωxi sin(τ ω) + ωvi sin(τ ω) .
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The term is defined by =
ω2
+
ω2 e4rτ
+
2r 2 e2rτ
1 . cos(2τ ω) − 2e2rτ r 2 + ω2
(13.25)
The boundary condition given in Eq.13.22 yields the classical action 4 1 xI MI J xJ . Sc = Sc (xf , vf , xi , vi ) = − 2 I,J =1
(13.26)
From Eq. 13.18, the classical action has the symmetry Sc [xf , vf ; xi , vi ] = S[xi , −vi ; xf , −vf ]
(13.27)
and hence the matrices MI J given in Eq. 13.26 satisfy symmetry, M11 = M44 ,
M22 = M33 ,
M12 = −M34 ,
M13 = −M24 .
The action can be simplified to 1 1 Sc (xf , vf , vi , xi ) = − M11 (xi2 + xf2 ) − M22 (v2i + v2f ) + M12 (xi vi − xf vf ) 2 2 (13.28) + M13 (xi vf − xf vi ) − M14 xi xf − M23 vi vf . The solutions for MI J are given by Yang (2012) and listed below: M11 = 2arω r 2 + ω2 ω e4rτ − 1 + 2re2rτ sin(2τ ω) , M12 = ω2 e4rτ 2ar 2 − b − 2r 2 e2rτ 2aω2 + b cos(2τ ω) − ω2 b − 2ar 2 + 2be2rτ r 2 + ω2 , M13 = 4arωerτ e2rτ − 1 r 2 + ω2 sin(τ ω) , M14 = − 4arωerτ r 2 + ω2 r e2rτ + 1 sin(τ ω) + ω e2rτ − 1 cos(τ ω) , M22 = − 2arω ω −e4rτ + 2re2rτ sin(2τ ω) + ω , M23 = 4arωerτ r e2rτ + 1 sin(τ ω) − ω e2rτ − 1 cos(τ ω) . 13.3 Propagator: path integral The quadratic potential given in Eq. 13.8 yields the Lagrangian given in Eq. 13.9, namely 1 L = − γ x¨ 2 + α x˙ 2 + βx 2 . 2
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A parameterization that is more suitable for studying the Hamiltonian and state space is given by Kleinert (1986), γ (13.29) L = − x¨ 2 + (ω12 + ω22 )x˙ 2 + ω12 ω22 x 2 . 2 The Lagrangian is completely symmetric in parameters ω1 and ω2 . For the case of real ω1 and ω2 , the entire parameter space is covered by choosing say ω1 > ω2 ; the roots are chosen accordingly and are ! 8 8 9 9 1 ω1 = √ α + 2 γβ + α − 2 γβ , 2 γ ! 8 8 9 9 1 ω2 = √ α + 2 γβ − α − 2 γβ , (13.30) 2 γ ω1 > ω2 for ω1 , ω2 real. Note that ω1 > ω2 real ω1 and ω2 . √ We define the critical value αc = 2 βγ . The parameters have three branches, namely the real and the complex, plus the critical branch separating them, and these are shown in Figure 13.1: • Complex branch α < αc . Frequencies ω1 , ω2 are complex, ω1 = ω2∗ = Reiφ : R > 0, φ ∈ [−π/2, π/2].
(13.31)
• Real branch α > αc .
Complex Branch
a=0
Real Branch
w1 = Reif
w1 = Reb
w2 = Re−if = w1*
w2 = Re−b
2 βγ
α
Figure 13.1 √ Parameter branches for the Euclidean Lagrangian. The critical value of αc = 2 βγ is equivalent to ω1 = ω2 .
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Frequencies ω1 , ω2 are real and ω1 > ω2 is chosen without any loss of generality, ω1 = Reb , ω2 = Re−b : R > 0, b ∈ [0, +∞]. • Equal frequency α = αc . This is a special case that is treated in detail in Chapter 15, ω1 = ω2 , b = 0 = φ. Note that φ ∈ [−π/2, π/2] for all α, β > 0, and this is also the range for which the path integral is convergent. The infinite time path integral is given by −T H = DxeS , Z = lim tr e T →∞ +∞
1 S=− γ dt x¨ 2 + (ω12 + ω22 )x˙ 2 + ω12 ω22 x 2 . 2 −∞ The propagator is given by the path integral 1 DxeS x(t)x(t ). G(τ ) = Z
(13.32)
The acceleration action is a quadratic functional of the paths x(t) and hence the propagator can be evaluated exactly. We define the Fourier transformed variables that diagonalize the action, namely +∞ dk ikx (13.33) x(t) = e xk , −∞ 2π +∞ dk 4 1 (13.34) k + (ω12 + ω22 )k 2 + ω12 ω22 x−k xk . ⇒S=− γ 2 −∞ 2π Using Gaussian path integration yields 1 +∞ dk eik(t−t ) G(τ ) = γ −∞ 2π (k 2 + ω12 )(k 2 + ω22 ) −ω2 τ 1 1 e e−ω1 τ = , τ = t − t , − 2γ ω12 − ω22 ω2 ω1
(13.35)
where the last equation has been obtained using counter integration. Figure 13.2a shows a single exponential and Figure 13.2b shows how the acceleration term in the Lagrangian smooths out the “kink” of the exponential at x = 0; this smoothing out holds for all branches, including the real and complex branches for ω1 , ω2 .
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(a)
(b)
Figure 13.2 (a) Single exponential. (b) Propagator for equal frequency exp{−ωτ }[1 + ωτ ].
13.4 Dirac constraints and acceleration Hamiltonian The acceleration Lagrangian belongs to the class of higher derivative Lagrangians for which the canonical derivation of the Hamiltonian from the Lagrangian does not hold. The reason is that the term (d 2 x/dt 2 )2 in the Lagrangian does not yield a pair consisting of a canonical coordinate and momenta, as is required by the canonical framework. This problem of higher derivative Lagrangians was addressed almost two centuries ago by Ostogradski and discussed by Simon (1990); but instead of following his approach, a more direct route is taken here, based on Dirac’s analysis of constrained systems. In this approach, the number of independent degrees of freedom is increased by imposing constraints that recast the acceleration Lagrangian into another equivalent form in which the term (dx 2 /dt 2 )2 is rewritten as a first derivative expression, namely as (dv/dt)2 , with a constraint that v = −dx/dt; the Lagrangian can now be analyzed using the canonical method. Dirac’s analysis shows how to impose a set of consistent constraints on the enlarged phase space on which the Hamiltonian is defined so that only the physical phase space is obtained for the higher derivative system. The Minkowski Hamiltonian for the action acceleration has been obtained by Mannheim and Davidson (2000) using the Dirac bracket approach and this analysis is now carried out for Euclidean space. The canonical transformation that takes the Lagrangian to the Hamiltonian makes no explicit reference to time and hence one needs to write the constraint equation for the Euclidean Lagrangian to obtain the correct Euclidean Hamiltonian.
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The Euclidean Lagrangian is given by Eq. 13.29 as follows (· = d/dt): L=−
γ 2 x¨ + (ω12 + ω22 )x˙ 2 + ω12 ω22 x 2 . 2
(13.36)
The first step is to rewrite the path integral; from Eqs. 5.17 and 13.1, Euclidean velocity v is made equal to −dx/dt by imposing a delta function constraint in the path integral; ignoring the boundary condition for now, one obtains ' '& δ v(t) + x(t) ˙ , Z = Dx exp dtL = DxDv exp dtL t
=
DxDvDλ exp
' dtLD ,
(13.37) (13.38)
with LD = −
γ 2 v˙ + (ω12 + ω22 )v2 + ω12 ω22 x 2 + iλ[x˙ + v]. 2
(13.39)
The equivalent Euclidean Lagrangian LD has only first order derivatives and hence can be treated by the canonical method for obtaining the Hamiltonian from the Lagrangian, as discussed in Section 5.1. Compared to the original Lagrangian L, which has one degree of freedom x(t), the equivalent Lagrangian LD has three degrees of freedom x(t), v(t), and λ(t). Dirac’s formalism of constraints removes the redundant degree(s) of freedom. For simplicity, the notation for Euclidean momenta px , E and pv , E has been abbreviated to px , pv . Let us define the canonical momenta for Euclidean time as given in Eq. 5.21, which yields px =
∂LD ∂LD ∂LD = 0. = iλ, pv = = −γ v˙ , pλ = ∂ x˙ ∂ v˙ ∂ λ˙
(13.40)
The canonical Euclidean Hamiltonian is given by expressing all time derivatives in terms of the canonical momenta; hence, using Eqs. 13.39 and 13.40 yields ˙ λ − LD ˙ x + v˙ pv + λp Hc = xp γ 2 γ 2 (ω1 + ω22 )v2 + ω12 ω22 x 2 − ivλ. = − pv + 2 2
(13.41)
Note the canonical momenta px and pλ do not appear in the Hamiltonian Hc ; hence these two momenta have to be imposed, following the notation and terminology of Dirac, as constraints φ1 = px − iλ, φ2 = pλ .
(13.42)
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Similarly to including a Lagrange multiplier λ to impose a constraint on the Lagrangian, as given in Eq. 13.39, the most general Euclidean Hamiltonian consistent with the constraints is given by H1 = Hc + u1 φ1 + u2 φ2 = Hc + u1 (px − iλ) + u2 pλ ,
(13.43)
where u1 , u2 are arbitrary functions of the degrees of freedom. Hamiltonian dynamics yields that the time dependence of any function F of the dynamical variables is given by its Poisson bracket with the Hamiltonian H1 , namely dF ∂F = {F, H1 }P + , dt ∂t
(13.44)
where the Poisson bracket for arbitrary functions F, G of the dynamical variables x, v, λ is given by {F, G}P ≡
∂F ∂G ∂G ∂F ∂F ∂G ∂G ∂F ∂F ∂G ∂G ∂F − + − + − ∂x ∂px ∂x ∂px ∂v ∂pv ∂v ∂pv ∂λ ∂pλ ∂λ ∂pλ
⇒ {F, GH }P = {F, G}P H + G{F, H }P .
(13.45)
In light of the derivation of Mannheim and Davidson (2000), consider the choice of functions u1 = −v, u2 = γ ω12 ω2 that, for the Hamiltonian given in Eq. 13.43, yields H1 = Hc − v(px − iλ) + γ ω12 ω2 xpλ γ γ 2 ω1 + ω22 )v2 + ω12 ω22 x 2 − vpx + γ ω12 ω2 xpλ . = − pv2 + 2 2
(13.46)
The constraints φ1 , φ2 must be conserved over time; hence, we need to evaluate the Poisson brackets of the constraints with H1 . A straightforward calculation using Eqs. 13.45 and 13.46 yields {φ1 , H1 }P = −γ ω12 ω2 pλ , {φ2 , H1 }P = 0.
(13.47)
The dynamical variable λ is cyclic since it does not appear in the Hamiltonian H1 . Hence for conjugate variables λ and pλ dpλ ∂pλ ∂H1 ∂H1 ∂pλ ∂H1 − =− = {pλ , H1 }P = =0 dt ∂λ ∂pλ ∂λ ∂pλ ∂λ ⇒ pλ = constant.
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To decouple the degree of freedom λ from the degrees of freedom x and v, choose pλ = 0; the resulting physical Hamiltonian, from Eq. 13.46, is γ H (v, pv , x, px ) = − pv2 + 2 γ 2 = − pv + 2
α 2 β 2 v + x − vpx 2 2 γ 2 ω1 + ω22 )v2 + ω12 ω22 x 2 − vpx . 2
(13.48)
It can be shown that the Poisson brackets amongst x, px , v, pv , and H form a closed subalgebra; on this subspace the constraint equations are all conserved. Hence, H is the requisite classical Hamiltonian for the acceleration Lagrangian.
13.5 Phase space path integral and Hamiltonian operator The classical Hamiltonian yields two pathways for quantization, namely one via the formalism of phase space path integration and the other by imposing commutation equations on the pairs of canonical coordinate and its canonical momentum. These two routes for quantization are discussed below. The Euclidean Hamiltonian obtained in Eq. 13.48 yields the Lagrangian L for Euclidean time, which from Eq. 5.31, is given by ˜ pv , x, px ] = v˙ pv + xp ˙ x − H (v, pv , x, px ). L[v,
(13.49)
As discussed in Eq. 5.22, the Euclidean momentum is pure imaginary and equal to ip, where p is real. Making the change of variables pv → ipv , px → ipx and keeping the same notation for simplicity yields, from Eqs. 13.48 and 13.49, the Euclidean Lagrangian1 ˜ ipv , x, ipx ] = i v˙ pv + i xp ˙ x − H (v, ipv , x, ipx ) ≡ L[v, pv , x, px ] L[v, γ α β ˙ x − pv2 − v2 − x 2 + ivpx , (13.50) ⇒ L[v, pv , x, px ] = i v˙ pv + i xp 2 2 2 and the finite time Euclidean action is given by τ S[v, pv , x, px ] = dtL[v, pv , x, px ].
(13.51)
0
The Euclidean path integral, from Eq. 5.34, is hence defined by Z = Dx exp{S} τ = DxDpx DvDpv exp{ dtL[v, pv , x, px ]},
(13.52)
0 1 Using the notation for the Lagrangian given in Eq. 13.9.
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with boundary conditions x = x(0), v = v(0) and x = x(τ ), v = v(τ ). Equation 13.50 yields τ ' γ 2 α 2 β 2 dt (i v˙ pv + i xp ˙ x − pv − v − x + ivpx ) Z = DxDpx DvDpv exp 2 2 2 0 τ ' 1 α β dt (i xp ˙ x − v˙ 2 − v2 − x 2 + ivpx ) = DxDpx Dv exp 2 2 2 τ 0 ' 1 2 α 2 β 2 & = DxDv exp dt (− v˙ − v − x ) δ[v + x], ˙ (13.53) 2 2 2 0 t and we have recovered the path given earlier in Eq. 13.37, thus verifying that the Hamiltonian is indeed correct. Note the sign of the constraint in the equation, namely that v = −x, ˙ above what expected, and the fact that Euclidean momentum is pure imaginary consistently yields the required result. To obtain the quantum Hamiltonian, since the system has constraints, the Dirac brackets are required to obtain the Heisenberg commutators. In particular, the equal time Euclidean commutation equations have to be obtained from the Dirac brackets. A set of constraints, given by φ , = 1, 2 . . . M, defines as per Eq. 5.55, the constraint matrix C, = {φ , φ }P .
(13.54)
The Dirac brackets are given by Eq. 5.56 for arbitrary function f (q, p), g(q, p), by {f, g}D = {f, g}P −
M
−1 {f, φ }P C, {φ , g}P .
, =1
For degrees of freedom x, px , v, pv , from Eq. 13.42 the constraints are given by φ1 = px − iλ, φ2 = pλ ,
(13.55)
and the constraint matrix is given by C12 = {φ1 , φ2 }P = −i{λ, pλ }P = −i = −C21 , ⇒
−1 C12
=i=
−1 −C21 .
(13.56) (13.57)
Evaluating the Dirac bracket for only the x, v sector yields {x, φ1 }P = 1, {px , φ1 }P = {x, φ2 }P = 0 = {px , φ2 }P ,
(13.58)
{v, φ1 }P = 0 = {pv , φ1 }P = {v, φ2 }P = 0 = {pv , φ2 }P .
(13.59)
From the result above, the Dirac brackets become equal to the Poisson bracket for all the conjugate variables, namely {x, px }D = {x, px }P , . . .
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The canonical quantization for equal Euclidean time is given by [x, px ] = −I{x, px }D = −I,
(13.60)
[v, pv ] = −I{v, pv }D = −I,
(13.61)
[x, v] = [x, pv ] = [v, px ] = 0. From the above, the nonzero commutation equations for the degrees of freedom x, v are given by [x, px ] = −I = [v, pv ].
(13.62)
Hence, from Eqs. 5.50 and 5.51, the Euclidean momentum operators are given by ∂ ∂ , pv = . ∂x ∂v In summary, the degrees of freedom for the acceleration Lagrangian are given by (setting = 1) ∂ ∂ x, px = , v, pv = . (13.63) ∂x ∂v Hence, from Eqs. 13.48 and 13.63, the quantum nonHermitian Hamiltonian for the acceleration Lagrangian is given by px =
∂ γ ∂2 γ 2 ω1 + ω22 v2 + ω12 ω22 x 2 − v , + 2 2 ∂v 2 ∂x and its Hermitian conjugate is H =−
H† = −
(13.64)
2 ∂ γ ∂2 γ 2 2 2 2 2 v +v ω + + ω + ω ω x = H, 1 2 1 2 2 ∂v2 2 ∂x
since (v∂/∂x)† = −v∂/∂x.
Noteworthy 13.1 Important features of the Euclidean Hamiltonian The following are some important features in the derivation and form of the Euclidean Hamiltonian H obtained in Eq. 13.64: • The Euclidean Hamiltonian is not Hermitian due to the v∂/∂x term. • The Euclidean analysis of the Poisson brackets is consistent and one has to explicitly keep track of the signatures of time that differ from the Minkowski derivation. • The acceleration Lagrangian (incorrectly) seems to have only one degree of freedom, namely x(t). On the other hand, the Hamiltonian H and its state space are each quantum systems with two degrees of freedom, namely x and v; this result can also be seen from the path integral since one needs two initial conditions and
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two final conditions to define the finite time path integral, reflecting that the initial and final state vectors have two degrees of freedom. The Dirac constraint analysis brings out this feature of the acceleration (higher derivative) Lagrangian. • The only difference between the initial canonical Hamiltonian Hc given in Eq. 13.41 and the final result given by H in Eq. 13.64 is that the term ivλ in Hc has been replaced by −v∂/∂x. The analysis of the constraints results in the replacement of the nonphysical degree of freedom iλ by the physical degree of freedom operator −∂/∂x. • It is shown in the path integral analysis of H , analyzed in Section 13.6 below, that the term −v∂/∂x has the remarkable property of constraining, in the path integral, v to be equal to −dx/dt, as required by Eq. 13.1. • The fact that −v∂/∂x is a constraint operator explains why the term is independent of the coupling constants, since the constraint is on the phase space itself and not on the forms of interaction allowed in this phase space.
13.6 Acceleration path integral The path integral derivation of K(xf , x˙f ; xi , x˙i ) given in Eq. 13.14 was done entirely in terms of the coordinate degree of freedom x(t); in particular, the velocity degree of freedom v(t) did not appear in the expression for the path integral. One would like to interpret the evolution kernel K(xf , x˙f ; xi , x˙i ) as the probability amplitude from an initial state of state space to a final state. Such an interpretation of course needs both a state space and a Hamiltonian. Based on the boundary conditions given in Eq. 13.6, it can be seen that the state space has to have two independent degrees of freedom, corresponding to the two initial conditions given by the initial position x and velocity x. ˙ Hence, to have a state space interpretation of the path integral, state space V is taken to have two degrees of freedom, namely (position) coordinate degrees of freedom x and a degree of function reflecting x˙ in the state vector. The Hamiltonian has to be chosen in such a manner that an independent velocity degree of freedom v is is introduced and it is constrained to be equal to the velocity −x˙ of the coordinate degree of freedom x. Consider two independent degrees of freedom x and v. The completeness equation for the basis states is given by +∞ dxdvx, vv, x, (13.65) I= −∞
x, vx , v = δ(x − x )δ(v − v ). A state space representation of the evolution kernel K(xf , x˙f ; xi , x˙i ) is derived below. It will be shown in this section that the evolution kernel is closely related
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to the probability amplitude of going, in time τ , from the initial state xi , vi to the final state xf , vf  and is given by KS (xf , vf ; xi , vi ) = xf , vf e−τ H xi , vi .
(13.66)
A similar definition is adopted for the path integral of a pseudoHermitian Hamiltonian in Kandirmaz and Sever (2011). It remains to be seen what is the relation of the probability amplitude KS (xf , vf ; xi , vi ) defined using the state space and Hamiltonian to the probability amplitude K(xf , x˙f ; xi , x˙i ) defined using the path integral. In particular, as it stands in Eq. 13.66, the initial vi and final velocity vf have no relation to the coordinate degree of freedom x. The Hamiltonian has to implement a constraint to set the initial and final state in Eq. 13.66 to have the same boundary conditions given in Eqs. 13.6 and 13.7 for K(xf , x˙f ; xi , x˙i ). The Hamiltonian, for infinitesimal time τ = , is given by the Dirac–Feynman formula as
x, ve−H x , v = C()e L(x,x ;v,v ) ,
(13.67)
where C() is a normalization constant that depends only on . In terms of v = −x, ˙ the discrete time Lagrangian, from Eq. 13.2, is given by ! γ x˙ − x˙ 2 α 2 1 − v − [(x) + (x )]. (13.68) L(x, x ; v, v ) = − 2 2 2 A straightforward generalization of the Hamiltonian given in Eq. 13.64 yields a Euclidean Hamiltonian with an arbitrary potential (x), ! ∂ ∂ 1 ∂2 1 2 ∂ . (13.69) −v H =− + αv + (x) = H x, , v, 2γ ∂v2 ∂x 2 ∂x ∂v The Hamiltonian and its conjugate both act on a state space V that has two degrees of freedom, namely position coordinate x and velocity degree of freedom v. The state function  is given by  ∈ V, x, v = (x, v).
(13.70)
The transition probability amplitude is given by defining = τ/N and inserting N − 1 complete sets of states given in Eq. 13.65. Hence, for boundary conditions given by x0 = xi , v0 = vi ; xN = xf , vN = vf , the transition amplitude is given by K(xf , vf ; xi , vi ) = xf , vf e−τ H xi , vi N N−1 & & dxn dvn xn , vn e−H xn−1 , vn−1 = n=1
n=1
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=
dxn xN , vN e−H xN−1 , vN−1
n=1
×
2 N−1 &
3 dvn xn , vn e−H xn−1 , vn−1 .
(13.71)
n=1
The differential operator H given in Eq. 13.69, for xn = x, v = vn ; x = xn−1 , v = vn−1 , yields ∂
∂
x, ve−H x , v = e−H (x, ∂x ;v, ∂v ) xx vv dp dq ip(x−x )+iq(v−v ) −H =e e 2π 2π dp dq − 2γ q 2 +iq(v−v )− α2 v2 ip(x−x +v)−(x) = e e 2π 2π 2 γ 3 α 2 = Cδ(x − x + v) exp − (v − v )2 − v − (x) . 2 2 (13.72) The appearance of the δfunction in Eq. 13.72 yields the following constraint in the path integral: δ(x − x + v) ⇒ v = − ⇒
lim v = −
→0
x − x ,
dx . dτ
(13.73) (13.74)
Equation 13.72 yields the remarkable result that the term −v∂/∂x in the Hamiltonian yields a constraint on the degree of freedom v so that it is constrained to be −x˙ degree of freedom, namely v = −dx/dt. It is the delta function constraint on the velocity degree of freedom that leads to its complete elimination in the path integral. Equation 13.72 yields xn , vn e−H xn−1 , vn−1 = Cδ(xn − xn−1 + vn ) exp{Ln },
(13.75)
where Ln , from Eqs. 13.72 and 13.73, is given by Ln = −
γ α 1 (v − v )2 − v2 − [(x) + (x )]. 2 2 2 2
(13.76)
The path integral and Lagrangian that appear in Eq. 13.4 make no reference " to the integration over the velocity variables. Hence, all the velocity integrations dvn need to be carried " out in order to obtain the expression in Eq. 13.4. Remarkably enough, all the dvn integrations can be done exactly using the δfunctions that appear in Eq. 13.72.
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Equations 13.71 and 13.75 yield N−1 N−1 & & −H N−1 dvn xn , vn e dvn δ(xn − xn−1 + vn ) xn−1 , vn−1 = C n=1
n=1
× exp{Ln }.
(13.77)
The full path integral, from Eqs. 13.71 and 13.77, can be heuristically written, in the continuum notation, as K(xf , vf ; xi , vi ) = xf , vf e−τ H xi , vi τ & δ[x(t) ˙ + v(t)] exp = DxDv t=0
τ
' dtL(x, v) , (13.78)
0
γ α L(x, v) = − v˙ 2 − v2 − (x), 2 2 and we recover the expression for the Lagrangian given in Eq. 13.68. To obtain the precise content of the continuum expression given in Eq. 13.78, one needs to go to the lattice and carefully address the issue of the boundary conditions. 13.7 Change of path integral boundary conditions The initial and final time steps need to be examined carefully to see how the initial and final velocity, v0 and vN respectively, can be expressed solely in terms of the coordinate degree of freedom. The four boundary conditions given in Eq. 13.11 are solely in terms of the coordinate degree of freedom x(t), whereas the boundary conditions given in Eq. 13.66, the defining equation for KS (xf , vf ; xi , vi ), are given in terms of final and initial positions and velocities xf , vf ; xi , vi respectively. Note that the integrand of Eq. 13.77, for n = 1, yields, from Eq. 13.76, dv1 δ(x1 − x0 + v1 ) exp{L1 } ! γ v1 − v0 2 α 2 1 L1 = − − v1 − [(x1 ) + (x0 )]. (13.79) 2 2 2 " On performing the dv1 integration, the delta function constrains v1 = −(x1 − x0 )/; hence, L1 has the value !2 2 γ x −x 3 α 1 0 exp{L1 } = exp − + v0 − (x1 − x0 )2 + O() 2 2 !2 2 γ x −x 3 α 2 1 0 = exp − + v0 − v0 + O() 2 2 = C()δ(x1 − x0 + v0 ) + O() = C()δ(x1 − xi + vi ), (13.80)
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where x0 = xi and v0 = vi are the initial position and velocity. The final time boundary term for the action, from Eq. 13.75, yields xN , vN e−H xN−1 , vN−1 = C()δ(xN − xN−1 + vN ) exp{LN } = C()δ(xf − xN−1 + vf ) exp{LN }, since xN = xf and vN = vf . The path integral over the velocity degrees of freedom yields, in addition to the expected acceleration action, two extra delta functions. These delta" functions are crucial in changing the boundary conditions for the path integral Dx over the coordinate degree of freedom. Collecting all the results yields the discrete time path integral for the transition probability amplitude expressed solely in terms of the coordinated degrees of freedom, namely KS (xf , vf ; xi , vi ) = xf , vf e−τ H xi , vi N−1 & dxn δ(xf − xN−1 + vf )δ(x1 − xi + vi ) = C˜ n=1
6
× exp
N
D Ln ,
(13.81)
n=1
where C˜ is a normalization. Note that the path integral given in Eq. 13.81, due to the two delta functions in the integrand, has four boundary conditions for the coordinate degree of freedom, namely "xi , x1 , xN−1 , xf ; the delta functions, in effect, remove two integrations, namely dx1 dxN−1 in the path integral given in Eq. 13.81 by fixing the value of x1 , xN−1 . In the derivation of the probability amplitude carried out by Kleinert (1986), the x˙ boundary conditions are finally implemented in the discretized path integral by constraining the variables near the end points, namely x1 and xN−1 . To take the continuum limit we define x(t) ˙ =
dx(t) xn − xn−1 = , t = n. dt
(13.82)
Hence, from Eq. 13.81 x1 − x0 → −x˙i , xN − xN−1 → −x˙f , vf =
vi =
(13.83) (13.84)
since x˙i = dx(0)/dt and x˙f = dx(τ )/dt.
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From Eq. 13.77, γ α 2 (x˙n − x˙n−1 )2 − x˙ − (xn ). 2 2 n Taking the limit of → 0 and using x¨ = [x˙n − x˙n−1 ]/ yields γ α L = − x¨ 2 − x˙ 2 − (x). 2 2 Ln = −
(13.85)
(13.86)
" The delta functions for the boundary values of the functional integral Dx are constraints that change the boundary conditions " on the path integral, converting the two boundary conditions "each for x and v in DxDv to four boundary conditions for x in the path integral Dx. Hence, one obtains the continuum result % % KS (xf , vf ; xi , vi ) = xf , vf e−τ H xi , vi = Dxδ(vi + x˙i )δ(vf + x˙f )eS % (xi ,xf ) % % = DxeS % = K(xi , x˙i = −vi ; xf , x˙f = −vf ) (xi ,x˙i =−vi ;xf ,x˙f =−vf )
⇒ KS (xf , vf ; xi , vi ) = K(xf , −x˙f ; xi , −x˙i ). Hence KS (xf , vf ; xi , vi ) = xf , vf e−τ H xi , vi = DxeS = K(xf , −x˙f ; xi , −x˙i ).
(13.87)
The representation of the path integral in terms of the transition amplitude KS (xf , vf ; xi , vi ) is necessary to have the amplitude obey the composition law derived in Section 4.8. This is discussed by Hawking and Hertog (2002) and Fontanini and Trodden (2011).
13.8 Evolution kernel The Hamiltonian in Eq. 13.69 [for the Lagrangian in Eq. 13.86] is given by H =−
∂ 1 ∂2 α −v + v2 + (x). 2 2γ ∂v ∂x 2
In terms of the path integrals, from Eqs. 13.4 and 13.78 γ α γ α L(x) = − x¨ 2 − x˙ 2 − (x), L(x, v) = − v˙ 2 − v2 − (x), 2 2 2 2 τ τ τ & Dx exp{ dtL(x)} = DxDv δ[x(t) ˙ + v(t)] exp{ dtL(x, v)}. 0
t=0
0
Implicit in the path integrals above are the appropriate boundary conditions.
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The symmetry of the evolution kernel given in Eq. 13.19 using the path integral, namely K(xf , x˙f ; xi , x˙i ) = K(xi , −x˙i ; xf , −x˙f ) has a Hamiltonian derivation. Note that H † can be obtained from Eq. 13.69 by conjugation and is given by H† = −
1 ∂2 ∂ α +v + v2 + (x) = H. 2 2γ ∂v ∂x 2
(13.88)
From Eq. 13.87 KS∗ (xf , vf ; xi , vi ) = xf , vf e−τ H xi , vi ∗ = xi , vi e−τ H xf , vf . †
(13.89)
For the Hamiltonian given in Eq. 13.69, the only difference in the path integral between H and H † is to change the sign of v∂/∂x to −v∂/∂x; this in turn yields the constraint δ(v − x) ˙ for the path integral representation of Eq. 13.89. Hence, on " performing the velocity path integral Dv for Eq. 13.89, the only change from the earlier case is that the boundary on velocity now has a minus sign and yields xi , vi e−τ H xf , vf = KS (xi , −vi ; xf , −vf ). †
(13.90)
For the case of KS being real, which is the case considered in Eq. 13.19, Eqs. 13.89 and 13.90 yield KS (xf , vf ; xi , vi ) = KS (xi , −vi ; xf , −vf ),
(13.91)
from which Eq. 13.19 follows due to Eq. 13.87. The symmetry of the evolution kernel has been shown in Hawking and Hertog (2002) to be correct by a direct evaluation of the classical action S[xc ]. The evolution kernel has an implicit time ordering; explicitly putting in the initial ti and final tf time coordinates, the kernel can be written out explicitly in the form KS (xf , vf , tf ; xi , vi , ti ) = xf , vf e−tf −ti H xi , vi .
(13.92)
Using the identity ∂ exp{−αt} = −α exp{−αt} + δ(t) ∂t yields, from Eq. 13.92, the following Schrödinger equation with a source: ∂ + H KS (xf , vf , tf ; xi , vi , ti ) = δ(xf − xi )δ(vf − vi )δ(tf − ti ). (13.93) ∂tf This result has been derived for the Euclidean case by Kleinert (1986) and for the Minkowski case by Mannheim (2011b).
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13.9 Summary
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13.9 Summary The Euclidean acceleration Lagrangian leads to a number of new results. The theory has a state space that has two degrees of freedom, namely x and its velocity v. The theory has three branches shown in Figure 13.1. The first branch, given by α > αc , has a pseudoHermitian Hamiltonian and a state space with a positive definite norm. The second branch, given by α < αc , has complex parameters in its Hamiltonian and the third critical branch α = αc – separating the other two branches – consists of an essentially nonHermitian Hamiltonian consisting of Jordan blocks. To obtain the Hamiltonian, the acceleration Lagrangian is written as a constrained system and the Dirac constraint method was utilized to obtain a nonHermitian Hamiltonian. The Hamiltonian in turn was then used for obtaining a path integral representation of the evolution kernel and was shown to reproduce the earlier result, confirming the result obtained for the Hamiltonian. In particular, a derivation was given of the change of boundary conditions in going from the x, v degrees of freedom for the Hamiltonian to the solely x degree of freedom in terms of which the acceleration action is directly defined.
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14 PseudoHermitian Euclidean Hamiltonian
There is no conservation of probability in the Euclidean formulation of the acceleration Lagrangian. A narrow definition of a Hamiltonian, and the one that holds in quantum mechanics but is not suitable for our case, is that the Hamiltonian defines a probability conserving unitary evolution. A broader definition of the Hamiltonian is adopted that is consistent with both statistical mechanics and quantum mechanics. Namely, the Hamiltonian – which is equivalent to the transfer matrix in statistical mechanics – is the generated of infinitesimal translations in time for both the Minkowski and Euclidean cases. The Hamiltonian derived for the acceleration Lagrangian in Chapter 13 is studied in this chapter using the concepts of the state space and operators. The Hamiltonian for unequal and real frequencies ω1 , ω2 is pseudoHermitian and is explicitly mapped to a Hermitian Hamiltonian using a nontrivial differential operator. An explicit expression is obtained for the matrix element of the differential operator. The mapping fails for a critical value of the coupling constants. The Hamiltonian and state space are shown to have real and complex branches that are separated by the critical point; both branches have a wellbehaved state space and algebra of operators. At the critical point, the Hamiltonian is inequivalent to any Hermitian operator, and is shown in Chapter 15 as being equal to an infinite dimensional block diagonal matrix, with each block being a Jordan matrix. For the real branch of the theory, the state space of a single quantum degree of freedom described by the acceleration action is shown to be a function of two distinct degrees of freedom, namely velocity v and position x, with the two being related by a constraint equation. The state vector for the acceleration Hamiltonian is given by ψ(x, v), and is quite unlike the state vector ψ(x) of a quantum degree of freedom described by a Lagrangian having only the velocity term – namely, without an acceleration term. The acceleration Lagrangian and its path integral are well defined for all values of ω1 , ω2 , including complex values. The Euclidean Hamiltonian obtained for the
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acceleration Lagrangian shows that the state space of the acceleration Lagrangian is only well defined for real ω1 , ω2 . For equal frequency ω1 = ω2 , the Hamiltonian becomes essentially nonHermitian, and this is analyzed at length in Chapter 15. For the case of complex ω1 , ω2 the Hamiltonian is neither Hermitian nor pseudoHermitian and the state space is no longer a positive normed space. In Section 14.1 a similarity transformation eQ/2 is discussed that shows that the Hamiltonian H is pseudoHermitian. In Section 14.2 the Hamiltonian H is mapped, by a similarity transformation, to a Hermitian Hamiltonian that is the sum of two decoupled oscillators. In Section 14.3 the matrix elements of the similarity transformation are evaluated and are shown in Section 14.4 to yield the expected results. The eigenfunctions of the equivalent Hermitian Hamiltonian are evaluated in Section 14.5 and a few of the excited states of the acceleration Hamiltonian are derived in Section 14.9. The state space for unequal frequencies constitutes a Hilbert space, with all state vectors having a positive norm. The general features of the state space of the pseudoHermitian Hamiltonian are discussed in Section 14.11, and in particular the necessity of introducing a metric on state space that is required for defining the conjugate state vector and scalar product for the state space. The eigenfunctions of the acceleration Hamiltonian are studied in Section 14.6: the propagator for the degree of freedom is evaluated in Section 14.12 using operator methods and in Section 14.13 using the state space analysis. It is shown that the propagator cannot be obtained from any Hermitian Hamiltonian. 14.1 PseudoHermitian Hamiltonian; similarity transformation The acceleration Hamiltonian, from Eq. 13.64, is given by H =−
∂ γ γ 1 ∂2 −v + (ω12 + ω22 )v2 + ω12 ω22 x 2 . 2 2γ ∂v ∂x 2 2
(14.1)
For H to be pseudoHermitian, similarity transformation eQ/2 is required such that e−Q/2 H eQ/2 = HO ,
(14.2)
where HO is a Hermitian Hamiltonian; it is shown in Section 14.2 that HO consists of a system of two decoupled harmonic oscillators, one each for degree of freedom x and v. The Hermitian conjugate Hamiltonian H † , from Eq. 14.2, is given by H † = e−Q/2 HO eQ/2 ⇒ H † = e−Q H eQ : pseudoHermitian.
(14.3)
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Note that
e−τ H
†
= e−τ H = e−Q e−τ H eQ . †
(14.4)
Equation 14.3 is a definition of a pseudoHermitian operator, that differs from its Hermitian conjugate by a similarity transformation. It can be shown that the energy eigenvalues of a pseudoHermitian Hamiltonian are either real or appear in complex conjugate pairs, and vice versa. All Hamiltonians that are equivalent to a Hermitian Hamiltonian up to a similarity transformation, as is the case for Eq. 14.2, are automatically pseudoHermitian. However, all pseudoHermitian Hamiltonians are not equivalent to a Hermitian Hamiltonian, and Chapter 15 analyzes such a case. There is another class of Hamiltonians that have real eigenenergies and the energy eigenstates are complete; these Hamiltonians can be brought to a Hermitian form by a similarity transformation. Such Hamiltonians are referred to by Scholtz et al. (1992) as being quasiHermitian. All quasiHermitian Hamiltonians are thus also pseudoHermitian, but not all pseudoHermitian Hamiltonians are quasiHermitian. A Q is obtained for Euclidean time by analytically continuing the remarkable result obtained by Bender and Mannheim (2008b), and this yields Q = axv − b
∂2 . ∂x∂v
(14.5)
For the real domain where ω1 , ω2 are real, both coefficients a, b are real and hence Q = Q† : Hermitian for ω1 , ω2 real.
(14.6)
The equation for the commutator e−Q OeQ =
∞ 1 [[[O, Q], Q] . . . Q] n! n=0
(14.7)
needs to be applied to O = x, v, ∂/∂x, ∂/∂v. To obtain the commutator, note that the nfold commutator of Q with x, v, ∂/∂x and ∂/∂v follows a simple pattern that repeats after two commutations. In particular, note that ∂ [x, Q] = b , ∂v ∂ , Q = av, ∂x ∂ [v, Q] = b , ∂x
[[x, Q], Q] = abx, ∂ ∂ , Q , Q = ab , ∂x ∂x
...
[[v, Q], Q] = abv,
...
...
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14.2 Equivalent Hermitian Hamiltonian HO
∂ , Q = ax, ∂v
297
∂ ∂ , Q , Q = ab , ∂v ∂v
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Carrying out the nested commutators to all orders and summing the result yields, for a, b > 0, √ √ b ∂ −τ Q τQ xe = cosh(τ ab)x + sinh(τ ab) , e a ∂v √ √ a ∂ −τ Q ∂ τ Q e e = cosh(τ ab) + sinh(τ ab)v, ∂x ∂x b √ √ b ∂ e−τ Q veτ Q = cosh(τ ab)v + sinh(τ ab) , a ∂x √ √ ∂ a ∂ (14.8) sinh(τ ab)x. e−τ Q eτ Q = cosh(τ ab) + ∂v ∂v b Note that the definitions of the values of a and b are chosen based on ω1 > ω2 , which makes the operator Q Hermitian the domain where ω1 , ω2 are real. The definition of Q continues to hold for the complex domain but Q is no longer Hermitian.
14.2 Equivalent Hermitian Hamiltonian HO The fundamental commutation relations given in Eq. 14.8 can be applied to the acceleration Hamiltonian, and the coefficients a, b can be chosen to decouple the x and v degrees of freedoms. Consider the equation e−Q/2 H eQ/2 = C1
∂2 ∂2 ∂ ∂ + C x v + C5 x 2 + C6 v2 . + C + C 2 3 4 ∂v2 ∂v ∂x ∂x 2
(14.9)
To obtain the factorization of the Hamiltonian into two decoupled oscillators, following Bender and Mannheim (2008b), we choose the following values for a and b: ! √ √ a 2ω1 ω2 ω1 + ω 2 . (14.10) ⇒ ab = ln = γ ω1 ω2 ; sinh( ab) = 2 b ω1 − ω2 ω1 − ω22 We define
#√ $ #√ $ ω1 ω2 ab ab =8 =8 A = cosh , B = sinh , 2 2 ω12 − ω22 ω12 − ω22 a (14.11) C= = γ ω1 ω2 . b
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Using the result of Eq. 14.8 and the definitions in Eq. 14.9 yields ! 1 2 γ B 2 2 2 ω1 ω2 C1 = − A + 2γ 2 C #√ $ # √ $/ . 1 ab ab 1 cosh2 − sinh2 =− . =− 2γ 2 2 2γ Similarly, after some simplifications CAB AB + γ ω12 ω22 C2 = − γ C C3 = −(A + B ) + 2
2
γ (ω12
+
! = 0,
ω22 )
AB C
! = 0.
The constants a and b were chosen so that C2 = C3 = 0; hence one has C2 = C 3 = 0 ⇒
determines a and b.
The remaining coefficients are given by ! AB B 2 1 γ 2 2 C4 = − =− , + (ω1 + ω2 ) C 2 C 2γ ω12 1 γ γ C5 = − B 2 C 2 + ω12 ω22 A2 = ω12 ω22 , 2γ 2 2 γ 2 γ C6 = −ABC + (ω1 + ω22 )A2 = ω12 . 2 2 Collecting all the results yields e−Q/2 H eQ/2 = HO ⇒ HO = −
1 ∂2 1 ∂2 γ γ − + ω12 v2 + ω12 ω22 x 2 . 2 2 2 2γ ∂v 2 2 2γ ω1 ∂x
(14.12)
14.3 The matrix elements of e−τ Q The Qoperator is given from Eq. 14.5 by Q = axv − b where, from Eq. 14.10, a = γ ω1 ω2 , b
∂2 , ∂x∂v
√ 2ω1 ω2 sinh( ab) = 2 . ω1 − ω22
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14.3 The matrix elements of e−τ Q
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The finite matrix elements of the Qoperator are required for many calculations involving the state vectors. The exact matrix elements can be obtained by noting that the Hermitian Qoperator exactly factorizes into two decoupled harmonic oscillators by an appropriate change of variables. Consider the change of variables given by 1 1 α = √ (x + v), β = √ (x − v), 2 2 ! ! ∂ ∂ 1 ∂ 1 ∂ ∂ ∂ , . =√ + =√ − ∂x ∂v 2 ∂α ∂β 2 ∂α ∂β
(14.13) (14.14)
In these coordinates the α, β sectors completely factorize and yield ! a 2 b ∂2 ∂2 2 Q = (α − β ) − , − 2 2 ∂α 2 ∂β 2 1 ∂2 1 1 ∂2 1 = − b 2 + aα 2 + b 2 − aβ 2 . 2 ∂α 2 2 ∂β 2
(14.15) (14.16)
Consider the Hamiltonian of a quantum oscillator given by Hsho = −
1 ∂2 1 + mω2 z2 , 2 2m ∂z 2
(14.17)
with the transition amplitude given by K(z, z ; τ ) = ze−τ H z ' mω mω 2 2 = exp − (z + z ) cosh(ωτ ) − 2zz . 2π sinh ωτ 2 sinh(ωτ ) (14.18) Comparing the α and βsectors of Q with Hsho shows that, for a, b real, the α sector is the usual quantum oscillator but the β sector yields a divergent transition amplitude. The result for the β sector is assumed to be given by the analytic continuation of the oscillator transition amplitude; this assumption will later be verified by an independent derivation. To exploit the quadratic form of the α and β sectors, consider extending the range of m to the real line. This yields α − sector : β − sector :
1 ; m 1 b=− ; m
b=
a = mω2
⇒
ω=
√ ab,
(14.19)
a = −mω2
⇒
ω=
√ ab.
(14.20)
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Hence Eqs. 14.18, 14.19, and 14.20 yield α, βe−τ Q α , β 6
D √ √
2 ab 1 (α + α 2 ) cosh(τ ab) − 2αα = Nα Nβ exp − √ 2b sinh(τ ab) D 6 √ √
2 ab 1 2 (β + β ) cosh(τ ab) − 2ββ × exp √ 2b sinh(τ ab) 2 1 3 = N (τ ) exp − G(τ )(α 2 + α 2 − β 2 − β 2 ) + H(τ )(αα − ββ ) , 2
where
G(τ ) =

√ a coth(τ ab), b
and
7
√
H(τ ) =
7
(14.21)
√
ab · √ √ 2πb sinh(τ ab) −2πb sinh(τ ab) % 1 a i %% i = ⇒ N (τ ) = H(τ )%. √ 2π b  sinh(τ ab) 2π N (τ ) =
ab
1 a √ b sinh(τ ab)
(14.22)
Hence a heuristic derivation for Q yields x, ve−τ Q x , v = N (τ ) exp −G(τ )(xv + x v ) + H(τ )(xv + vx ) . (14.23) The normalization constant N (τ ) will be seen to play a crucial role in the normalization of all the state vectors of the Hamiltonian H . The result obtained by the heuristic method will be verified to be indeed exactly correct, including the normalization constant. For the real branch, both coefficients G(τ ) and H(τ ) are real and hence Q = Q† is Hermitian. Note that the form obtained for e−τ Q in Eq. 14.23 is a major simplification since in general, for Hermitian Q, one would need to evaluate 4 + 12 = 16 real coefficients. Instead, the form that has been heuristically derived in Eq. 14.23 has reduced the determination of e−τ Q to that of computing two real coefficient functions G(τ ) and H(τ ) and a normalization constant N (τ ). The operator Q is unbounded and many of the manipulations are only formally valid. For example, the identity e−τ Q eτ Q = I holds for the matrix elements only in a formal sense. Explicitly evaluating the matrix elements of the product e−τ Q eτ Q using Eq. 14.23 yields
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x, ve−τ Q eτ Q x , v = dξ dζ x, ve−τ Q ξ, ζ ξ, ζ eτ Q x , v 2 = N (τ ) dξ dζ e−G (τ )(xv+ξ ζ )+H(τ )(xζ +vξ ) eG (τ )(ξ ζ +x v )−H(τ )(ξ v +ζ x ) 2 −G (τ )(xv−x v ) dξ dζ exp{iH(τ )iξ(v − v) + iH(τ )iζ (x − x)} = N (τ )e !2 2π = N 2 (τ ) δ(x − x )δ(v − v ) = δ(x − x )δ(v − v ) iH(τ ) ⇒ e−τ Q eτ Q = I. To make the above derivation more rigorous one can analytically continue τ back to Minkowski time t = −iτ , do the computation and then analytically continue back to Euclidean time τ . This would give the result above.
14.4 e−τ Q and similarity transformations The heuristic derivation for G(τ ) and H(τ ) was obtained by an analogy with the oscillator Hamiltonian and cannot be assumed to be correct since the βsector yields an unstable Hamiltonian. The result needs to be independently verified. That the matrix elements of e−τ Q given by Eq. 14.23 are in fact correct is now verified in this section. The fundamental similarity transformations of e−τ Q Oeτ Q for operators O = x, ∂/∂x, v and ∂/∂v are directly obtained using the result given in Eq. 14.23 and shown to be identical to the defining equations for Q given in Eq. 14.8. Recall from Eq. 14.8 that eτ Q yields the following similarity transformations: I. e
−τ Q
I I. e−τ Q
xe
τQ
∂ τQ e ∂x
I I I. e−τ Q veτ Q I V . e−τ Q
∂ τQ e ∂v

√
√ b ∂ sinh(τ ab) , a ∂v √ √ a ∂ = cosh(τ ab) + sinh(τ ab)v, ∂x b √ √ b ∂ = cosh(τ ab)v + sinh(τ ab) , a ∂x √ √ a ∂ = cosh(τ ab) + sinh(τ ab)x. ∂v b = cosh(τ ab)x +
Consider the operator equation e
−τ Q
xe
τQ
√
= cosh(τ ab)x +

√ b ∂ sinh(τ ab) . a ∂v
(14.24)
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The matrix element of e−τ Q xeτ Q is given by x, ve−τ Q xeτ Q x , v # $ √ √ b ∂ = cosh(τ ab)x + sinh(τ ab) δ(x − x )δ(v − v ). a ∂v
(14.25)
Equation 14.23 yields x, ve−τ Q x , v = N exp{−g(xv + x v ) + h(xv + vx )},
(14.26)
where √ a cosh(τ ab) g = G(τ ) = , √ b sinh(τ ab) 1 a h = H(τ ) = , √ b sinh(τ ab) i . N = N (τ ) = 2π H(τ ) 
The left hand side of Eq. 14.25 yields −τ Q τQ xe x, v = dξ dζ x, ve−τ Q ξ, ζ ξ, ζ ξ eτ Q x , v x, ve = N dξ dζ e−g(xv+ξ ζ )+h(xζ +vξ ) ξ ξ, ζ eτ Q x , v . But ξ exp{−g(xv + ξ ζ ) + h(xζ + vξ )} / .√ √ b ∂ sin(τ ab) + cosh(τ ab)x e−g(xv+ξ ζ )+h(xζ +vξ ) . = a ∂v
(14.27)
Hence, we obtain the expected result, given in Eq. 14.25, that . x, ve
−τ Q
xe
τQ
√
x , v = cosh(τ ab)x +
⇒ I. e−τ Q xeτ Q

√ b ∂ sinh(τ ab) a ∂v
/
× x, ve−τ Q eτ Q x , v √ √ b ∂ = cosh(τ ab)x + sinh(τ ab) . a ∂v
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14.4 e−τ Q and similarity transformations
Consider −τ Q
yielding the expected result that I I. e
−τ Q
ve
303
ve x , v = dξ dζ x, ve−τ Q ξ, ζ ξ, ζ ζ eτ Q x , v x, ve = dξ dζ ζ x, ve−τ Q ξ, ζ ξ, ζ eτ Q x , v $ #√ √ b ∂ sinh(τ ab) + cosh(τ ab)v δ(x − x )δ(v − v ), = a ∂x τQ
i
τQ
√ = cosh(τ ab)v +

(14.28)
√ b ∂ sinh(τ ab) . a ∂x
The matrix element has the representation ∂ −τ Q ∂ τ Q e x , v = dξ dζ x, ve−τ Q ξ, ζ ξ, ζ  eτ Q x , v x, ve ∂x ∂ξ ! ∂ −g(xv+ξ ζ )+h(xζ +vξ ) e = N dξ dζ − ∂ξ =
× ξ, ζ eτ Q x , v dξ dζ gζ − hv x, ve−τ Q ξ, ζ ξ, ζ eτ Q x, v.
Using Eq. 14.28 to replace ζ in the above expression yields, #$ √ √ ∂ b ∂ e−τ Q eτ Q = g sinh(τ ab) + cosh(τ ab)v − hv ∂x a ∂v √ √ ∂ b ∂ ⇒ I I I. e−τ Q eτ Q = cosh(τ ab) + sinh(τ ab)v. ∂x ∂x a And lastly, similarly to the above derivation ∂ −g(xv+ξ ζ )+h(xζ +vξ ) −τ Q ∂ τ Q ξ, ζ eτ Q x , v x, ve e x , v = N dξ dζ − e ∂v ∂ζ = dξ dζ gξ − hx x, ve−τ Q ξ, ζ ξ, ζ eτ Q x , v . Using Eq. 14.27 to replace ξ in the equation above yields #$ √ √ b ∂ −τ Q ∂ τ Q e e =g sinh(τ ab) + cosh(τ ab)x − hx ∂v a ∂v √ √ ∂ b ∂ ⇒ I V . e−τ Q eτ Q = cosh(τ ab) + cosh(τ ab)x. ∂v ∂v a
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Hence we have verified that the expression for x, ve−τ Q x , v given in Eq.14.23 is in fact correct and reproduces all the basic similarity transformations, from I to I V .
14.5 Eigenfunctions of oscillator Hamiltonian HO The Euclidean Hamiltonian has two distinct and independent state spaces, namely the state space V of the nonHermitian Hamiltonian H and the state space of the Hermitian Hamiltonian HO , namely VO . The oscillator state space VO is a Hilbert space, with the norm of two state vectors  and χ given by χ = dxdvχ ∗ (x, v)(x, v). (14.29) The oscillator Hamiltonian HO , from Eq. 14.12, is given by HO = −
1 ∂2 1 ∂2 γ γ − + ω12 v2 + ω12 ω22 x 2 . 2 2 2 2γ ∂v 2 2 2γ ω1 ∂x
The oscillator structure of the Hamiltonian yields two sets of creation and destruction operators, given by 1 ∂ 1 ∂ γ ω1 γ ω1 † v+ , av = v− , av = 2 γ ω1 ∂v 2 γ ω1 ∂v 7 7 1 1 γ ω12 ω2 ∂ γ ω12 ω2 ∂ † x+ , ax = x− , ax = 2 2 γ ω12 ω2 ∂x γ ω12 ω2 ∂x ⇒ [ai , aj† ] = δi−j , i, j = 1, 2.
(14.30)
Note that, similarly to the case for Minkowski time, the creation operator ai† , for i = 1, 2, is the Hermitian conjugate of the destruction operator ai . From the above 7 1 γ ω1 ∂ † (av + av ), = (av − av† ), v= 2γ ω1 ∂v 2 7 7 1 γ ω12 ω2 ∂ † (a + a ), = (ax − ax† ). x= x x ∂x 2 2γ ω12 ω2 The oscillator Hamiltonian is given by 1 HO = ω1 av† av + ω2 ax† ax + (ω1 + ω2 ). 2
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The vacuum is defined, as usual, by requiring that there be no excitations, namely av 0, 0 = ax 0, 0 = 0, 1 HO 0, 0 = E0 0, 0, E0 = (ω1 + ω2 ). 2 The coordinate representation of the oscillator vacuum state is given by !1/4 ' 1 γ 2 ω13 ω2 2 2 2 exp − [γ ω1 v + γ ω1 ω2 x ] . x, v0, 0 = π2 2 The energy eigenfunctions n, m and eigenenergies Enm of HO are given by HO n, m = Enm n, m, n, mHO = Enm n, m, av†n
√ 0, 0, x, vm, n = m, nx, v : real, m! 1 Enm = nω1 + mω2 + E0 = nω1 + mω2 + (ω1 + ω2 ). 2 The dual eigenfunctions n, m satisfy the orthonormality equation n, m = √
(14.31)
ax†m
n!
(14.32)
n , m n, m = δn−n δm−m .
(14.33)
Hence, the spectral representation of HO is given by Emn m, nm, n. HO =
(14.34)
mn
All the state vectors and operators have been defined entirely on state space VO with no reference to state space V. The operator eQ/2 maps state space VO into V. 14.6 Eigenfunctions of H and H† The Euclidean Hamiltonian is given, from Eq. 14.1, by H =−
1 ∂2 γ ∂ γ + (ω12 + ω22 )v2 + ω12 ω22 x 2 , −v 2γ ∂v2 ∂x 2 2
and from Eq. 14.12, the equivalent Hermitian Hamiltonian HO is given by H = eQ/2 HO e−Q/2 , H † = e−Q/2 HO eQ/2 , HO = −
1 ∂2 1 ∂2 γ γ − + ω12 v2 + ω12 ω22 x 2 . 2γ ∂v2 2γ ω12 ∂x 2 2 2
(14.35) (14.36)
The left and right eigenfunctions of H are different since H is pseudoHermitian; let us denote the right eigenfunctions by mn and left dual eigenD D . The notation mn  is used to denote the dual of the state functions by mn
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mn to differentiate it from the state mn  that is obtained from mn by complex conjugation. The energy eigenfunctions of Hamiltonian H and H † , from Eqs. 14.31 and 14.36, are given by H mn = Emn mn , mn = eQ/2 m, n, D D D H = Emn mn , mn  = m, ne−Q/2 = mn e−Q . mn
(14.37)
The eigenfunctions are orthonormal since D n m = nm e−Q n m = n, meQ/2 e−Q eQ/2 n , m nm
= n, mn , m = δn−n δm−m .
(14.38) (14.39)
Since the eigenfunctions of HO are complete, the completeness equation for the Hilbert space V on which the pseudoHermitian Hamiltonian H acts is given by I=
∞
D mn mn .
(14.40)
m,n=1
The state space for the Euclidean Hamiltonian H has all positive norm eigenstates. For many of the computations, it is convenient to separate out the overall normalization constants of the eigenfunctions. We define D D  = Nmn ψmn , mn = Nmn ψmn , mn D 2 D ∗ mn = Nmn ψmn ψmn = 1, Nmn = Nmn > 0 : real, positive. mn
(14.41)
One can write the completeness equation given in Eq. 14.40 as I=
∞
2 D ψmn Nmn ψmn .
(14.42)
m,n=1
All the states m, n are real, as are all the matrix elements of eQ/2 given in Eq. 14.52; hence the coordinate representation of mn is real and is given by ∗ (x, v) : real. mn (x, v) = x, vmn = mn
From Eqs. 14.12 and 14.34, the Hamiltonian H has the spectral decomposition x, vH x , v = x, veQ/2 HO e−Q/2 x , v = Emn x, veQ/2 m, nm, ne−Q/2 x , v mn
=
D Emn mn (x, v)mn (x , v ).
(14.43)
mn
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The evolution kernel has the spectral decomposition given by D e−τ Emn mn (x, v)mn (x , v ). x, ve−τ H x , v =
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307
(14.44)
mn
14.6.1 Dual energy eigenstates Note that unlike the case for a Hermitian Hamiltonian, the dual eigenfunction  D (x, v) are given by is not the complex conjugate of . The dual eigenstates mn D D D  = m, ne−Q/2 ⇒ mn (x, v) = mn x, v = m, ne−Q/2 x, v. mn
To obtain the matrix element of a nonHermitian operator, it is important to note that all operators act only on the dual space. This fact is unimportant for Hermitian operators, since acting on the state space or its dual space is equivalent. But this is not true for nonHermitian operators, since the result depends on which space the ˆ operators act on. In particular, for operator O(x, v, ∂x , ∂v ) one has ˆ ˆ x, vO = O(x, v, ∂x , ∂v )(x, v). Since all operators are defined by their action only on the dual space, the matrix ˆ element of the operator O(x, v, ∂x , ∂v ) acting on the dual state vector  is given in terms of the conjugate operator Oˆ † (x, v, ∂x , ∂v ) as ∗ ˆ Ox, v ≡ x, vOˆ † 
∗ = O † (x, v, ∂x , ∂v )(x, v) . Given the general rule that all operators for a nonHermitian system must act on D H x, v consider the the dual coordinate x, v, to obtain the matrix element mn eigenfunction equation D D D D (x, v)∗ = mn H x, v∗ = x, vH † mn = H † mn (x, v). Emn mn
Note that the general form of the Hamiltonian H given in Eq. 14.1 implies that H † (x, v, ∂/∂v, ∂/∂) = H (x, −v, −∂/∂v, ∂/∂x) = H (−x, v, ∂/∂v, −∂/∂x). Since all the eigenfunctions are real, there are two possibilities, D (x, v) = H † mn (x, −v) = Emn mn (x, −v) H † mn
or H
†
D mn (x, v)
= H † mn (−x, v) = Emn mn (−x, v).
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D Hence, the only difference between mn (x, v) and mn (x, v) is that v → −v or x → −x and one obtains the general result that ⎧ ⎨mn (x, −v) D (x, v) = or mn . (14.45) ⎩ mn (−x, v)
As will be seen for the first two excited states given in Eq. 14.82, the dual eigenvectors are determined precisely in a manner that guarantees a positive norm (Hilbert) state space for the pseudoHermitian Hamiltonian H . This remarkable feature of how the dual state vector is defined can be expected to hold for all the eigenstates. Of the two possibilities stated in Eq. 14.45, it is the Q operator that determines which choice is made for the dual state vector. The probability amplitude, from Eq. 13.91, is given by KS (x, v; x , v ) = KS (x , −v ; x, −v) ⇒ x, ve−τ H x , v = x , −v e−τ H x, −v.
(14.46)
From Eqs. 14.46 and14.44 D D e−τ Emn mn (x, v)mn (x , v ) = e−τ Emn mn (x , −v )mn (x, −v) mn
mn
⇒
D (x , v ) mn (x, v)mn
D = mn (x , −v )mn (x, −v).
(14.47)
To prove that the duality relation given in Eq. 14.45 yields Eq. 14.47 we need to discuss the parity operator P, defined by P : x → −x, v → −v, P 2 = I ⇒ Pmn (x, v) = mn (−x, −v).
(14.48)
It can be readily shown that P commutes with H , [P, H ] = 0. Hence all the eigenfunctions mn (x, v) are also eigenfunctions of parity with eigenvalue s; Eq. 14.48 yields Pmn (x, v) = mn (−x, −v) = smn (x, v), s = ±1.
(14.49)
Let us consider the two distinct cases for the dual vector given in Eq. 14.45, and Eq. 14.47 is verified for each case: D (x, v) = mn (x, −v). Eq. 14.47 yields • mn D Left hand side : mn (x, v)mn (x , v ) = mn (x, v)mn (x , −v ), D (x, −v) = mn (x , −v )mn (x, v), Right hand side : mn (x , −v )mn
hence, Eq. 14.47 is verified.
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D • mn (x, v) = mn (−x, v). Eq. 14.47 yields, using Eq. 14.48, D (x , v ) = mn (x, v)mn (−x , v ), Left hand side : mn (x, v)mn D (x, −v) = mn (x , −v )mn (−x, −v), Right hand side : mn (x , −v )mn
= s 2 mn (−x , v )mn (x, v). Since s 2 = 1, Eq. 14.47 is verified. It is the nontrivial behavior of the dual state vector, as given in Eq. 14.45, that leads to results that are not allowed for Hermitian Hamiltonians, such as the expression obtained for the propagator and discussed in Section 14.13.
14.7 Vacuum state; eQ/2 The vacuum state and the dual vacuum state of the Euclidean Hamiltonian are obtained using the expressions for eQ/2 and e−Q/2 respectively. The vacuum can be verified to be correct by the direct application of the Hamiltonians H and H † and hence provides an independent verification of the matrix elements obtained for eτ Q in Section 14.3. The vacuum state is given by 1 00 = eQ/2 0, 0, H 00 = E0 00 , E0 = [ω1 + ω2 ]. 2
(14.50)
The coordinate representation of the vacuum state can be directly obtained from the Hamiltonian H by inspection and is given by 00 (x, v) = x, v00 = x, veQ/2 0, 0 2 γ 3 γ = N00 exp − (ω1 + ω2 )ω1 ω2 x 2 − (ω1 + ω2 )v2 − γ ω1 ω2 xv . 2 2 (14.51) The vacuum state 00 is real valued and normalizable, with N00 the normalization constant. The dual vacuum state obeys D D D  = 0, 0e−Q/2 ⇒ 00 (x, v) = 00 x, v = 0, 0e−Q/2 x, v. 00
It can be directly verified that the dual ground state, from Eq. 14.51, is 2 γ 3 γ D 00 (x, v) = N00 exp − (ω1 + ω2 )ω1 ω2 x 2 − (ω1 + ω2 )v2 + γ ω1 ω2 xv . 2 2
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For τ = −1/2, Eq. 14.23 yields the matrix elements x, veQ/2 x , v ⎫ ⎧ #√ $ ⎨ ⎬ 1 a ab a (xv + vx ) √ . (xv + x v ) − = N ( ) exp coth ⎩ b 2 2 b sinh ab ⎭
(14.52)
2
Note that #√ $ 8 1 a ab a 2 √ = γ ω1 ω12 − ω22 , coth = γ ω1 , b 2 b sinh ab 2 8 1 a i 1 i √ = N( ) = γ ω1 ω12 − ω22 . 2 2π b sinh ab 2π 2
(14.53)
(14.54)
Hence, from Eq. 14.52 x, ve
Q/2
' 8 1 2 2 2 x , v = N ( ) exp γ ω1 (xv + x v ) − γ ω1 ω1 − ω2 (xv + vx ) . 2
From Eq. 14.27, the oscillator vacuum state is given by !1/4 2 γ 3 γ 2 ω13 ω2 2 2 2 exp − v + ω ω x ) . (ω x, v0, 0 = 1 2 1 π2 2 Hence
00 (x, v) = x, veQ/2 0, 0 = 1 γ 2 ω13 ω2 = N( ) 2 π
dξ dζ x, veQ/2 ξ, ζ ξ, ζ 0, 0
!1/4 dξ dζ eS ,
where γ S = − (ω1 ζ 2 + ω12 ω2 ξ 2 − 2c1 ξ ζ ) − γ c2 (ζ x + vξ ) + γ c1 xv, 2 8 c1 = ω12 ,
c2 = ω1 ω12 − ω22 .
Performing the Gaussian integrations over ξ and ζ yields 00 (x, v) = N00 eF , where γ 2 c22 [ω1 v2 + ω12 ω2 x 2 + 2c1 xv] + γ c1 xv 3 2 2γ (ω1 ω2 − c1 ) γ = − [(ω1 + ω2 )ω1 ω2 x 2 + (ω1 + ω2 )v2 ] − γ ω1 ω2 xv, 2 which is the expected result. F =
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To determine N00 note that the Gaussian integration has the matrix 9 2 ω1 ω2 −c1 3/2 √ , det(M) = iγ ω1 ω1 − ω2 . M=γ −c1 ω1 The determinant of matrix M is negative and hence leads to the correct sign for the exponent F of the vacuum state vector. Using Eq. 14.54, Gaussian integration yields !1/4 2π 1 γ 2 ω13 ω2 N00 = N ( ) √ 2 π det(M) !1/4 8 2 3 2π i 2 2 γ ω1 ω2 = γ ω1 ω1 − ω2 √ 3/2 2 2π π iγ ω1 ω1 − ω2 γ ⇒ N00 = (ω1 ω2 )1/4 (ω1 + ω2 ). π
(14.55)
To derive the dual vacuum state, note from Eq. 14.52 ' 8 1 −Q/2 2 2 2 x, ve x , v = N ( ) exp −γ ω1 (xv + x v ) + γ ω1 ω1 − ω2 (xv + vx ) . 2 A derivation similar to the one for the vacuum state 00 yields the dual vacuum state D (x, v) = 0, 0e−Q/2 x, v 00 γ γ = N00 exp{− (ω1 + ω2 )ω1 ω2 x 2 − (ω1 + ω2 )v2 + γ ω1 ω2 xv}. 2 2 (14.56)
The norm of a state is defined by the scalar product of the state with its dual and yields the following norm for the vacuum state: D D (x, v)00 (x, v) 00 00 = dxdv00 2 dxdv exp{−γ (ω1 + ω2 )ω1 ω2 x 2 − γ (ω1 + ω2 )v2 } = 1. = N00 Recall that Euclidean velocity is related to Minkowski velocity by v = dx/dτ = −idx/dtM = −ivM . Hence, the vacuum state 00 when analytically continued to Minkowski time has a divergent norm. Furthermore, the eigenstates generated by applying the creation operators on the vacuum state all have a divergent norm. The problem of rendering the Minkowski time state space convergent has been addressed by Bender and Mannheim (2008b).
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14.8 Vacuum state and classical action The definition of the probability amplitude given in Eq. 13.66, KS (xf , vf ; xi , vi ) = xf , vf e−τ H xi , vi , yields, from Eq. 14.44, the infinite τ limit given by lim KS (xf , vf ; xi , vi ) e−τ E0 xf , vf 00 00 e−Q xi , vi + O(e−τ (E1 −E0 ) )
τ →+∞
D (xi , vi ). = e−τ E0 00 (xf , vf )00
(14.57)
Hence, from Eq. 14.57, the infinite limit of transition amplitude is the product of D (xi , vi ) and the vacuum state 00 (xf , vf ). On the other the dual vacuum state 00 hand, the classical solution yields lim K(xf , vf , xi , vi ) = lim N eSc (xf ,vf ,xi ,vi ) .
τ →+∞
τ →+∞
(14.58)
The infinite τ limits of matrix elements M, given in Section 13.2, are given by lim M11 = 2ra(r 2 + ω2 ),
τ →+∞
lim M22 = 2ra,
τ →+∞
lim M12 = 2r 2 a − b,
τ →+∞
lim M13 = lim M14 = lim M23 = 0.
τ →+∞
τ →+∞
τ →+∞
(14.59)
Therefore, the vacuum state is obtained from Eqs. 13.28 and 14.59 and yields ! 1 1 2 2 (x, v) = lim N exp − M11 x − M22 v − M12 xv τ →+∞ 2 2 (14.60) = N exp −ra 2 (r 2 + ω2 )x 2 − rav2 − (2r 2 a − b)xv , where xf = x, vf = v, and N is fixed by normalizing (x, v). To make connection with the earlier parameterization, Let us rewrite the Lagrangian in ω1 and ω2 parameterization (the only difference from the parameterization given in Eq. 13.29 is in the overall factor of a instead of γ ) ! b 2 c 2 1 2 L = − a x¨ + 2 x˙ + x 2 a a γ 2 = − x¨ + (ω12 + ω22 )x˙ 2 + (ω12 ω22 )x 2 , (14.61) 2 where ω1 and ω2 are1 1 ω1 = √ 2γ
! 8 8 √ √ b + ac + b − ac ,
1 Note that ω and ω in Eq. 13.30 have a different parameterization, given by 1 2
1 ω1 = √ 2 γ
8
! 8 9 9 1 α + 2 γβ + α − 2 γβ , ω2 = √ 2 γ
8
! 8 9 9 α + 2 γβ − α − 2 γβ .
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1 ω2 = √ 2γ
! 8 8 √ √ b + ac − b − ac .
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The definition variables r and ω in terms of ω1 and ω2 are given by ω1 = r + iω, ω2 = r − iω. The two parameterizations have the relationship 1 γ r 2 + ω2 = ω12 ω22 , r = (ω1 + ω2 ), b = (ω12 ω22 ). (14.62) 2 2 Equations 14.60 and 14.62 hence yield – replacing a by γ to conform to the notation given in Eq. 13.30 – the vacuum state given in Eq. 14.51, namely γ γ 00 (x, v) = N exp − (ω1 + ω2 )ω1 ω2 x 2 − (ω1 + ω2 )v2 − γ ω1 ω2 xv . 2 2 The definition of the evolution kernel KS (xf , vf ; xi , vi ) = xf , vf e−τ H xi , vi given in Eq. 13.66 is seen to be correct since the evolution kernel obtained from the classical solution also gives the same result for the vacuum state as given by the Hamiltonian in Eq. 14.51. In particular, there is no need to include the state space metric e−Q in the definition of KS (xf , vf ; xi , vi ), since the expression xf, vf e−τ H xi, vi for the probability amplitude is the matrix elements in a complete basis, and not the probability amplitude between physical states. However, in obtaining the vacuum state the operator e−Q appears, since it defines the dual of the vacuum state 00 via the expression 00 e−Q . 14.9 Excited states of H To illustrate the general features of the eigenfunctions, the first few eigenfunctions are evaluated. Note ! 1 ∂ γ ω1 † v− , (14.63) av = 2 γ ω1 ∂v 7 ! 1 γ ω12 ω2 ∂ † x− . (14.64) ax = 2 γ ω12 ω2 ∂x In general, one can find the explicit coordinate representation of any eigenfunction by the following procedure: †n †m ' a a Q/2 nm (x, v) = x, ve 0, 0 √v √x n! m! a †n a †m = x, veQ/2 √v √x e−Q/2 eQ/2 0, 0 n! m!
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a †n a †m = eQ/2 √v √x e−Q/2 00 (x, v), n! m!
(14.65)
where, using Eqs. 14.69 and 14.76 given below, one can explicitly evaluate n m eQ/2 av†n ax†m e−Q/2 = eQ/2 av† e−Q/2 eQ/2 ax† e−Q/2 . (14.66) 14.9.1 Energy ω1 eigenstate 10 (x, v) The ω1 single excitation energy eigenstate is the following: 3 2 10 (x, v) = x, veQ/2 av† 0, 0 = x, veQ/2 av† e−Q/2 eQ/2 0, 0 ⇒ 10 (x, v) = eQ/2 av† e−Q/2 00 (x, v).
(14.67)
The fundamental similarity transformation given in Eq. 14.8 yields eQ/2 ve−Q/2 = Av −
B ∂ , C ∂x
∂ −Q/2 ∂ e = A − BCx, ∂v ∂v where, from Eq. 14.11 the coefficient functions are given by ω2 ω1 , B=8 , C = γ ω 1 ω2 . A= 8 ω12 − ω22 ω12 − ω22 eQ/2
Hence, from Eq. 14.63 1 ∂ γ ω1 Q/2 Q/2 † −Q/2 e−Q/2 e av e = e v− 2 γ ω1 ∂v BC γ ω1 B ∂ A ∂ Av + = x− − 2 γ ω1 C ∂x γ ω1 ∂v γ ω ∂ 1 1 ∂ 1 Q/2 † −Q/2 2 . ⇒ e av e = ω1 v + ω2 x − − γ ω1 ∂x γ ∂v 2(ω12 − ω22 )
(14.68)
(14.69)
Equations 14.67 and 14.69 yield γ ω1 1 ∂ 1 ∂ 2 ω1 v + ω2 x − 00 (x, v). 10 (x, v) = − γ ω1 ∂x γ ∂v 2(ω12 − ω22 ) Using the explicit representation of the vacuum state 00 (x, v) given in Eq. 14.50 yields the final result 7 2γ ω1 (ω1 + ω2 ) [v + ω2 x] 00 (x, v). (14.70) 10 (x, v) = ω12 − ω22
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The dual energy eigenstate is defined by D 10  = 1, 0e−Q/2 = 0, 0av e−Q/2 D ⇒ 10 = e−Q/2 av† 0, 0 = e−Q/2 av† eQ/2 e−Q/2 0, 0 D D D (x, v) = x, v10 = e−Q/2 av† eQ/2 e−Q/2 00 (x, v). ⇒ 10
Similarly to the derivation of Eq. 14.69, Eq. 14.8 yields γ ω1 1 ∂ 1 ∂ −Q/2 † Q/2 2 ω1 v − ω2 x + , e av e = − γ ω1 ∂x γ ∂v 2(ω12 − ω22 )
(14.71)
(14.72)
and, from Eq. 14.71, we obtain the dual eigenfunction γ ω1 1 ∂ 1 ∂ D 2 D ω1 v − ω2 x + 00 10 (x, v) = (x, v). − γ ω1 ∂x γ ∂v 2(ω12 − ω22 ) D (x, v) given in Eq. 14.56 yields Using the representation of the vacuum state 00 the final result 7 2γ ω1 D D (x, v) = (ω1 + ω2 ) [v − ω2 x] 00 (x, v). (14.73) 10 ω12 − ω22
14.9.2 Energy ω2 eigenstate 01 (x, v) The ω2 one excitation energy eigenstate is given by 3 2 01 (x, v) = x, veQ/2 ax† 0, 0 = x, veQ/2 ax† e−Q/2 eQ/2 0, 0 ⇒ 01 (x, v) = eQ/2 ax† e−Q/2 00 (x, v).
(14.74)
The similarity transformation given in Eq. 14.8 yields eQ/2 xe−Q/2 = Ax − eQ/2
B ∂ , C ∂v
∂ −Q/2 ∂ =A e − BCv. ∂x ∂x
(14.75)
Hence, from Eq. 14.64 7 1 γ ω12 ω2 Q/2 ∂ Q/2 † −Q/2 e−Q/2 x− = e e ax e 2 γ ω12 ω2 ∂x 7 BC γ ω12 ω2 ∂ B ∂ A Ax + = v− − 2 C ∂v γ ω12 ω2 ∂x γ ω12 ω2 γ ω2 1 ∂ 1 ∂ 2 ⇒ eQ/2 ax† e−Q/2 = . (14.76) v + ω x − ω − 2 1 γ ω2 ∂x γ ∂v 2(ω12 − ω22 )
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Hence, from Eqs. 14.74 and 14.76 γ ω2 1 ∂ 1 ∂ 2 ω2 v + ω1 x − 00 (x, v). 01 (x, v) = − γ ω2 ∂x γ ∂v 2(ω12 − ω22 ) Using the representation of the vacuum state 00 (x, v) given in Eq. 14.50 yields the final result 7 2γ ω2 01 (x, v) = (ω1 + ω2 ) [v + ω1 x] 00 (x, v). (14.77) ω12 − ω22 The dual energy eigenstate, similarly to Eq. 14.78, is defined by D D 01 (x, v) = e−Q/2 av† eQ/2 e−Q/2 00 (x, v).
Equations 14.64 and 14.8 yield γ ω2 1 ∂ 1 ∂ −Q/2 † Q/2 2 −ω2 v + ω1 x − . e ax e = + γ ω2 ∂x γ ∂v 2(ω12 − ω22 )
(14.78)
(14.79)
D Using the explicit representation of the vacuum state 00 (x, v) given in Eq. 14.56 with Eq. 14.78 yields the dual eigenfunction 7 2γ ω2 D D (ω1 + ω2 ) [−v + ω1 x] 00 (x, v). (14.80) 01 (x, v) = 2 2 ω1 − ω2
We collect the results for the first two one excitation states; the normalization constants are separated out for later convenience; using the value of N00 given in Eq. 14.55 yields D D (x, v) = N10 [v − ω2 x] ψ00 (x, v), 10 (x, v) = N10 [v + ω2 x] ψ00 (x, v), 10 D D (x, v) = N01 [−v + ω1 x] ψ00 (x, v), 01 (x, v) = N01 [v + ω1 x] ψ00 (x, v), 01 √ (ω1 + ω2 ) √ (ω1 + ω2 ) 3/4 1/4 1/4 3/4 N10 = γ 2 √ ω1 ω2 , N01 = γ 2 √ ω1 ω2 , π(ω1 − ω2 ) π(ω1 − ω2 ) E10 = ω1 + E00 , E01 = ω2 + E00 . (14.81)
The eigenstates are orthogonal and normalized, namely D D D 10 = 1 = 01 01 , 10 01 = 0. 10
Note the remarkable result that under a duality transformation, the dual eigenstates have a transformation that depends on the eigenstate, as discussed in Eq. 14.45; in particular D D (x, v) = 10 (−x, v), 01 (x, v) = 01 (x, −v). 10
(14.82)
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14.10 Complex ω1 , ω2
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This feature generalizes to all the energy eigenstates and guarantees that the state space, for ω1 > ω2 , always has a positive norm. The first two energy eigenstates of the pseudoHermitian Hamiltonian H are the first excitation of the position degree of freedom x, given by 01 , and the velocity degree of freedom v, given by 10 . Note that in the limit of ω1 → ω2 , the eigenstates 01 and 10 become degenerate. This important property of the energy eigenspectrum will be studied in some detail when the limit of ω1 → ω2 is taken in Section 15.1. 14.10 Complex ω1 , ω2 The application of the acceleration Lagrangian to the study of equities, as shown by Baaquie et al. (2012), requires that the parameters ω1 , ω2 are complex. From Eq. 13.31, consider the parameterization given by ω1 = Reiφ , ω2 = Re−iφ : R > 0, φ ∈ [−π/2, π/2], ⇒ ω1 + ω2 = 2R cos(φ), ω1 ω2 = R . 2
(14.83) (14.84)
The eigenenergies are given by 1 E00 = (ω1 + ω2 ) = R cos φ, 2 Emn = mω1 + nω2 + E00 = mReiφ + nRe−iφ + R cos φ. As discussed in Section 14.2, the eigenenergies of a pseudoHermitian Hamiltonian come in conjugate pairs and are given by Emn = mReiφ + nRe−iφ + R cos φ, ∗ . Enm = nReiφ + mRe−iφ + R cos φ = Emn
In other words, energy eigenvalues Emn and Enm are complex conjugates of each other. From Eq. 14.10, the parameters are given by √ a −i 2ω1 ω2 = = γ ω1 ω2 = γ R 2 , sinh( ab) = 2 . 2 b cos(2φ) ω1 − ω2 Since a, b are now complex, the Qoperator is given from Eq. 14.5, Q = axv − b
∂2 ∂x∂v
is no longer Hermitian. Since the eigenenergies are complex, there is no sense of the ordering of energies and the concept of a ground state is no longer valid. One can still consider the
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real part of the eigenenergies and the lowest real energy E00 has the following eigenstate given by the continuation of the vacuum state (14.85) 00 (x, v) = N00 exp −γ R 3 cos(φ)x 2 − γ R cos(φ)v2 − γ R 2 xv . Equation 14.85 yields a positive norm state that is well defined for cos(φ) > 0 ⇒
− π/2 < φ < π/2.
Since ω1 , ω2 are complex, the algebra of the creation and destruction operators given in Eq. 14.30 is no longer valid. Hence, for the complex branch of the parameters, these operators can no longer generate the spectrum of states for H .
14.11 State space V of Euclidean Hamiltonian The state space of the nonHermitian Hamiltonian H , namely V, has a nontrivial metrical structure. This is most clearly seen in the manner in which the dual state vectors are defined as well as the rules of conjugation for operators, called Qconjugation, that is different from the usual Hermitian conjugation. To define orthonormality of the eigenfunctions nm , one needs to define the scalar product of a state vector  with a dual vector χ , that are elements of V. In particular, one needs to define a dual that is consistent with the similarity transformation given in Eq. 14.12 that maps the nonHermitian Hamiltonian H to the Hermitian Hamiltonian HO . Figure 14.1 schematically shows the mapping from the state space of HO denoted by V0 to the state space of H denoted by VH . V0
VH
e+Q
Figure 14.1 A mapping between two state spaces.
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A consistent quantum mechanics can be defined by generalizing the scalar product by using a metric W on state space in the following manner [Ballentine (1998)]: χ W = χ W , W † = W.
(14.86)
In analogy with the result for Minkowski time [Mannheim and Davidson (2005)], the metric for the state space V is given by W † = W = e−Q ⇒ χ Q = χ e−Q  : scalar product.
(14.87)
Hence two state vectors in V are orthogonal if χ e−Q  = 0 : orthogonal.
(14.88)
The norm of a state vector, as exemplified by Eq. 14.38, is given by 2Q = e−Q .
(14.89)
The definition of the scalar product given in Eq. 14.87 is equivalent to defining the dual vector of  by the rule  → Dual → e−Q : dual vector.
(14.90)
In particular, using the definition of the scalar product given in Eqs. 14.87 and 14.90, it follows that all the eigenfunctions nm are orthonormal as shown in Eq. 14.39. The completeness equation for state space V is +∞ I= dxdveQ/2 x, vx, ve−Q/2 −∞ +∞ = dxdveQ/2 x, v x, veQ/2 e−Q , −∞
with completeness and orthogonality obtained using Eq. 13.65 14.11.1 Operators acting on V The conjugation of operators in state space V depends on the state space metric e−Q . Qconjugation for operator O is defined by χ e−Q O∗ = O† e−Q χ = e−Q [eQ O† e−Q ]χ = e−Q OQ† χ, where the Qconjugate of operator O is defined by OQ† = eQ O† e−Q and O† is the usual Hermitian conjugation.
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All operators in state space V must be Qconjugated under conjugation. In particular, a Qselfconjugate operator is given by OQ† = eQ O† e−Q = O ⇒ χ e−Q O∗ = e−Q Oχ : Qselfconjugate operator.
(14.91)
A Qselfconjugate operator has real eigenvalues since, from Eq. 14.91, for a normalized eigenfunction  E = O, OQ† = O ⇒ Ee−Q  = E = e−Q O, E ∗ = e−Q O∗ = e−Q OQ†  = e−Q O = E ⇒ E : real. An obvious but important result is that the state space metric e−Q is Qconjugate, namely that eQ−Q = eQ/2 e−Q e−Q/2 = e−Q † ⇒ eQ−Q = e−Q/2 e−Q eQ/2 = e−Q : Qselfconjugate operator. The pseudoHermitian Hamiltonian H is Qselfconjugate; from Eq. 14.12 H = eQ/2 HO e−Q/2 ⇒ HQ† = eQ H † e−Q = eQ e−Q/2 HO† eQ/2 e−Q = eQ/2 HO e−Q/2 = H : Qselfconjugate operator.
(14.92) (14.93)
All operators acting on state space V need to be Qconjugated to obtain the conjugated form of any operator equation. The presence of the operator Q indicates that the operators and state vectors belong to state space V. Examples of operator equations that are only valid on state space V are given in Eq. 14.8. It can be seen that taking the Hermitian conjugate of any the equations in Eq. 14.8 leads to an inconsistency, whereas taking the Qconjugate of these same equations leads to a consistent result. Note that all the properties of the state space V depend on the operator Q being well behaved. From Eq. 14.10 it can be seen that for ω1 = ω2 , the operator Q diverges and hence the Hamiltonian and state space need to be studied from first principles. To illustrate the role of the Hilbert space metric e−Q consider the timeordered vacuum expectation value of the Heisenberg operators at two different (Euclidean) times τ > 0. The vacuum state is given by 00 with H 00 = E00 00 ;
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E00 ≡ E0 = (ω1 + ω2 )/2; using the rule for forming the dual vector of the pseudoHermitian Hamiltonian yields D xH (τ )xH (0)00 = 00 e−Q xH (τ )xH (0)00 G(τ ) = 00
= 00 e−Q eτ H xe−τ H x00 = 00 eτ H e−Q xe−τ H x00 †
= 00 e−Q xe−τ (H −E0 ) x00 .
(14.94)
The term G(τ ) in Eq. 14.94 is the propagator and is analyzed in detail in Section 14.12. Recall that the probability amplitude is given in Eq. 13.66, KS (x, v; x , v ) = x, ve−τ H x , v , and the matrix elements of the the Hilbert space metric e−Q are given in Eq. 14.23, x, ve−τ Q x , v = N exp −G(τ )(xv + x v ) + H(τ )(xv + vx ) . (14.95) For both the operators e−τ H and e−τ Q , there is no need for an extra metric e−τ Q since Eqs. 13.66 and 14.23 are the matrix elements of the operators in a complete basis x, v and its dual x , v . If the matrix elements of operators e−τ H and so on are determined for eigenstates and dual eigenstates of H , then the metric e−Q is required. using the basis One can evaluate the matrix elements of e−τ H in Hilbert space, states of V, namely eQ/2 x, v and the dual basis x, veQ/2 e−Q = x, ve−Q/2 , which yields x, ve−Q/2 e−τ H eQ/2 x , v = x, ve−τ H0 x , v .
(14.96)
The symmetry of the matrix elements of x, ve−Q/2 e−τ H eQ/2 x , v – given above in Eq. 14.96 – is not the symmetry given in Eq. 13.91 for x, ve−τ H x , v ; hence the correct expression for the kernel is given by KS (x, v; x , v ) = x, ve−τ H x , v , which in turn yields the correct matrix elements given by the path integral as seen in the results obtained in Section 14.8.
14.11.2 Heisenberg operator equations In Schrödinger’s formulation of quantum mechanics, the time dependence of a quantum system arises solely due to the time evolution of the state vector (t) – with the operators O being taken to be timeindependent. For a pseudoHermitian Hamiltonian, the Schrödinger equation yields, for Minkowski time tM , (t) = exp{−itM H }, χ (t) = χ exp{itM H † }.
(14.97)
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For Euclidean time t = itM , one has the expressions (t) = e−tH , χ (t) = χ  etH . †
To obtain the Heisenberg operator equations of motion for the pseudoHermitian Hamiltonian, it is necessary to define the dual state vectors using Qconjugation. In particular, for Euclidean time, the time dependent expectation value, using Eq. 14.4, yields the result E[O; t] = χ (t)e−Q O(t) = χ etH e−Q Oe−tH  †
= χ e−Q etH Oe−tH  = χe−Q OH (t).
(14.98)
Hence, from Eq. 14.98, due to the choice of the Hilbert space metric e−Q , the Heisenberg timedependent operator OH (t) for the pseudoHermitian Hamiltonian is given by the same expression as for the Hermitian Hamiltonian, namely ∂OH (t) = [H, OH (t)]. ∂t In particular, for the Hamiltonian of the acceleration action given, from Eq. 13.69, by OH (t) = etH Oe−tH ⇒
H =−
1 ∂2 ∂ 1 −v + αv2 + (x), 2γ ∂v2 ∂x 2
the Heisenberg equation for the coordinate operator xH (t), with x being the Schrödinger coordinate operator, yields xH (t) = etH xe−tH , ∂xH (t) (14.99) x˙H (t) = = [H, OH (t)] = −vH (t). ∂t Hence, the identification made in the path integral derivation, namely x(t) ˙ = −v(t), is seen to hold also as an operator equation for the Heisenberg operators x˙H (t), vH (t), as shown by Eq. 14.99. 14.12 Propagator: operators Constructing the propagator by inserting the complete set of states yields a realization of the propagator in terms of the state space and Hamiltonian. The state space definition of the propagator is given by Eq. 11.6 and yields 1 G(τ ) = lim tr e−(T −τ )H xe−τ H x , τ = t − t . T →∞ Z Note that lim e−T H e−T E0 00 00 e−Q = e−T E0 eQ/2 0, 00, 0e−Q/2 ,
T →∞
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Z = tr(e−T H ) = e−T E0 . Since H = eQ/2 HO e−Q/2 , 00 = eQ/2 0, 0, 00  = 0, 0eQ/2 , the propagator is given by 1 −(T −τ )H −τ H xe x , τ = t − t , tr e T →∞ Z = 00 e−Q xe−τ (H −E0 ) x00 ,
G(τ ) = lim
= 0, 0e
−Q/2
xe
Q/2 −τ (H0 −E0 ) −Q/2
e
e
xe
Q/2
0, 0.
(14.100) (14.101)
Note that Eq. 14.100 has been obtained earlier in Eq. 14.94 based on a Hilbert space derivation of the propagator. From Eq. 14.8, ∂ e−Q/2 xeQ/2 = Ax + BC ∂v 7 1 γ ω1 † (ax + ax ) + BC (av − av† ). =A 2 2 2γ ω1 ω2
(14.102)
Hence, from Eq. 14.102 7
1 γ ω1 e xe 0, 0 = A 0, 1 − BC 1, 0, 2 2γ ω12 ω2 7 1 γ ω1 −Q/2 Q/2 xe =A 0, 1 + BC 1, 0. 0, 0e 2 2 2γ ω1 ω2 −Q/2
Q/2
(14.103)
Equation 14.101 yields 1 γ ω1 0, 1e−τ (H0 −E0 ) 0, 1 − (BC)2 1, 0e−τ (H0 −E0 ) 1, 0 2 2 2γ ω1 ω2 γ ω1 1 A2 e−ω2 τ − = (BC)2 e−ω1 τ . 2 2 2γ ω1 ω2
G(τ ) = A2
Note that all the operators and state functions in the equation above are defined solely in state space V0 . However, the coefficients of the various matrix elements, in particular the negative sign on the second matrix element, are a result of the properties of the conjugation operator eQ and reflect the presence of the underlying state space V; the result obtained could not have been generated by working solely in Hilbert space V0 .
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Equation 14.11 yields #√
$ ω2 ab = 2 1 2, A = cosh 2 ω1 − ω2 #√ $ 1 ab b ω2 1 1 = 2 2 2· 2 2 2 = 2 2· 2 (BC)2 = sinh2 . a 2 γ ω1 ω2 ω1 − ω2 γ ω1 ω1 − ω22 2
2
Hence, collecting all the results yields the expected result that −ω2 τ 1 e e−ω1 τ 1 . − G(τ ) = 2γ ω12 − ω22 ω2 ω1
(14.104)
14.13 Propagator: state space Recall from Eq. 14.100, that the propagator is given by G(τ ) = 00 e−Q xe−τ (H −E0 ) x00 D xe−τ (H −E0 ) x00 . = 00
(14.105)
The completeness equation for H , from Eq. 14.40, is given by ∞
I=
D mn mn ,
(14.106)
m,n=1
and yields, from Eq. 14.105, G(τ ) =
∞
D D 00 xe−τ (H −E0 ) mn mn x00
m,n=1 D D D D = e−τ ω1 00 x10 10 x00 + e−τ ω2 00 x01 01 x00 , (14.107)
= e−τ ω1 G1 + e−τ ω2 G2 .
(14.108)
The vacuum state and its normalization, from Eqs. 14.50, 14.55, and 14.56 is D (x, v) = 00 (x, −v) = 00 (−x, v), 00 (x, v) = N00 ψ00 (x, v), 00 γ γ ψ00 (x, v) = exp{− (ω1 + ω2 )ω1 ω2 x 2 − (ω1 + ω2 )v2 − γ ω1 ω2 xv}, 2 2 γ N00 = (ω1 ω2 )1/4 (ω1 + ω2 ). π
Recall from Eq. 14.81, D D (x, v) = N10 [v − ω2 x] ψ00 (x, v), 10 (x, v) = N10 [v + ω2 x] ψ00 (x, v), 10 D D (x, v) = N01 [−v + ω1 x] ψ00 (x, v), 01 (x, v) = N01 [v + ω1 x] ψ00 (x, v), 01
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N10 = γ
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√ (ω1 + ω2 ) √ (ω1 + ω2 ) 3/4 1/4 1/4 3/4 2√ ω1 ω2 , N01 = γ 2 √ ω1 ω2 . π(ω1 − ω2 ) π(ω1 − ω2 )
Using the coordinate representation for the state vectors yields D D x10 10 x00 G1 = 00 2 2 D dxdv x(v + ω2 x)ψ00 = N10 N00 (x, v)ψ00 (x, v) D × dxdv x(v − ω2 x)ψ00 (x, v)ψ00 (x, v) 2 π 2 2 2 = −N10 N00 ω2 2γ (ω1 + ω2 )2 (ω1 ω2 )3/2 1 =− . 2 2γ (ω1 − ω22 )2 ω1
(14.109)
Similarly D D x01 01 x00 G2 = 00 2 2 D dxdv x(v + ω1 x)ψ00 = N10 N00 (x, v)ψ00 (x, v) D (x, v)ψ00 (x, v) × dxdv x(−v + ω1 x)ψ00 2 π 2 2 2 = N10 N00 ω1 2γ (ω1 + ω2 )2 (ω1 ω2 )3/2 1 = . 2 2γ (ω1 − ω22 )2 ω2
(14.110)
Hence, Eqs. 14.108, 14.109, and 14.110 yield the expected result given in Eq. 14.104, namely that G(τ ) =
∞
D D 00 xe−τ (H −E0 ) mn mn x00
m,n=1 D D D D x10 10 x00 + e−τ ω2 00 x01 01 x00 = e−τ ω1 00
= e−τ ω1 G1 + e−τ ω2 G2 −ω1 τ 1 1 e−ω2 τ e = + − . 2γ ω12 − ω22 ω1 ω2
(14.111)
There are a number of remarkable features of the state space derivation. The negative sign that appears in the propagator for the term G1 is usually taken to be a proof that no unitary theory can yield this result. The reason for this is the following:
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consider any arbitrary Hermitian Hamiltonian such that HA = HA† ; the spectral resolution of this Hamiltonian in terms of its eigenstates χmn is given by I=
∞
χmn χmn .
(14.112)
mn=1 D , since for a Hermitian Hamiltonian HA the left and right Note that χmn  = χmn eigenstates are complex conjugates of each other. Hence, the propagator for the Hermitian Hamiltonian HA is given by
GA (τ ) =
∞
D D χ00 xe−τ (HA −E0 ) χmn χmn xχ00
m,n=1 D D D D = e−τ ω1 χ00 xχ10 χ10 xχ00 + e−τ ω2 χ00 xχ01 χ01 xχ00 % % % % 2 2 = e−τ ω1 %χ00 xχ10 % + e−τ ω2 %χ00 xχ01 % .
The result above shows that a Hermitian Hamiltonian defined on a Hilbert space cannot have %a propagator such as the one given in Eq. 14.104, except by allowing % %χ00 xχ10 %2 < 0, which implies that χ10 is a ghost state that has a negative norm. In contrast, the pseudoHermitian Euclidean Hamiltonian H has a positive norm for all the states in its state space; the duality transformation in going from mn to D  provides the negative signs that allow for the propagator given in Eqs. 14.104 mn and 14.111.
14.14 Many degrees of freedom Consider the generalization of a Hamiltonian with a quadratic potential given in Eq. 13.69 to many degrees of freedom. For degrees of freedom xn , n = 1, 2, . . . N the acceleration action, in matrix notation, is chosen to be H =−
1 1 ∂ T1 ∂ ∂ 1 S S −v + vS T αSv + xS T βSx, 2 ∂v γ ∂v ∂x 2 2
(14.113)
that yields the Lagrangian L=−
1 T xS ¨ γ S x¨ + xS ˙ T αS x˙ + xS T βSx . 2
(14.114)
A very special choice has been made for H and L in that all the couplings have the same similarity transformation S connecting the different degrees of freedom. In particular, choosing S = I would decouple all the different degrees of freedom.
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14.14 Many degrees of freedom
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The (real) orthogonal matrix S and the diagonal matrices are given in matrix notation as SS T = I, γ = diag(γ1 , γ2 , . . . , γN ), α = diag(α1 , α2 , . . . , αN ), β = diag(β1 , β2 , . . . , βN ).
(14.115)
Let us define new variables, in matrix notation x = S T z, v = S T u.
(14.116)
The Hamiltonian in Eq. 14.113 is given by ∂ 1 1 1 1 ∂2 2 2 − u + α u + β n zm . n n n 2 n=1 γn ∂u2n n=1 ∂zn 2 n=1 2 n=1 N
H =−
N
N
N
(14.117)
The many degrees of freedom Lagrangian, from Eq. 14.114 and similar to Eq. 13.29, is given by 1 2 2 2 2 2 2 γn z¨ n + (ω1n + ω2n )˙zn2 + ω1n ω2n zn . L=− 2 n=1 N
The parameterization is a generalization of Eq. 13.30 and is given by ! 8 8 9 9 1 αn + 2 γn βn + αn − 2 γn βn , ω1n = √ 2 γn ! 8 8 9 9 1 ω2n = √ αn + 2 γn βn − αn − 2 γn βn , 2 γn
(14.118)
(14.119) (14.120)
ω1n > ω2n for ω1n , ω2n real. The Hamiltonian H given in Eq. 14.113 is diagonalized by the following generalization of the Qoperator given in Eq. 14.5: Q=
N
(14.121)
Qn ,
n=1
Qn = an zn un − bn
∂2 , ∂zn ∂un
with the following values for an and bn : ! 9 an ω1n + ω2n . = γn ω1n ω2n , an bn = ln bn ω1n − ω2n
(14.122)
(14.123)
The diagonal Hamiltonian HO is given similarly to the earlier case. The ground state is given by generalizing Eq. 14.51 and yields
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PseudoHermitian Euclidean Hamiltonian
00 (x, v) = N
2 γ 3 γn n exp − (ω1n + ω2n )ω1n ω2n zn2 − (ω1n + ω2n )u2n − γn ω1n ω2n zn un . 2 2 n=1
N &
In terms of the original coordinates, D 6 N 1 Pmn xm xn + Qmn vm vn + 2Rmn xn vn , 00 (x, v) = N exp − 2 m,n=1 (14.124) where, again in matrix notation P = SpS T , pn = γn (ω1n + ω2n )ω1n ω2n , Q = SqS T , qn = γn (ω1n + ω2n ), R = SrS T , rn = γn ω1n ω2n .
14.15 Summary The Euclidean acceleration Hamiltonian, for the real branches, was shown to be pseudoHermitian and was mapped to a Hermitian Hamiltonian, which consists of two decoupled harmonic oscillators. The similarity transformation Q was shown to be an unbounded differential operator and the matrix elements of e±τ Q were exactly evaluated. All the defining commutation equations were evaluated using the matrix elements of e±τ Q to confirm the result obtained. The state space of the pseudoHermitian Euclidean Hamiltonian has a state space metric that is a natural generalization of the state space of quantum mechanics. The Heisenberg operator equations were analyzed to conclude that a state space metric is required for making the theory consistent and leads to a generalized scalar product and to the concept of Qconjugation discussed by Mostafazadeh (2002). The state vector and its dual vector were analyzed. It was shown that the asymmetry of the evolution kernel obtained using the Lagrangian and classical action can be derived using the parity symmetry of the Hamiltonian and duality property of the state functions. The propagator was evaluated using the algebra of the creation and destruction operators. The first two excited states were computed and the propagator was evaluated a second time using the properties of state space. It was seen that the propagator has a form that is forbidden for a Hermitian Hamiltonian and exists for the acceleration Hamiltonian due to the properties of Qconjugation required for a positive norm state space.
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As was seen in many of the derivations in this chapter, the results from Minkowski time serve as a useful guide for the derivations; but – given a plethora of i and various + and − signs that differ between the Euclidean and Minkowski results – all the derivations for the Euclidean have to be done from first principles and independently from the Minkowski case.
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15 NonHermitian Hamiltonian: Jordan blocks
The Euclidean action with acceleration has been analyzed in Chapters 13 and 14 for its path integral and Hamiltonian. In this chapter, the acceleration Hamiltonian is analyzed for the case when it is essentially inequivalent to a Hermitian Hamiltonian: for critical values of the parameters, given by ω1 = ω2 and shown in Figure 13.1, the mapping of the Hamiltonian to an equivalent Hermitian Hamiltonian becomes singular. The Hamiltonian continues to be pseudoHermitian but is no longer equivalent to a Hermitian Hamiltonian. The Hamiltonian for real ω1 > ω2 is pseudoHermitian as well as being equivalent to a Hermitian operator H0 = H0† , due to the existence of a similarity transformation Q such that H = e−Q/2 H0 eQ/2 ⇒ H † = e−Q H eQ , ω1 > ω2 . For the case ω1 = ω2 the coefficients in Q diverge and the Hamiltonian H can no longer be mapped to a Hermitian Hamiltonian. The Hamiltonian is essentially nonHermitian and is shown in this chapter to be equal to a direct sum of block diagonal matrices, with each block being a Jordan block matrix. The Jordan block itself can be shown to be pseudoHermitian, but the transformation cannot be obtained from the diverging Q. The case of the Hamiltonian being a Jordan block is analyzed in detail in this chapter as it forms a quantum mechanical system with its own unique features that is not encountered in Hermitian quantum mechanics. In Section 15.1 the equal frequency Hamiltonian is obtained; in Sections 15.2 and 15.3 propagator is analyzed and the two lowest lying state vectors for the singular case are evaluated. In Section 15.5 the equal frequency propagator is evaluated using the state vectors of the Jordan block. In Sections 15.6 and 15.7 the Hamiltonian for the Jordan block is derived and the Schrödinger equation for this system is studied.
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15.1 Hamiltonian: equal frequency limit The critical Hamiltonian is given for the equal frequency limit. In the equal fre√ quency limit of ω1 = ω2 (α = 2 βγ ) the parameters of the Qoperator, namely a and b, given in Eq. 14.10 become divergent and a well defined Qoperator no longer exists. Although the Hamiltonian has special properties for the equalfrequency point, the path integral is well behaved for all (complex and real) values of ω1 , ω2 , including the equalfrequency critical point at ω1 = ω2 . Moreover, the nonHermitian Hamiltonian H remains well defined at the critical point. The singularity for the Qoperator is due to the fact that the acceleration Hamiltonian H cannot be mapped to an equivalent Hermitian Hamiltonian H0 . For the case of ω1 = ω2 , nonHermitian Hamiltonian H is no longer pseudoHermitian but instead, H is essentially nonHermitian and has been shown to be expressible as a Jordanblock matrix by Bender and Mannheim (2008a). The general analysis of the equal frequency Hamiltonian has been carried out for Minkowski time in the pioneering work of Bender and Mannheim (2008a) and the analysis for Euclidean time is similar to their analysis, but with many details that are quite different.
15.2 Propagator and states for equal frequency To illustrate the general features of the equal frequency limit, the propagator is analyzed from the point of view of the underlying state space. As mentioned at the end of Section 14.9, in the limit of ω1 = ω2 the single excitation eigenstates 10 , 01 become degenerate, with both eigenstates having energy 2ω. The purpose of analyzing the propagator is to extract the state vectors that emerge in the limit of ω1 → ω2 . Since ω1 > ω2 , consider the limit of → 0+ with ω1 = ω + , ω2 = ω2 − ,
(15.1)
which yields, from Eq. 14.32 1 E00 = (ω1 + ω2 ) → ω, E10 → 2ω + , E01 → 2ω − . 2 Consider the limit of ω1 → ω2 for the state vector expansion of the propagator given by Eq. 14.107, D D D D x10 10 x00 + e−τ ω2 00 x01 01 x00 G(τ ) = e−τ ω1 00
= e−τ ω [G10 + G01 ], " " where, defining dxdvdx dv = x,v,x ,v yields
(15.2)
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NonHermitian Hamiltonian: Jordan blocks D D G10 = e−τ 00 x10 10 x00 2 2 D = N00 N10 x(v + ω2 x)ψ00 (x, v)ψ00 (x, v)
× x (v =
2 2 N00 N10
x,v,x ,v D − ω2 x )ψ00 (x , v )ψ00 (x , v )
xx (v + ω2 x)(v − ω2 x )P (x, v)P (x , v )
x,v,x ,v
(15.3)
and D D x01 01 x00 G01 = eτ 00 2 2 D = N00 N01 x(v + ω1 x)ψ00 (x, v)ψ00 (x, v) x,v,x ,v
D (x , v )ψ00 (x , v ) × x (−v + ω1 x )ψ00 2 2 = N00 N01 xx (v + ω1 x) (−v + ω1 x )P (x, v)P (x , v ). x,v,x ,v
(15.4)
In the limit of ω1 → ω2 the vacuum state has the welldefined limit given by lim ψ00 (x, v) = ψˆ 00 (x, v) = exp{−γ ω3 x 2 − γ ωv2 − γ ω2 xv}
→0
(15.5)
and yields D (x, v)ψˆ 00 (x, v) = exp{−2γ ω3 x 2 − 2γ ωv2 }. P (x, v) = ψˆ 00
To leading order in the normalization constants yield the following:1 ω1 ω2 4γ 2 ω3 2 →C , C= , N10 , π
(15.6)
2γ ω2 2 2 = lim N00 = . Nˆ 00 →0 π
(15.7)
2 N10 →C
and from Eq. 14.55
Hence, collecting the above equations, 2 xx F(x, v; x , v )P (x, v)P (x , v ), G10 + G01 = C Nˆ 00 x,v,x ,v
(15.8)
where F(x, v; x , v ) = e−τ
ω1 ω2 (v + ω2 x)(v − ω2 x ) + eτ (v + ω1 x)(−v + ω1 x ). (15.9)
1 The definition of the constant C in this chapter is different from the constant with the same notation used in
Chapter 14.
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Expanding F(x, v; x , v ) to leading order in yields 1 F(x, v; x , v ) = (1 − τ )(ω + ){v + (ω − )x}{v − (ω − )x }
+ (1 + τ )(ω − ){v + (ω + )x}{−v + (ω + )x } + O() = 2 (ωτ − 1)(v + ωx)(−v + ωx ) + ωx(−v + ωx ) + ω(v + ωx)x + O() = 2 − (v + ωx)(−v + ωx ) + ω{x + τ (v + ωx)}(−v + ωx ) (15.10) + ω(v + ωx)x .
The expression for F(x, v; x , v ) given in Eq. 15.10 carries information on the state vectors that determine the propagator; to extract this information, the state vectors need to read off from the equation. We recall that the state vectors and their duals are polynomials of x, v multiplied into the vacuum state. Hence, Eqs. 15.2, 15.8, and 15.10 yield the following [the factor of e−2ωτ has been included in the definitions of the state functions for convenience later]: ψ1 (x, v; τ ) = x, vψ1 (τ ) = (v + ωx)ψˆ 00 (x, v)e−2ωτ , D ψ1D (x, v; τ ) = ψ1D (τ )x, v = (−v + ωx)ψˆ 00 (x, v)e−2ωτ ,
(15.11)
ψ2 (x, v; τ ) = x, vψ2 (τ ) = ω{x + τ (v + ωx)}ψˆ 00 (x, v)e−2ωτ , D (x, v)e−2ωτ . ψ2D (x, v; τ ) = ψ2D (τ )x, v = ω{x + τ (−v + ωx)}ψˆ 00
(15.13)
(15.12) (15.14)
The dual state vector is defined by v → −v, namely2 ψ1D (x, v; τ ) = ψ1 (x, −v; τ ), ψ2D (x, v; τ ) = ψ2 (x, −v; τ ). Note that the subtlety of conjugation for the unequal frequency case – with a different rule for each excited state as given in Eq. 14.82 – has been lost for the equal frequency case since the two excited states have become degenerate. Collecting the results from Eqs. 15.2, 15.8, 15.9–15.14 yields 2 ωτ ˆ D e ψ00 x − ψ1 (τ )ψ1D (0) G(τ ) = 2C Nˆ 00 + ψ2 (τ )ψ1D (0) + ψ1 (τ )ψ2D (0) xψˆ 00 . (15.15) 2 Bender and Mannheim (2008a), for the case for Minkowski time, define the dual state vector by x → −x,
which gives an incorrect result for Euclidean time.
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The result in Eq. 15.15 shows that the eigenstates 10 , 01 that gave the result for the propagator in Eq. 15.2 have been replaced, in the limit of ω1 → ω2 , by new state vectors that are well defined and finite for = 0. In state vector notation, Eqs. 15.2 and 15.15 yield D D D e−τ E10 10 10  + e−τ E01 00 01  2 ωτ e − ψ1 (τ )ψ1D (0) + ψ2 (τ )ψ1D (0) + ψ1 (τ )ψ2D (0) . = 2C Nˆ 00
lim
ω1 →ω2
(15.16) 15.3 State vectors for equal frequency The Hamiltonian for the equal frequency case, from Eq. 14.1, is given by H =−
1 ∂2 ∂ γ −v + ω2 v2 + ω4 x 2 . 2 2γ ∂v ∂x 2
(15.17)
The vacuum state is an energy eigenstate with D ˆ H ψˆ 00 (x, v) = ωψˆ 00 (x, v), ψˆ 00 ψ00 =
1 π = . 2 ˆ 2γ ω2 N00
(15.18)
The state vectors ψ1 (τ ), ψ2 (τ ) were obtained by analyzing the equal frequency propagator. The state vectors have the interpretation set out below. 15.3.1 State vector ψ1 (τ ) The state vector ψ1 (τ ) is an energy eigenstate given by the average of the two unequal frequency eigenstates that become degenerate, namely 1 ψ1 (x, v; τ ) = lim [e−τ E10 ψ10 (x, v) + e−τ E01 ψ01 (x, v)] →0 2 ⇒ ψ1 (x, v; τ ) ≡ x, vψ1 (τ ) = e−2τ ω (v + ωx)ψˆ 00 (x, v) H ψ1 (x, v; τ ) = 2ωψ1 (x, v; τ ).
(15.19)
The first sign of the irreducible nonHermitian nature of the equal frequency Hamiltonian appears with ψ1 (τ ); unlike the norm of all the energy eigenstates, the norm of ψ1 (τ ) is zero; namely D −4ωτ ˆ 2 D N00 dxdv(−v + ωx)(v + ωx)ψˆ 00 (x, v)ψˆ 00 (x, v) ψ1 (τ )ψ1 (τ ) = e ⇒ ψ1D (τ )ψ1 (τ ) = 0.
(15.20)
The norm of the eigenstate being zero is a general feature of a Hamiltonian that is of the form of a Jordanblock and, in particular, is not pseudoHermitian (Bender
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and Mannheim, 2008a). The fact that the eigenstate has zero norm does not prevent the eigenstate being included in the collection of state vectors that, taken together, yields a resolution of the identity operator. 15.3.2 State vector ψ2 (τ ) The second state vector ψ2 (τ ) that appears for the equal frequency case can be written as the difference of the two unequal frequency eigenstates that become degenerate; for dimensional consistency, the prefactor of ω is introduced in the → 0; hence ω −τ E01 ψ01 (x, v) − e−τ E10 ψ10 (x, v) ψ2 (x, v; τ ) = lim e →0 2
ω = e−2ωτ (1 − τ )(v + (ω − )x) − (1 − τ )(v + (ω − )x) 2 × ψˆ 00 (x, v)
(15.21) ⇒ ψ2 (x, v; τ ) = x, vψ2 (τ ) = e−2τ ω ω x + τ (v + ωx) ψˆ 00 (x, v). Timedependent state vector ψ2 (τ ) is not an (energy) eigenstate of H ; however, since it results from the superposition of two energy eigenstates, it can be explicitly verified that ψ2 (τ ) satisfies the time dependent Schrödinger equation, namely ∂ψ2 (x, v; τ ) = H ψ2 (x, v; τ ) ⇒ ψ2 (x, v; τ ) = exp{−τ H }ψ2 (x, v; 0). ∂τ Initial value ψ2 (x, v; 0) = ωx ψˆ 00 (x, v). (15.22) −
Note that ψ2 (τ ) has a finite norm and a nonzero overlap with ψ1 (τ ); namely, using Eq. 15.6, ψ2D (τ )ψ2 (τ ) =
e−4τ ω 4γ 2 ω3 = ψ2D (τ )ψ1 (τ ), C = . 2C π
(15.23)
The equal frequency state space continues to have a nonnegative norm; in particular, the norm of the time dependent state ψ2 (τ ) is positive definite. Of course, since one is working in Euclidean time, probability is not conserved and one can see from Eq. 15.23 that the norm of the states decays exponentially to zero. In summary, on taking the equal frequency limit, the two energy eigenstates 10 , 01 coalesce to yield a single energy eigenstate ψ1 (τ ); a second time dependent state ψ2 (τ ) appears in this limit, also from the two eigenstates, and takes the place of the loss of one of the eigenstates. An analysis similar to that carried out for the single excitation level holds for all levels, and has been discussed by Bender and Mannheim (2008a). At each level, all the energy eigenstates collapse into a single energy eigenstate of the
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y10 w1 → w2 y01
Time dependent y2
Figure 15.1 The equal frequency limit yields two new states from two energy eigenstates.
equal frequency Hamiltonian; the eigenstates that are “lost” are replaced by timedependent state vectors that are a superposition of the eigenstates of the unequal frequency Hamiltonian. The timedependent states together with the single eigenstate provide a resolution of the identity. This structure of the equal frequency state space is illustrated in Figure 15.1.
15.4 Completeness equation for 2 × 2 block We now discuss how the timedependent state replaces the lost energy eigenstate to provide the complete set of states for the equal frequency case. The example of the single excitation states, created by applying a creation operator av† or ax† to the harmonic oscillator vacuum state 0, 0, showed that in the limit of ω1 = ω2 the two energy eigenstates 10 , 01 were superposed to create new states ψ1 (τ ), ψ2 (τ ). Since the orthogonality of the eigenstates is maintained in the superposition, the mixing of states is only amongst states of a fixed excitation; in other words, states having two excitations consisting of applying the creation operator twice, namely (av† )2 , (ax† )2 , or av† ax† yield three eigenstates that only mix with each other in the limit of ω1 = ω2 , and so on for all the higher excitation states. Hence, the resolution of the identity – which is an expression of the completeness of a set of basis states – as shown in Figure 15.2, breaks up into a blockdiagonal form, with states of a given excitation mixing with each other and not with the states of the other blocks. To illustrate the general result, consider the 2 × 2 block for the single excitation states. In light of the result obtained in Eq. 15.15, consider the following Hermitian
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ansatz for the 2 × 2 block identity operator, with all the state vectors taken at initial time τ = 0. For notational simplicity, let ψ1 (0) = ψ1 = (v + ωx)ψˆ 00 , ψ1D (0) = ψ1D , ψ2 (0) = ψ2 = xψˆ 00 , ψ2D (0) = ψ2D .
(15.24) (15.25)
Then the identity operator, which is Hermitian, has the following representation for the 2×2 block of Hilbert space: I2×2 = −P ψ1 ψ1D  + Q ψ2 ψ1D  + ψ1 ψ2D  = I†2×2 . (15.26) Recall from Eqs. 15.20 and 15.23, ψ1D ψ1 = 0, ψ2D ψ2 =
1 = ψ2D ψ1 . 2C
Hence, from Eq. 15.26 and the above equations I2×2 = I22×2 3 1 2 = (−2P Q + Q2 )ψ1 ψ1D  + Q2 ψ2 ψ1D  + ψ1 ψ2D  , 2C and this yields from Eqs. 15.26 and 15.23 1 2 1 2P Q − Q2 , Q = Q 2C 2C 8γ 2 ω3 ⇒ P = Q = 2C = . π
P =
Hence, the completeness equation for the 2 × 2 block single excitation states is given by D D D I2×2 = 2C − ψ1 (0)ψ1 (0) + ψ2 (0)ψ1 (0) + ψ1 (0)ψ2 (0) . (15.27) The completeness equation above is equal, up to a normalization, to Eq. 15.16.
15.5 Equal frequency propagator The defining equation for the propagator is, from Eq. 14.105, D G(τ ) = 00 xe−τ (H −ω) x00 2 ˆD ˆ ) = lim G(τ ) = Nˆ 00 ψ00 xe−τ (H −E0 ) xψˆ 00 . ⇒ G(τ →0
(15.28)
The completeness equation can be used to give a derivation of the equal frequency propagator from first principles. Inserting the completeness equation given
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in Eq. 15.27 into the expression for the equal frequency propagator given in Eq. 15.28 yields 2 ˆD ˆ ) = 2C Nˆ 00 ψ00 xe−τ (H −ω) G(τ × − ψ1 (0)ψ1D (0) + ψ2 (0)ψ1D (0) + ψ1 (0)ψ2D (0) xψˆ 00 .
(15.29) Note that Eq. 15.29 is equivalent to the earlier expression given in Eq. 15.15. It follows from Eqs. 15.19 and 15.21 that x, ve−τ H ψ1 (0) = x, vψ1 (τ ) = e−2τ ω (v + ωx)ψˆ 00 (x, v), x, ve
−τ H
ψ2 (0) = x, vψ2 (τ ) = e
−2τ ω
(15.30)
ω{x + τ (v + ωx)}ψˆ 00 (x, v). (15.31)
It can be shown that the first and last terms inside the square bracket in Eq. 15.29 cancel. Hence, from Eqs. 15.29, 15.30, and 15.31 2 ˆD ˆ ) = e−τ ω 2C Nˆ 00 ψ00 xψ2 (τ )ψ1D (0)xψˆ 00 G(τ −τ ω 2 ˆ = e 2C N00 dxdv ωx{x + τ (v + ωx)}P (x, v) × dxdv x(−v + ωx)P (x, v) 2 −τ ω 2 ˆ2 2 = e 2Cω N00 [1 + ωτ ] × dxdv x P (x, v) .
Performing the Gaussian integrations yields dxdv x 2 P (x, v) =
1 . 2ω2 C
(15.32)
(15.33)
Hence ˆ2 ˆ ) = e−τ ω N00 [1 + ωτ ] = 1 1 e−ωτ [1 + ωτ ] , G(τ 2ω2 C 4γ ω3
(15.34)
2 = where C = (4γ 2 ω3 )/π is given in Eq. 15.6 and the normalization constant Nˆ 00 2 2γ ω /π is given in Eq. 15.7. To verify the equal frequency result obtained for the propagator, consider taking the limit of ω1 → ω2 in Eq. 14.104, the result being shown in Figure 15.2b. The propagator has the welldefined and finite limit τ 1 e 1 e−τ −ωτ ˆ e G(τ ) = − 4γ ω1 + ω2 ω− ω+ 1 1 −ωτ = e (15.35) [1 + ωτ ] , 4γ ω3
and agrees with the result obtained in Eq. 15.34.
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1 1
1
2x2 11
1
1
3x3
0 11
0 4x4
1 1
1
w1
w2
1
NxN
0
0
1
(a)
(b)
Figure 15.2 (a) Completely diagonal Hamiltonian H for the unequal frequency case. (b) Blockdiagonal structure of the Hamiltonian in the equal frequency limit, with each N × N block being given by an N × N Jordan block.
The path integral yields a propagator for all values of ω1 , ω2 . The state space approach requires a lot more effort to find the propagator and, in particular, the equal frequency result needs a calculation quite distinct from the unequal case. Furthermore, it is not clear how the state space approach can be used to evaluate the propagator for the case when ω1 , ω2 are complex. These results show the utility of the path integral which, among other things, allows us to have a deeper understanding of the underlying state space and operator structure of quantum mechanics.
15.6 Hamiltonian: Jordan block structure In the limit of equal frequencies ω1 = ω2 , there is a reorganization of state space into a direct sum of finite dimensional subspaces, one subspace for each block diagonal component of H , as shown in Figure 15.2. The breakdown of the pseudoHermitian property of the Hamiltonian H is due to the fact that, for equal frequencies, H becomes a direct of sum of Jordan blocks. The total Hilbert space V, for the equal frequency case, breaks up into a direct sum of finite dimensional vector spaces Vn , and is given by V = ⊕∞ n=1 Vn ,
(15.36)
where V1 is one dimensional, V2 is two dimensional and so on. The coordinate x and velocity v operators are not block diagonal in this representation; the matrix elements of these operators connect the vectors of different subspaces Vn . This feature of the coordinate operator comes to the fore in the calculation of the propagator in the block diagonal basis.
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Denoting the finite dimensional matrix representation of the Hamiltonian by Hn , as shown in Figure 15.2, yields the block diagonal decomposition ∞ H = ⊕∞ n=1 Hn = ⊕n=1 an Jλn ,n .
(15.37)
The coefficients an are real constants; Jλn ,n is an n × n Jordan block, specified by its size n and eigenvalue λn , and given by3 ⎡ ⎤ .. .. . . 0 λ ±1 0 ⎢ n ⎥ .. .. ⎥ ⎢ . . ⎥ ⎢ 0 λn ±1 0 ⎢ . .. ⎥ ⎢ .. . ⎥ 0 λn ±1 0 ⎢ ⎥ Jλn ,n = ⎢ . (15.38) ⎥. . . . . . ⎢ .. .. .. .. .. .. ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ . . . . . . . . 0 λ ±1 ⎥ n ⎣ ⎦ .. .. .. .. . . . . 0 λn The Hamiltonian is analyzed for the first two blocks; H1 is one dimensional and H2 is a 2×2 matrix. The ground state forms an invariant subspace V1 with a single element e0 proportional to ψˆ 00 ; for dimensional consistency and to preserve the correct normalization, the mapping is e0 = Nˆ 00 ψˆ 00 , e0 e0 = 1.
(15.39)
The eigenvalue equation H ψˆ 00 = ωψˆ 00 yields the Hamiltonian on V1 given by H1 = ω, H1 e0 = ωe0 .
(15.40)
15.7 2×2 Jordan block A derivation is given of the 2×2 Jordan block structure of the Hamiltonian and state space. The result given in Eq. 15.40 together with Eq. 15.37 yields H = H1 ⊕ H2 ⊕ . . . = ω ⊕ 2ωJ2 ⊕ . . .
(15.41) (15.42)
It will be shown in this section that
1 −1 J2 = . 0 1
(15.43)
3 The ±1 terms in the superdiagonal in Eq. 15.38 are allowed, since multiplying J λn ,n by −1 can switch the
sign of the superdiagonal from 1 to −1, and in so doing redefine the eigenvalue to be −λn .
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The Jordan block Hamiltonian has been shown by Bender and Mannheim (2008a) to be pseudoHermitian; for the case of J2 , it can be seen that 1 0 0 1 † = J2 , σ1 = . σ1 J2 σ1 = −1 1 1 0 The result above demonstrates that although the Jordan block matrix is pseudoHermitian, it is essentially inequivalent to a Hermitian matrix. Bender and Mannheim (2008a) derive the 2×2 Jordan block for the Minkowski Hamiltonian by defining creation and destruction operators that have a finite limit when → 0. In this section, the 2×2 Jordan block for the Euclidean Hamiltonian is directly derived from the state vectors, and the completeness equation is obtained by taking → 0 following the procedure in Sections 15.3 and 15.4. Recall from Eqs. 15.17, 15.24, and 15.25, that the Hamiltonian and state vectors for the equal frequency limit are given by ∂ γ 1 ∂2 −v + ω 2 v2 + ω 4 x 2 , 2 2γ ∂v ∂x 2 ψ1 = ψ1 (0) = (v + ωx)ψˆ 00 , ψ1D  = ψ1D (0), ψ2 = ψ2 (0) = ωxψˆ 00 , ψ2D  = ψ2D (0). H =−
The fact that the state vectors ψ1 , ψ2 form a closed subspace under the action of H points to an invariant 2×2 subspace of the total Hilbert space. In the 2×2 block space, the Hamiltonian can be represented by a 2×2 Jordan block in a basis fixed by the representation of ψ1 , ψ2 by twodimensional column vectors. To obtain this finite dimensional representation, note that H ψˆ 00 = ωψˆ 00 and from Eq. 15.19 H ψ1 = 2ωψ1 ; hence, the action of H on the state vectors ψ1 , ψ2 is given by H ψ1 = 2ωψ1 , (15.44) 2 H ψ2 = −ωvψˆ 00 + ω xψˆ 00 = −ω(v + ωx)ψˆ 00 + 2ωψ2 ⇒ H ψ2 = −ωψ1 + 2ωψ2 .
(15.45)
Since ψ1 is an eigenvector of the Jordan block it is natural to make the identification 1 ψ1 ∝ . (15.46) 0 Recall from Eqs. 15.20 and 15.23, ψ1D ψ1 = 0, ψ2D ψ2 =
1 = ψ2D ψ1 . 2C
(15.47)
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Since ψ1 has zero norm, its normalization is fixed by its overlap with ψ2 . Choosing the normalization consistent with Eq. 15.47 yields √ √ 1 1/2 Cψ1 = e1 = , Cψ2 = e2 = , (15.48) 0 1/2 with the dual vectors given by √
√
Cψ1D  = e1D  = 0, 1 , Cψ2D  = e2D  = 1/2, 1/2 .
(15.49)
Note that e1D  is not the transpose of e1 . The completeness equation for the state space of the 2×2 block has a discrete realization; recall from Eq. 15.27 I2×2 = 2C − ψ1 ψ1D  + ψ2 ψ1D  + ψ1 ψ2D  ⇒ I2×2 = 2 − e1 e1D  + e2 e1D  + e1 e2D  . (15.50) The completeness equation for the Jordan block shows that there is a nontrivial metric on the discrete state space V2 . Using Eqs. 15.48 and 15.49, Eq. 15.50 yields ' 1 0 1 1 1 1 1 0 0 1 + = , I2×2 = 2 − + 0 1 0 0 2 0 0 2 0 1 and we have obtained the expected result. 15.7.1 Hamiltonian Let H2 denote the realization of the Hamiltonian as a discrete and dimensionless matrix acting on the twodimensional state space of the 2×2 Jordan block. Applying Eq. 15.48 to Eqs. 15.44 and 15.45 yields the 2×2 representation H2 e1 = 2ωe1 ⇒ e1D H2 = 2ωe1D  H2 e2 = −ωe1 + 2ωe2 ⇒ e2D H2 = −ωe1D  + 2ωe2D . The Hamiltonian H2 – in the e1 and e2 basis – is proportional to the 2×2 Jordan block matrix and is given by4 1 −1 H2 = 2ω . (15.51) 0 1 The definition √ of the discrete vectors e1 and e1 given in Eq. 15.48 requires a rescaling by C because of dimensional consistency; in contrast, there is no need to rescale H2 since it has the correct dimension set by ω. 4 The Euclidean Hamiltonian given in Eq. 15.43 has a −1 for the superdiagonal, unlike the case for the
Minkowski Hamiltonian (Bender and Mannheim, 2008a), where it is +1.
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The Jordan block Hamiltonian given in Eq. 15.43 has only one eigenvalue and this is the reason that the two different eigenstates for the unequal frequencies collapsed into a single eigenstate. The Jordan block limit of H2 (for equal frequency) shows that H2 continues to be pseudoHermitian but is inequivalent to any Hermitian Hamiltonian since the Jordan block is inequivalent to any Hermitian matrix. The right eigenvector of H2 is e1 and the left eigenvector of H is the dual e1D ; namely H2 e1 = 2ωe1 , e1D H2 = 2ωe1D , ⇒ e1D e1 = 0 = ψ1D ψ1 . Hence, the Jordan block structure shows why the equal frequency eigenstate has a zero norm.
15.7.2 Schrödinger equation for Jordan block The Schrödinger equation for an arbitrary vector e is given by −
∂ e(τ ) = H2 e(τ ). ∂τ
For eigenvector e1 the timedependent solution is ∂ e1 (τ ) = H2 e1 (τ ) = 2ωe1 (τ ), ∂τ 1 . ⇒ e1 (τ ) = e−2ωτ e1 , e1 = 0
−
The time dependence of the state vector e2 (τ ) is given by ∂ 1/2 e2 (τ ) = H2 e2 (τ ), e2 (0) = e2 = − . 1/2 ∂τ
(15.52)
(15.53)
In the 2×2 block representation e2 (τ ) is given from the solution obtained in Eq. 15.21, which yields
ψ2 (x, v; τ ) = e−2τ ω ω x + τ (v + ωx) ψˆ 00 (x, v) −2τ ω −2τ ω 1/2 + ωτ ⇒ e2 (τ ) = e e2 + ωτ e1 = e . (15.54) 1/2 It can be directly verified using the explicit form for the Hamiltonian given in Eq. 15.43 that the solution for e2 (τ ) given in Eq. 15.54 satisfies the Schrödinger equation given in Eq. 15.53.
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15.7.3 Time evolution The Jordan block Hamiltonian is given by Eq. 15.43; a simple calculation yields the evolution operator 1 2ωτ . (15.55) e−τ H2 = e−2ωτ 0 1 The time dependence of the state vectors follows directly from the evolution operator. For eigenvector e1 the timedependent solution is e1 (τ ) = e−τ H2 e1 = e−2ωτ e1 , which is the expected result as in Eq. 15.52. The time dependence of the state vector e2 (τ ) is given by −τ H2 −2ωτ 1/2 + ωτ , e2 = e e2 (τ ) = e 1/2
(15.56)
which is the expected result as in Eq. 15.54.
15.8 Jordan block propagator The equal frequency propagator is given in Eq. 15.28, 2 ˆD ˆ ) = Nˆ 00 G(τ ψ00 Xe−τ (H −ω) Xψˆ 00 .
The position operator X, unlike the Hamiltonian, is not block diagonal for the equal frequency case and connects different subspaces Vn . To determine the propagator, the representation of the position operator X needs to be determined in the 3×3 subspace given by V1 ⊕ V2 , which includes the ground state and the 2×2 Jordan block. The operator X has the following matrix elements: D ψˆ 00 Xψˆ 00 = 0 = ψ1D Xψ1 = ψ2D Xψ2 = ψ2D Xψ1 , 1 D D Xψ1 = Xψ2 . ψˆ 00 = ψˆ 00 2ωC
(15.57)
Note that the matrix elements of the operator X are zero within a block and are nonzero only for elements that connect vectors from two different blocks. Since X acts on the V1 ⊕ V2 we need to extend the vectors defined on the subspaces e0 ∈ V1 and e1 , e2 ∈ V2 to the larger space; let us define the following threedimensional vectors: ⎡ ⎤ 1 ⎣ e0 = 1, E0 = e0 ⊕ 0 = 0⎦ , 0
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⎡ ⎤ ⎡ ⎤ 0 0 ⎣ ⎣ ⎦ E1 = 0 ⊕ e1 = 1 , E2 = 0 ⊕ e2 = 1/2⎦ . 0 1/2 The dual vectors are given by the transpose, except for E1D  given by
E1D  = 0, 0, 1 .
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(15.58)
(15.59)
Let the position operator in the block diagonal space be denoted by X ; from Eq. 15.57, since all the elements are dimensionless in the Jordan block representation, E0D X E0 = 0 = E1D X E1 = E2D X E2 = E1D X E2 , E0D X E1 = 1 = E0D X E2 ,
(15.60)
which yields the following representation for the Hermitian matrix X : ⎡ ⎤ 0 1 1 X = ⎣1 0 0⎦ . 1 0 0
(15.61)
Since X is dimensionless, its mapping to the coordinate position operator X needs a dimensional scale; let X = ζ X. From Eqs. 15.39, 15.49, 15.57, and 15.60, √ Nˆ 00 D Xψ1 = ζ √ , 1 = E0D X E1 = ζ Nˆ 00 Cψˆ 00 2ω C √ ˆ N00 2ω C . ⇒X= √ X, ζ = 2ω C Nˆ 00
(15.62) (15.63)
Extending the Hamiltonian to the V1 ⊕ V2 space yields, from Eq. 15.43, ⎡ ⎤ 1 0 0 1 −1 = ω ⎣0 2 −2⎦ . H = H1 ⊕ H2 = ω ⊕ 2ω 0 1 0 0 2 The evolution kernel is given by ⎡
eωτ −2ωτ ⎣ exp{−τ H} = e 0 0
⎤ 0 0 1 2ωτ ⎦ . 0 1
(15.64)
The completeness equation from Eq. 15.50 has the following extension to V 1 ⊕ V2 : (15.65) I3×3 = E0 E0D  + 2 − E1 E1D  + E2 E1D  + E1 E2D  .
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In the blockdiagonal basis, the propagator is given by 2 ˆD ˆ ) = Nˆ 00 ψ00 Xe−τ (H −ω) Xψˆ 00 G(τ $2 # Nˆ 00 E0D X e−τ (H−ω) X E0 . = √ 2ω C
(15.66)
Using the completeness equation given in Eq. 15.65 yields ˆ ) G(τ =
2 Nˆ 00 E0D X e−τ (H−ω) 2C 4ω
× E0 E0D  + 2 − E1 E1D  + E2 E1D  + E1 E2D 
X E0
2 Nˆ 00 D −ωτ D ωτ D −ωτ D X − e E E  + e E (τ )E  + e E E  X E0 E 1 2 1 1 1 2 2ω2 C 0 2 2 Nˆ 00 Nˆ 00 ωτ D D E X E (τ )E X E = (15.67) = e eωτ E0D X E2 (τ ), 2 0 0 1 2ω2 C 2ω2 C =
since E1D X E0 = 1 = E2D X E0 . From Eq. 15.53, the time dependence of E2 (τ ) is given by ⎡ ⎤ 0 E2 (τ ) = e−τ H E2 = e−2ωτ ⎣1/2 + ωτ ⎦ ⇒ E0D X E2 (τ ) = 1 + ωτ, 1/2 which yields, from Eq. 15.67, the expected result for the propagator, namely ˆ2 ˆ ) = N00 e−ωτ (1 + ωτ ) = 1 e−ωτ (1 + ωτ ). G(τ 2ω2 C 4γ ω3 A direct derivation can be given using the matrix representation of the evolution kernel; from Eq. 15.66 $2 # ˆ 00 N ˆ )= E0D X e−τ (H−ω) X E0 . G(τ √ 2ω C Using ⎡ ⎤ 0
⎣ X E0 = 1⎦ , E0D X = 0, 1, 1 1
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yields, from Eq. 15.64, the expected answer $2 # ˆ 00 N 1 −ωτ ˆ )= G(τ e−ωτ (2 + 2ωτ ) = e (1 + ωτ ). √ 4γ ω3 2ω C All the N × N blocks for the Hamiltonian can be analyzed one by one and it can be shown that they are all equal to a corresponding Jordan block matrix. However, the higher order blocks may not be as simple as J2 as they can include the direct sum of lower order Jordan blocks. 15.9 Summary The equal frequency case of the acceleration Lagrangian leads to a Hamiltonian that is essentially inequivalent to any Hermitian Hamiltonian. A carefully chosen limit for the equal frequency leads to a Hamiltonian that is block diagonal, with each block consisting of Jordan block matrices. The state space has zero norm state vectors, even for the Euclidean theory, showing that the fundamental inequivalence of the acceleration Hamiltonian to a Hermitian Hamiltonian holds for both Minkowski and Euclidean time. The equal frequency propagator was evaluated using various techniques to highlight the different aspects of the Jordan block system that provide a representation of the essentially nonHermitian sector of the theory. The specific form of the propagator for equal frequency reflects the presence of timedependent states that are essential in evaluating the propagator. In particular, these time dependentstates appear in the completeness equation and hence are required for spanning out a complete basis for the state space of the equal frequency case. The quantum mechanics of the 2×2 Jordan block was worked out, with the solution of the Schrödinger equation having only one eigenstate and another time dependent state. A calculation for the propagator was done, block by block, based on the discrete subspaces and finite Jordan block matrices.
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Part six Nonlinear path integrals
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16 The quartic potential: instantons
The simple harmonic oscillator is exactly soluble because it has a quadratic potential and yields a linear theory in the sense that the classical equation of motion is linear. Nonlinear path integrals have potentials that typically have a quartic or higher polynomial dependence on the degree of freedom, or the potential can be a transcendental function, such as an exponential. The techniques discussed for quadratic Gaussian path integrals needed to be further developed for nonlinear path integrals. In general, to solve these nonlinear systems, one usually uses either a perturbation expansion or numerical methods. A perturbation expansion is useful if the theory has a behavior that is smooth about the dominant piece of the action or Hamiltonian; in practice a smooth expansion of the physical quantities yields an analytic series in some expansion parameter, say a coupling constant g, around g = 0. There are, however, cases of physical interest for which nonperturbative effects change the qualitative behavior of the theory Two examples where nonperturbative effects dominate are the following: • Tunneling through a finite barrier. If one perturbs about the lowest lying eigenstates inside a well, one cannot produce the tunneling amplitude. • The spontaneous breaking or restoration of a symmetry cannot be produced by perturbing about an incorrect ground state. Tunneling and symmetry breaking are nonperturbative because these effects depend on g as exp{−1/g}, which is nonanalytic about g = 0, and hence cannot be obtained by perturbation theory. The semiclassical expansion is an approximation that is a useful tool for studying nonperturbative effects. The general features of this approximation scheme are discussed for a nonlinear Lagrangian in Section 16.1. Section 16.3 covers nontrivial classical solutions of the doublewell potential, called instantons. In Section 16.4 the so called zero mode features of the instantons are discussed, and in Section
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16.5 the Faddeev–Popov analysis is applied to the zero mode problem. In Section 16.6 the multiinstanton solutions are obtained, and the transition amplitude, correlators and the dilute gas approximation are then discussed in Sections 16.7, 16.8 and Section 16.9. The doublewell potential, in the strong coupling limit, is shown in Section 16.10 to be equivalent to an Ising model, and in Section 16.11 a nonlocal Ising model is shown to produce the doublewell potential. Sections 16.12–16.14 discuss spontaneous symmetry breaking and symmetry restoration.
16.1 Semiclassical approximation Consider the evolution kernel for Euclidean time, K(x, x ; τ ) = DxeS[x]/ , where
τ
S[x] =
Ldt
(16.1)
0
" and Dx is the path integration measure. For → 0, the classical trajectory, for which S is a maximum, that is δS[xc ] = 0, δx(t)
(16.2)
dominates the path integral with the next leading term yielding an expansion in a power series in . One can expand the paths about the classical path, that is x(t) = xc (t) + η(t),
η(0) = 0 = η(τ ),
and expand the action about the classical path. Thus, we have S[x] = S[xc + η]
δ 2 S[xc ] 1 dt1 dt2 ηt1 ηt2 = S[xc ] + + 0(η3 ) 2 δx(t1 )δx(t2 ) = Sc + S2 , Sc = S[xc ].
The classical action is a maximum of the Euclidean action, namely δ 2 S[xc ] ≤ 0. δx(t1 )δx(t2 )
(16.3)
Hence the quadratic approximation of the action yields a convergent expansion for the path integral,
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16.2 A onedimensional integral
K(x.x ; τ ) N eSc /
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DηeS2 / E7 1 δ 2 S[xc ] Sc / . = Ne det δx(t)δx(t )
Note the approximation breaks down if det δ 2 S[xc ]/δx(t)δx(t ) = 0. Consider a typical Euclidean action ! τ 1 2 mx˙ + V (x) . dt S=− 2 0
(16.4)
(16.5)
Then ! d2 δ 2 Sc = − −m + V (x) δ(t − t ), δx(t)δx(t ) dt 2
(V = ∂V /∂x),
(16.6)
and hence 1 δ 2 Sc det δxδx
!
1 d2 ∝ det −m 2 + V (x) dt
!! .
(16.7)
To evaluate the determinant, the earlier discussion on the simple harmonic oscillator needs to be generalized.
16.2 A onedimensional integral The semiclassical expansion is identical to the saddle point method for finite dimension integrals. To see this, consider the integral ' +∞ 1 2 g2 4 dx 2 I (g ) = √ exp − x − x . 2 8 2π −∞ The integral is well defined for g 2 > 0, but can it be analytically continued to g 2 < 0? A perturbation expansion around g 2 > 0 cannot answer this question and a saddle point expansion will be used to go beyond the perturbation expansion. The minimum is at x0 = 0, and expanding the integrand about this yields ! +∞ dx − 1 x 2 g2x 4 g4x 6 I + 1− √ e 2 8 128 2π −∞ 3 105 4 = 1 − g2 + (16.8) g + 0(g 5 ) 8 128 : analytic about g = 0.
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Let us rewrite I as
dxdy − 1 x 2 − 1 y 2 − i gyx 2 e 2 e 2 e 2 2π dy 1 2 1 = √ e− 2 y − 2 ln(1+igy) . 2π
I=
(16.9)
If one expands about g 2 = 0 one obtains the same result as Eq. 16.8. However, consider expanding about the minimum of 1 1 S(y) = − y 2 − ln(1 + igy). 2 2
(16.10)
The minimum y0 is given by ig 2(1 + igy0 ) 9 1 −1 ± 1 + 2g 2 . ⇒ iy0 (±) = 2g
S (y0 ) = 0 = y0 +
For g → 0, the limiting values are iy0 (−) → − iy0 (+) →
1 g − → ∞, g 2
g → 0. 2
The action about the saddle point y0 (−) yields a negative divergent action, namely ! 1 1 g2 lim S(y0 (−)) → − 2 + ln − g→0 2g 2 2 → −∞ divergent. Hence, the y0 (+) branch is chosen for doing the expansion since it yields the finite action # $ 9 2 1 + 1 + 2g 1 9 1 , (16.11) S(y0 (+)) = 2 ( 1 + 2g 2 − 1)2 − ln 8g 2 2 ≡ S+(0) → 0 as g → 0.
(16.12)
Furthermore, the action has a Taylors expansion about y0 (+) given by 1 1 S(y) = − y 2 − ln(1 + igy) 2 2 ∞ (n) S+ = (y − y0 (+))n , n! n=0
(16.13)
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where the notation S+(n) = S+(0) S+(2) S+(4)
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∂n S(y0 (+)) ∂y n
is being used: $ # 9 2 1 + 2g 1 9 1 1 + 2 , = 2 ( 1 + 2g 2 − 1) − ln 8g 2 2 9 2 1 + g2 8ig 3 =− > 0, S+(3) = − , 9 9 1 + 1 + 2g 2 (1 + 1 + 2g 2 )3 48g 4 = , S+(n) ∼ 0(g n ). 9 2 4 (1 + 1 + 2g )
Hence, shifting y by y0 (+), Eqs. 16.9 and 16.13 yield . / +∞ (4) (0) dy S 1 (2) 2 I (g 2 ) eS+ √ e 2 S+ y 1 + + y 4 + 0(g 6 ) 4! 2π −∞ √ . / 1 ( 1+2g 2 −1)2 4 8g 2 e 3g ∼ 1+ + ··· . 9 = (1 + 2g 2 )1/4 2(1 + 2g 2 )(1 + 1 + 2g 2 )2
(16.14)
Note that the second term above is O(g 4 ), whereas in the expansion given in Eq. 16.8 the second term is only O(g 2 ). The nonanalytic structure of the integral about g 2 = −1/2, namely that it has a branch cut, is captured in the semiclassical expansion, but is missed in the expansion about g 2 = 0 as given in Eq. 16.8. In particular, from Eq. 16.14 it can be seen that the semiclassical expansion provides an analytic continuation of the integral I (g 2 ) to the range −1/2 ≤ g 2 ≤ +∞, whereas the perturbation expansion cannot be extended to g 2 ≤ 0. The semiclassical expansion is a technique that can capture nonperturbative properties of the path integral and is very useful in quantum mechanics and in quantum field theory. 16.3 Instantons in quantum mechanics Consider the nonlinear action and Lagrangian for Euclidean time given by 1 S = dtL, L = − mx˙ 2 − V , (16.15) 2 g2 (16.16) V = (x 2 − a 2 )2 . 8 The potential term V is shown in Figure 16.1. The Lagrangian L has the parity symmetry of being invariant under the transformation x → −x. Note that if one expands the potential into a polynomial, the quadratic term has a coefficient that is positive; hence any expansion about the quadratic Lagrangian
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a
+a
x
Figure 16.1 The doublewell potential for Euclidean time.
will be unstable and lead to a divergent expansion. The reason is that the minima of the Lagrangian are at ±a, as shown in Figure 16.1, and hence the Lagrangian needs to be expanded about one of these minima. The transition amplitude for Euclidean time τ is given by (16.17) K(x, x ; τ ) = DxeS = xe−τ H x , where H is the Euclidean Hamiltonian. Writing the time integration in a symmetric manner yields the action ! +τ 2 1 2 g2 2 S=− (16.18) dt mx˙ + (x − a 2 )2 . 2 8 − τ2 The action is expanded about one of the classical solutions, say x = a, similarly to the exercise carried out for a single variable in Section 16.2. Consider the shift of variable x = y − a; the action is given by ! +τ 2 1 2 g2 2 2 dt my˙ + y (y + 2a) S=− 2 8 − τ2 ! +τ 2 1 2 g2 2 2 dt (16.19) − my˙ + a y + 0(y 3 ). τ 2 2 −2 The theory has small oscillations about the minimum at x = a, as shown in Figure 16.2; these oscillations spontaneously break the symmetry of x → −x, as is evident from Figure 16.2 and Eq. 16.19. It is known that a onedimensional quantum mechanical system cannot spontaneously break any continuous symmetry. However, the system can, in principle, break discrete symmetries. The question that needs to be addressed is whether the perturbative breaking of discrete symmetry of x → −x by the ground states centered at x = a or x = −a, referred to as x = ±a, is only an approximation – with
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x=+a
x=a
Figure 16.2 The doublewell potential for Euclidean time.
nonperturbative effects restoring the parity symmetry? And if symmetry restoration takes place, what is the mechanism by which, in particular for the x 4 potential, the discrete parity symmetry is restored? The semiclassical expansion for the path integral reveals that there are large quantum fluctuations that are encoded in the neighborhood of an instanton configuration. [A fluctuation is defined as one possible path for the degree of freedom.] It is these large quantum fluctuations – far from the perturbative vacuum defined by x = ±a – that are responsible for restoring the parity symmetry that is apparently broken by the perturbative vacuum. Consider the equation of motion [x˙ = dx/dt, x¨ = d 2 x/dt 2 ] 0=
δS[xc ] = mx¨c − V (xc ) δx(t)
(16.20)
or g2 (16.21) xc (xc2 − a 2 ) = 0. 2 The minimum at x = ±a are two solutions of the classical equations of motion, and expanding about either of them gives an analytical expansion around g = 0. The equation of motion given in Eq. 16.21 can be rewritten as the conservation of energy in the following manner: dE d 1 2 g2 2 2 2 mx˙c − (xc − a ) = 0 = dt 2 8 dt 2 1 g ⇒ E = mx˙c2 − (xc2 − a 2 )2 . (16.22) 2 8 Hence, from Eq. 16.22, the Euclidean energy is given by mx¨c −
1 E = mx˙ 2 − V (x). 2
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–V(x)
classical
–a
a
0
x
x 0 –a
(a)
a
(b)
Figure 16.3 The potential for Euclidean time (a), and for Minkowski time (b).
In Minkowski time, the potential is a double well, as shown in Figure 16.3(b). The equations of motion have no classical solutions that go from one well to another when the classical particle does not have an energy E sufficient to cross the potential barrier given by g 2 a 4 /8. In particular, for E = 0, which corresponds to the vacuum solutions, there is no trajectory from one well to another. The Euclidean equations of motion have a nontrivial solution, since it is −V that appears in eS ; hence, the sign of the potential for the Euclidean case has been reversed from the Minkowski case. The potential wells correspond to maxima of two “hills,” as shown in Figure 16.3(a), and the classical Euclidean solution corresponds to the particle starting at x = −a and rolling down the hill to reach x = a; there is no energy barrier separating the two maxima. Two zero energy classical solutions are given by the particle being stationary at the two wells x = ±a. The question is, are there other zero energy classical solutions to the equations of motion that contribute to the path integral as well as the x = ±a solution? The answer is in the affirmative, and we look for these solutions. For zero energy the classical trajectory, from Eq. 16.22, is given by 1 g2 1 E = 0 = mx˙ 2 − V (x) ⇒ mx˙ 2 − (x 2 − a 2 )2 = 0. 2 2 8
(16.23)
The trivial classical solution is given by xc = ±a with x˙c = 0 for all time. The nontrivial solution has nonzero velocity and is given by
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g g x˙ = ± √ xc2 − a 2  = ± √ (a 2 − xc2 ), x 2 ≤ a 2 2 m 2 m t x± dy g ⇒ =± √ dt. a2 − y 2 2 m tc 0 Integrating the above equation yields x 1 g ± tanh−1 = ± √ (t − tc ). a a 2 m Hence the instanton and antiinstanton classical solutions xc are given by ! ω(t − tc ) g2a2 , ω2 = xc = x± = ±a tanh . (16.24) 2 m The particle starts off with zero velocity at ±a at t = −∞ and travels very slowly until time tc when it rapidly crosses over towards ∓a and then slowly rolls to ∓a at t = +∞. Figure 16.4 shows the shape of the instanton for two different values of ω. As ω becomes large, the particle almost “jumps” in an instant from −a to +a at time tc . Figure 16.5(a) shows a kink (instanton) solution that tunnels from −a to +a, and Figure 16.5(b) shows an antikink (antiinstanton) solution that tunnels from a to −a. For this reason, the classical solution is called an instanton or an antiinstanton, depending on whether it tunnels from left to right or from right to left; it also called a kink or an antikink due to the kinklike shape of the classical solution. t
ω: large ω: small
tc
a
+a
x
Figure 16.4 The classical instanton solution for different values of ω2 = g 2 a 2 /m.
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AntiKink
t
t
–a
a
x
–a
(a)
a
x
(b)
Figure 16.5 Classical solution for large coupling constant g that is (a) a kink or instanton, and (b) an antikink or antiinstanton.
The instanton classical action – from the zero energy classical equation given in Eq. 16.23 – is the following:
±τ/2 m 2 S[xc ] = dt[− x˙c − V (xc )] = −m dt x˙c2 2 ∓τ/2 ∓τ/2 ±τ/2 ±τ/2 dxc == −m dt x˙c x˙c = −m dt x˙c dt ∓τ/2 ∓τ/2 ±a = −m dxc x˙c . ±τ/2
∓a
Hence, from Eq. 16.23, the classical action for one kink or one antikink is given by
ω dxc ∓ (xc2 − a 2 ) 2a ∓a mω 1 3 2 2 =± ( xc − xc a 2 )±a ∓a = − mωa . 2a 3 3
S[xc ] = −m
±a
Hence, for one kink or antikink Sc = −
2m2 ω3 . 3g 2
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The evolution kernel for the particle to travel from one well to the other is given by 2 ω3 3g 2
− 2m
K(±a, ∓a; τ ) = ±ae−τ H  ∓ a N eSc = N e 2 − c ≡ N e g2 , c = m2 ω3 3 : essential singularity around g = 0.
A perturbation expansion of the path integral about g = 0 to any order cannot produce an essential singularity at g = 0. The classical Lagrangian for a kink or an antikink is given by 1 Lc = − mx˙c2 − V (xc ) = −mx˙c2 2 ! ma 2 ω2 t − tc 4 . =− sech ω 4 2 The classical kink Lagrangian, as shown in Figure 16.6, is sharply localized around t = tc with a width of √ m 1 . t ∼ = ω ga
–L
∼ 1/ω
t0
t
Figure 16.6 Classical Lagrangian for a kink.
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The time at which the tunneling occurs, namely tc , is arbitrary and reflects the fact that the theory, for τ → ∞, is invariant under translation in time, namely under a shift given by t → t+const. 2 At first glance, it seems that, for g 0, +ae−τ H  − a ∼ eSc = e−c/g is negligibly small and hence cannot contribute significantly to the transition amplitude. However, this is not true since there is a whole collection of classical solutions, one for each tc , that are all close to each other, and they all contribute to the transition amplitude. We will show that in fact a more accurate analysis yields +τ 2 2 −τ H −c/g 2 +ae  − a ∼ N e dtc = N τ e−c/g . − τ2
Hence, as long as τ < e−c/g the contribution from the classical trajectory is indeed negligible; but for large τ the contribution becomes very large. There are also multikink and multiantikink solutions with multiple tunneling across the barrier that all contribute in the limit of large time. There is no exact solution known due to the kink–antikink interaction, but for g → 0, an approximate solution is to compose the multiple kink–antikink solutions from the product of single kink and antikink solutions. 2
16.4 Instanton zero mode The doublewell action is given by ' 1 2 g2 2 2 2 mx˙ + (x − a ) , S = − dt 2 8
(16.25)
with the kink, antikink classical solution given by (ω2 = a 2 g 2 /m) ! ω(t − tc ) . xc = ±a tanh 2 Consider the semiclassical expansion for the transition amplitude. Let the (false) vacua at ±a be denoted by ± . Then −τ H + e − = dbdb + bbe−τ H b b  ∼ = + (a)+ae−τ H  − a− (a).
(16.26)
The last equation has been obtained using the fact that the false vacua ± are approximately deltafunctions (well localized) at points ±a. Let x(t) = xc (t) + η(t); then, since η(τ/2) = 0 = η(−τ/2), one has from Eq. 16.26 −τ H  − a = DηeS[xc +η] . (16.27) +ae
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The action has the following semiclassical expansion:1 S[x] = S[xc + η]
δ2S 1 dtdt ηt ηt + 0(η3 ) 2 δxt δxt 1 g2 2 dt (6xc2 η2 − 2a 2 η2 ) + 0(η3 ) = Sc − m dt η˙ − 2 8 ! 1 +τ/2 d2 3 2 2 g2a2 η + 0(η3 ) dtη −m 2 + g xc − = Sc − 2 −τ/2 dt 2 2 +τ/2 1 dtdt η(t)M(t, t )η(t ) + 0(η3 ), (16.28) = Sc − 2 −τ/2 = S[xc ] +
where the matrix elements of the operator M are defined by ! d2 3 2 2 g2a2 δ(t − t ). M(t, t ) = t Mt ≡ −m 2 + g xc − dt 2 2 " To do the Dη path integration one diagonalizes the action using the normal ( mode expansion of η(t) = ∞ n=0 ψn (t)ηn . The normal modes ψn are defined by the eigenfunction equation
Mψn = λn ψn . The explicit eigenfunction equation, with ψn ≡ tψn , is given by ! 3g 2 2 g2a2 xc ψn − ψn λn ψn = −mψ¨ n + 2 2 ! 2 2 2 g2a2 3g a d 2 ω(t − tc ) − ψn . = −m 2 + tanh dt 2 2 2
(16.29)
(16.30)
We recall that the classical equation of motion given in Eq. 16.21 yields mx¨c −
g2 3 g2a2 x + xc = 0. 2 c 2
Differentiating the above equation with respect to t, and defining eigenfunction ψ0 as 7 +τ/2 1 ψ0 = dt x˙c2 , (16.31) x˙c , x˙c  = x˙c  −τ/2 1 Note that (x 2 − a 2 )2 = (x 2 − a 2 )2 + 6x 2 η2 − 2a 2 η2 + 0(η3 ). c c
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yields 3g 2 2 g2a2 xc ψ0 + ψ0 = 0. 2 2 Hence, from Eqs. 16.30 and 16.32, ψ0 is an eigenfunction such that mψ¨ 0 −
(16.32)
λ0 = 0 for ψ0 . The zero eigenvalue eigenfunction, also called the zero mode ψ0 , arises in the action due to the time translation invariance of the action: the instanton can tunnel from −a to +a at any time tc . This is the reason that the eigenvalue equation for ψ0 is a direct result of the equations of motion, which is also time translation invariant. The operator M, since λ0 = 0, has the spectral representation M= det M =
∞ n=0 ∞ &
λn ψn ψn  =
∞
λn ψn ψn ,
(16.33)
n=1
λn = 0.
n=0
Note that M is singular, having no inverse, since det M = 0 due to the zero eigenvalue λ0 = 0. Hence, we obtain the path integral given in Eq. 16.27, since N e Sc −τ H +ae  − a = DηeS[xc +η] √ = ∞. (16.34) det M 16.5 Instanton zero mode: Faddeev–Popov analysis The Faddeev–Popov method is of great generality and is indispensable in the quantization of Yang–Mills gauge fields. The elimination of a divergence in the path integral due to asymmetry is analyzed using the Faddeev–Popov approach; the advantage of carrying out this analysis is that it illustrates the main features of the Faddeev–Popov approach in a relatively simple context. The analysis yields what are called the collective coordinates. The fundamental ingredient in the Faddeev–Popov approach is to explicitly introduce a constraint into the path integral that breaks the symmetry, thus removing the singularity in the action; a term has to be introduced to compensate for constraint so as to leave the path integral invariant. We recall that the zero mode arises due to the invariance of the action under shifting the time coordinate, namely t → t+constant. This symmetry is strictly an invariance of the action only for τ → ∞, with corrections that are exponentially small and can be ignored in our analysis. In particular, from Eq. 16.18 and for τ >> t∗ , the doublewell action has the symmetry
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S=− −
365
!
+τ/2
dt
i
−τ/2 +τ/2
dt −τ/2
2 1 2 g2 2 mx˙ (t) + x (t) − a 2 2 8 ! 1 ˙2 g2 2 2 2 , x(t) ˜ = x(t + t∗ ) mx˜ (t) + x˜ (t) − a 2 8
⇒ S[x] = S[x]. ˜
(16.35)
For the double well potential, in the Faddeev–Popov approach, one breaks the invariance under t → t + t∗ by directly introducing a fixed value for time, denoted by t0 directly into the action in the following manner. Following ZinnJustin (2005), consider the identity " +τ/2
" dt x ˙ dt δ (t)x(t + t ) − λ 0 c 0 −τ/2 (16.36) 1 = " +τ/2
" . dt x˙c (t)x(t + t0 ) − λ −τ/2 dt0 δ To perform the integration in Eq. 16.36, consider the following change of variable: ξ = dt x˙c (t)x(t + t0 ) − λ, (16.37) ˙ + t0 ), (16.38) dξ = dt0 dt x˙c (t)x(t which yields " +τ/2 dξ δ[ξ ] dt0 δ dt x˙c (t)x(t + t0 ) − λ = " +τ/2 −τ/2 ˙c (t)x(t ˙ + t0 ) −τ/2 dt x = " +τ/2 −τ/2
Hence, from Eqs. 16.36 and 16.39, +τ/2 1= dt x˙c (t)x(t ˙ + t0 ) · δ −τ/2
+τ/2 −τ/2
1
(16.39)
.
dt x˙c (t)x(t ˙ + t0 )
dt0 δ[ dt x˙c (t)x(t + t0 ) − λ] . (16.40)
The path integral given in Eq. 16.27 is rewritten, using Eq. 16.40, in the following manner: −τ H  − a = K(a, −a; τ ) = DxeS +ae +τ/2 S = Dxe dt x˙c (t)x(t ˙ + t0 ) ×
−τ/2
+τ/2 −τ/2
dt0 δ
dt x˙c (t)x(t + t0 ) − λ
.
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The crucial step is to define a change of integration variables as follows:2 x(t) → x(t) ˜ = x(t − t0 ).
(16.41)
Due to Eq. 16.35, the action is invariant under this change of variables and S[x] = S[x]. ˜
(16.42)
In particular, note that S[x] ˜ is independent of t0 . Hence, dropping the tilde on x yields +τ/2 +τ/2 S K(a, −a; τ ) = Dxe dt x˙c (t)x(t) ˙ × dt0 δ dt x˙c (t)x(t) − λ =
−τ/2
+τ/2 −τ/2
S dt0 · Dxe
−τ/2
+τ/2 −τ/2
dt x˙c (t)x(t)δ ˙
dt x˙c (t)x(t) − λ .
The zero mode, namely the variable t0 , has been completely factorized out of the path integral, which no longer has any divergence due to the deltafunction constraint. The zero mode is a physical quantity that reflects that tunneling can take place at any instant in the interval [−τ/2, +τ/2]. In contrast, for the case of path integrals for gauge fields, the Faddeev–Popov procedure leads to a factorization of the nonphysical gauge degrees of freedom. It is more convenient to rewrite the deltafunction constraint as another term in the action. To do this note that the above equation is valid for all λ; hence, multiplying both sides with exp{−αλ2 /2) and integrating over λ yields +τ/2 " +τ/2 2 α S− α2 −τ/2 dt x˙c (t)x(t) Dx K(a, −a; τ ) = τ dt x˙c (t)x(t)e ˙ . (16.43) 2π −τ/2 Note that since α is an arbitrary parameter, the evolution kernel K(a, −a; τ ) must be independent of it. The classical field equation for the path integral given in Eq. 16.43 is given by % +τ/2 % δS dt x˙c (t )x(t ) %% = 0. (16.44) − α x˙c (t) δx(t) −τ/2 x=xc From Eq. 16.20, δS[xc ] =0 δx(t)
(16.45)
2 Note x (t) is unchanged since, unlike x(t), it is not an integration variable. c
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and
+τ/2
−τ/2
dt x˙c (t)xc (t) = xc2 (τ/2) − xc2 (−τ/2) = 0.
Hence, the classical trajectory xc (t) defined by the doublewell action given by Eq. 16.45 is also a classical solution (maximum) of the modified action given in Eq. 16.44. Note that, from Eq. 16.31, the modified action can be written as λ˜ 0 xψ0 ψ0 x, λ˜ 0 = αx˙c 2 , 2 +τ/2 1 ψ0 = dt x˙c (t)x(t). x˙c , ψ0 x ≡ x˙c  −τ/2
S = S −
(16.46)
Expanding the path integral about the classical solution, x(t) = xc (t) + η(t),
(16.47)
yields, from Eq. 16.28, the action 1 S[x] = Sc − 2 = Sc −
1 2
+τ/2
−τ/2 +τ/2
dtdt η(t)M(t, t )η(t ) −
λ˜ 0 ηψ0 ψ0 η, 2
˜ t )η(t ), dtdt η(t)M(t,
(16.48) (16.49)
−τ/2
where M˜ = λ˜ 0 ψ0 ψ0  +
∞
λn ψn ψn .
n=1
Note from Eq. 16.29 that ψ0 is a vector that is orthogonal to all the eigenfunctions of operator M and yields the completeness equation I = ψ0 ψ0  +
∞
ψn ψn .
n=1
Hence M˜ is a nonsingular operator with determinant given by det M˜ = λ˜ 0
∞ &
λn = 0.
n=1
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Equation 16.43 yields 
α Sc e 2π
1 Dη exp{− 2
+τ/2
˜ t )η(t )} dtdt η(t)M(t, −τ/2 α α Sc 1 1 2 Sc 2 = τ x˙c  = τ x˙c  e ×9 e ×8 C 2π 2π det M˜ λ˜ 0 ∞ n=1 λn
K(a, −a; τ ) = τ x˙c 
2
1 = τ x˙c eSc × 8 C , finite, 2π ∞ λ n n=1
(16.50)
where Eq. 16.46 has been used for the value of λ˜ 0 . The evolution kernel obtained above is indeed convergent, unlike the earlier result derived in Eq. 16.34 for which it was divergent. Furthermore, K(a, −a; τ ) given in Eq. 16.50 is indeed independent of α, as expected, since all the eigenvalues λn and the constant are independent of α.
16.5.1 Instanton coefficient N The one instanton transition amplitude is given by −ae
−τ H
a = N e
Sc
+τ/2 −τ/2
dtc = N τ eSc ,
which is obtained from τ e Sc . −ae−τ H a = 8 2 det(−m dtd 2 + V (xc )) The determinant is apparently zero due to the zero mode giving rise to a zero eigenvalue. Instead of the Faddeev–Popov approach, another way of factoring out the singularity due to the zero mode is to do an expansion of η that directly takes into account the zero mode of the action. This is accomplished by the expansion x(t) = xc (t − tc ) +
∞
ψn (t − tc )cn ,
n=0
where tc is now considered an arbitrary variable. Note that there is no coefficient c0 multiplying the classical solution xc (t − tc ); this is because c0 is replaced by an arbitrary variable tc , namely the instant at which the particle tunnels from one
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vacuum to the other. The change of variables from x(t) to cn , tc yields [Das (2006)] ∞ ∞ & & Dx = x˙c dtc dcn . dcn ⇒ Dx = x˙c  dtc (16.51) n=1
n=1
By eliminating the zero eigenvalue in the eigenfunction expansion of η(t) the zero mode in the determinant has been eliminated and the nonzero and finite determinant is given by ∞ & d2 det(−m 2 + V (xc )) = λn . dt n=1
(16.52)
Hence, the semiclassical expansion in the one instanton approximation yields .∞ / +τ/2 2 & −λn cn −τ H Sc dcn e 2 a = const e dtc −ae n=1
−τ/2
= N τ e Sc .
(16.53)
The coefficient N that appears in the transition amplitude is evaluated. For the trivial classical solution given by xc = ±a in which the particle sits at one of the wells, the classical action is zero. The effective action about one of the wells is given by the simple harmonic oscillator action ! +τ/2 1 2 g2 2 2 dt mη˙ + a η , S0 = − 2 2 −τ/2 boundary conditions: η(−τ/2) = 0 = η(+τ/2). The zeroth order transition amplitude is given by Eq. 11.21 for the simple harmonic oscillator,3 mω −τ H Sc (±a) S0 Dηe = 0e 0(0) = e , (16.54) 2π sinh ωτ mω 1/2 + O(e−ωτ ) e−ωτ/2 π ≡ A, (16.55) where Sc (±a) = 0 and we recall that ω2 = g 2 a 2 /2. A careful analysis by Das (2006) shows that +τ/2 dtc = Arτ, −ae−τ H a = Ar −τ/2
3 Recall that for the simple harmonic oscillator – restoring the dependence on – is given by
K(x , x; τ ) =

3 2 mω mω exp − (x 2 + x 2 ) cosh ωτ − 2xx . 2π sinh ωτ 2 sinh ωτ
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where A is the coefficient given in Eq. 16.55. The factor r comes from the instanton determinant given in Eq. 16.52 and has been shown by Das (2006) to be 2m 3/2 Sc ω e . r = 2a 16.6 Multiinstantons Exact multiinstanton classical solutions, characterized by multiple tunneling, have not yet been found. However, for tunneling at times that are widely separated, and for coupling g → ∞ one can build approximate multipleinstanton solutions from the single kink/antikink solutions. Consider the two kink–antikink ansatz ! ! ω(t − t2 ) ω(t − t1 ) tanh , t2 t1 (16.56) xc(2) (t) = ±a 2 tanh 2 2 as shown in Figure 16.7. It has been shown by Das (2006) that xc(2) satisfies the equations of motion up to terms of O(e−ω(t2 −t1 ) ). In general a string of widely separated instantons and antiinstantons satisfies the classical equations of motion. In the dilute instanton gas approximation, which is valid for strong coupling g 11, an N instanton classical solution is approximately composed of the product of N instantons and antiinstantons, xcN (t)
=
N &
xc (t − ti )
i=1
with −
τ τ ≤ tn ≤ tn−1 · · · ≤ t1 ≤ . 2 2
For g 2 → ∞, one can make further approximations of the N instanton solution: the cross over from one well to another is instantaneous. Furthermore, the N instanton configuration, similarly to Eq. 16.56, is constructed by multiplying
t2 ≈
t2
=
t1
a
+a
t1
a
+a
a
+a
Figure 16.7 A twokink configuration for the particle’s trajectory.
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XN (t)
–t/2 ≤ t N ≤ t N–1 ≤ ...... ≤ t 1 ≤ t/2
t/2
t/2
t
t8
t7
t6
t5
t4
t3
t2
t1
Figure 16.8 A multiple kink–antikink configuration for the particle’s classical trajectory.
the configuration of N single instantons, with each instanton tunneling taken to be independent of the other tunnelings. In this approximation, since −1 θ < 0 lim tanh(ωθ) = sgn(θ), sgn(θ) = , ω→∞ 1 θ >0 the N instanton classical solution is given by xc(N ) (t) = ±a N
N &
sgn(t − tj ).
(16.57)
j =1
A typical Ninstanton configuration is shown in Figure 16.8. 16.7 Instanton transition amplitude In the dilute instanton gas approximation, the transition amplitude is given by τ/2 t1 tn−1 −τ H N aN = Ar dt1 dt2 · · · dtn −ae −τ/2
−τ/2
= Ar N =A
1 N!
!N
τ/2
−τ/2
dt −τ/2
1 N N r τ , N − instanton contribution. N!
(16.58)
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Hence, the full transition matrix element is given by summing over odd N multiple instanton configurations, and yields ∞
∞
r 2N+1 τ 2N+1 (2N + 1)! N=0 N=0 1 mω −( ω −r )τ ω = A sinh(rτ ) = − e−( 2 +r )τ , e 2 2 π
−ae−τ H a =
−ae−τ H a2N+1 = A
(16.59)
where A is given by Eq. 16.55, A=(
mω 1/2 − ωτ ) e 2. π
16.7.1 Lowest energy states For the double well potential, ae−τ H a is given by the sum over all even N multiple instanton configurations, starting with N = 0. Hence, from Eq. 16.58 ae
−τ H
∞ (rτ )2N a = A = A cosh(rτ ) (2N)! N=0
1 mω 1/2 −( ω −r)τ ω + e−( 2 +r)τ ]. ) [e 2 = ( 2 π Let the two lowest excited states, as given in Figure 16.9, be denoted by − and +; the completeness equation in this twostate sector is given by I −− + ++ + O(e−4/g )2 . 2
(16.60)
Hence, the completeness equation given in Eq. 16.60 yields ae−τ H a ≈ ae−τ H ++a + ae−τ H −−a = e−τ E+ −a2 + e−τ E− +a2 1 mω 1/2 −( ω −r)τ ω = ( + e−( 2 +r)τ ]. ) [e 2 2 π Hence, the two lowest energy eigenvalues are given by ω ω E− ( − r), E+ ( + r), 2 2
(16.61)
+Ú
–Ú
Figure 16.9 Ising configuration and an instanton.
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yielding the level splitting given by √ E = E+ − E− = 2r = 4ω3/2 2m eSc .
16.8 Instanton correlation function The correlation of the doublewell potential can be obtained far from the linearized theory by expanding the path integral about the instanton solutions [Polyakov (1987)]. Consider the two point correlator 1 Dx x(T1 )x(T2 )eS . G(T1 , T2 ) = Z A normal mode expansion of the correlation function about xc() , the instanton configuration, yields x(t) = η (t − tc ) +
xc() (t
− tc ) =
∞
cn ψn (t − tc ) + xc() (t − tc ).
n=1
The fluctuations about the xc() are given by η . In the semiclassical expansion, the correlator is given by G(T1 , T2 ) ( " () () S[η+x () ] Dx[η(T1 − tc ) + x (T1 − tc )][η(T2 − tc ) + x (T2 − tc )]e . ( " S[η+x () ] Dxe ( Note that is a sum over the instanton and antiinstanton classical solutions, denoted by x () . The expansion of the action yields, to leading order S[x] = S[η + xc ] = S[xc ] + S[η ] + . . . The first two classical solutions are S[xc0 ] = 0, S[xc1 ] = Sc . Summing over the trivial classical ( = 0) solution given by x = ±a and the = 1 one instanton and antiinstanton yields, to leading order in , " a 2 Z0 + eSc Z1 dtc xc (T1 − tc )xc (T2 − tc ) " G(T1 , T2 ) Z0 + eSc Z1 dtc " a 2 + BeSc dtc xc (T1 − tc )xc (T2 − tc ) " = , (16.62) 1 + BeSc dtc +∞ 2 Sc dtc xc (T1 − tc )xc (T2 − tc ) − a 2 . ⇒ G(T1 , T2 ) a + Be (16.63) −∞
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Coefficient Z0 is given by ∞ & 1 2 2 dcn e− 2 (λn,0 ) cn , Z0 = Dη0 eS[η0 ] = n=1
where the classical solution is given by xc(0) = ±a. The effective action S[η0 ] is given in Eq. 16.19 and λn,0 are the eigenvalues for the effective action about the trivial classical solution ±a. From Eq. 16.51, for the one instanton solution the path integral measure is C Dη1 = x˙c dtc ∞ n=1 dcn = dtc D η1 . The tc integration couples to the classical solutions and is included in Eq. 16.62 Coefficient Z1 is given by the remaining integrations, Z1 = D η1 eS[η1 ] . xc(0)
The classical solution xc(1) is the one instanton solution and yields ∞ ∞ & & 1 2 2 dcn eS[η1 ] = x˙c  dcn e− 2 (λn ) cn , Z1 = x˙c  n=1
n=1
where the eigenvalues λn for the single instanton are given in Eq. 16.30. Hence, from Eq. 16.62 C∞ " − 12 λ2n cn2 Z1 n=1 dcn e B= = x˙c  · C∞ " . − 12 (λn,0 )2 cn2 Z0 n=1 dcn e
(16.64)
16.9 The dilute gas approximation The dilute gas approximation, from Eq. 16.57, is based on an approximate description of the doublewell potential for large coupling, namely g 1 and leads to the simplification that xc (ti − tc ) = ±a sgn(ti − tc ). The dilute gas approximation considers the N instanton solution to consist of a collection of N noninteracting instanton and antiinstanton solutions. With these assumptions, the double well potential can be solved exactly. From Eq. 16.63, for T2 > T1 , the integral over the one instanton contribution is given by4 +∞ 2 a B dtc [sgn(T1 − tc )sgn(T2 − tc ) − 1] −∞ T1 T2 +∞ +∞ = a2B dtc − dtc + dtc − dtc −∞
T1
T2
−∞
4 The case of T > T is obtained in a straightforward manner by exchanging T with T . 1 2 1 2
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375
T1 − T2  ω−1 .
(16.65)
= −2a 2 BT1 − T2 ,
The integration range is shown in Figure 16.7. Hence, in the one instanton approximation, the correlator is given by G(T1 , T2 ) = a 2 (1 − 2BT1 − T2 eSc ). In the dilute instanton gas approximation, the contributions of all the multiinstantons are simply products of the contribution of a single instanton and antiinstanton. The multiinstanton tunneling times tc,n yield xcN (t)
=
N &
xc (t − tc,i )
i=1
with T1 ≤ tN ≤ tN−1 · · · ≤ t1 ≤ T2 . Note that the instantons are well localized; for large time t, the single instanton has the behavior xc (t) = ±a(1 + O(e−ωt ). The action for the superposition of an instanton (I) and an antiinstanton (AI), with separation T12 , is given by S2 Sc (I ) + Sc (AI ) + O(e−ωT12 ) = 2Sc ,
(16.66)
and hence the action for N instantons is SN NSc .
(16.67)
Similarly to the derivation of Eq. 16.59, the correlator for the dilute instanton gas is given by ∞ G(T1 , T2 ) a 2 (−2B)N eN Sc dtc,N dtN−1 · · · dt1 = a2
N=0 ∞
(−2B)N eN Sc
N=0 2 −T1 −T2 /ξ
=a e
(T2 − T1 )N N!
,
where correlation time ξ is given by ξ
2 3' 1 −Sc 1 2m ω → ∞ as g → 0. = e exp 2B 2B 3g 2
The nonlinear double well potential gives rise to a nonperturbative correlation length, with an essential singularity at g = 0. Any perturbation about the trivial
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vacua x = ±a would not yield the correlation length; in contrast, the semiclassical expansion yields a welldefined order by order procedure for evaluating the path integral. 16.10 Ising model and the double well potential The N instanton solution, in the dilute gas approximation, can be constructed from N single instanton configurations. Since the overlap of the component one instanton configurations is taken to be negligible, the action is given by c S = NSc = −N 2 , g with tunneling times given by t1 < t2 < t3 · · · < tN . From Eq. 16.57 we find that the N instanton solution, for strong coupling given by g 2 1, reduces to xc(N ) (t) = ±a
N &
sgn(t − tj ).
j =1
A single instanton configuration is equivalent to a configuration of the Ising spins, as shown in Figure 16.10 The multiinstanton configuration is now equivalent to the onedimensional Ising model, as the the trajectory of the particle has a value of either +a or −a, and the tunneling between these two configurations is equal to a spin flip for the Ising spin. To make the connection with the Ising model more explicit, consider the limit √ of ω = ag/ m → ∞, or equivalently g 2 → ∞; up to irrelevant constants, the potential given in Eq. 16.15 yields g2
lim e− 8 (x
g→∞
2 −a 2 )2
→ δ(x 2 − a 2 ) =
1 [δ(x − a) + δ(x + a)]. 2a
t1
a
+a
t1
a
+a
Figure 16.10 Ising configuration and an instanton.
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Discretizing time t = n, the path integral given in Eq. 16.18 yields & m ( 2 δ(xi2 − a 2 ) Z = Dxe− 2 i (xi+1 −xi ) i
m
(
2 i (xi xi+1 −xi )
δ(x 2 − a 2 ) Dxe ( = eK i μi μi+1 , Ising model, =
{μi =±1}
where μi = xi /a, K =
ma 2 .
Recall from Eq. 8.22, the Ising model’s transfer matrix is ! eK e−K . L = −K e eK The eigenstates and eigenvalues, from Eqs. 8.27 and 8.25, are given by ! 1 1 , λ1 = eK + e−K , λ1 = √ 1 2 ! 1 1 λ2 = √ , λ2 = eK − e−K . −1 2 Since λ1 > λ2 , the two states have the following identification: ! 1 1 ≡ −, vacuum state, λ1 = √ 2 1 ! 1 1 λ2 = √ ≡ +. 2 −1
(16.68)
The vacuum state is symmetric under the exchange of a → −a and hence preserves parity symmetry.
16.11 Nonlocal Ising model Let us consider the periodic and nonlocal Ising model N K μi e−αi−j  μj S= 2 i,j =1
=
K μi μj Aij . 2 ij
(16.69)
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Using Gaussian integration, the action S has the representation e = S
N &
dxi exp[−
i=1
1 xi A−1 μxi ], ij xj + 2 ij i
Aij = Ke−αi−j  . It can be verified that the inverse of Aij is given by A−1 ij =
1 [Cδij − B(δi−j −1 + δi−j +1 )]. K
Then S=− =−
N N 1 xi Cδij − B(δi−j −1 + δi−j +1 ) xj + μi xi 2K i,j =1 i=1 N N 1 {B(xi − xi+1 )2 + (C − 2B)xi2 } + μi xi . 2K i=1 i=1
Therefore Z=
{μ}
e
S[μ]
=
N &
dxi eS[x] ,
i=1
where C − 2B 2 B (xi − xi+1 )2 − ( xi + ln(2 cosh xi ) ) 2K i 2K i i B =− (xi − xi+1 )2 − V (x), 2K i
S[x] = −
and the potential, shown in Figure 16.11, is given by ⎧ C−2B 2 1 2 ⎨− 2K x + 2 x ,
C − 2B 2 − V = −( )x + ln(2 cosh x) ⎩ 2K
x0
x 2 + x. x 1 − C−2B 2K
For (C − 2B)/2K > 0, V is a double well, as shown in Figure 16.12. Hence, the Ising model, for a certain choice of parameters, is equivalent to the double well potential.
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Full range view
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Small scale of V
Figure 16.11 The potential for different m = −(C −2B)/2K: coupling strengths. –V
Figure 16.12 Double well potential from an Ising model of a lattice site for (C − 2B)/2K > 0.
To determine coefficients B and C, we note that N
−αi−j  Ail A−1 − B(e−αi−j +1 + e−αi−j −1 ). lj = Ce
l=1
For the case of i = j , RH S = C − B(e−α + eα ) = 1 ⇒ C = 1 + 2Be−α .
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For the case of i > j , RH S = Ce−α(i−j ) − B(e−α(i−j +1) + e−α(i−j −1) ) = e−α(i−j ) [C − B(e−α + eα )] = 0, and for i < j , RH S = e−α(i−j ) [C − B(eα + e−α )] = 0. Therefore, the case for both i > j and i < j yields C 1 1 ⇒ C= , B= . 2 cosh α tanh α 2 sinh α (C − 2B)/2K = (cosh α − 1)/2K sinh α > 0 for cosh α > 1 and hence the nonlocal Ising model exhibits symmetry breaking for all values of α > 0. B=
16.12 Spontaneous symmetry breaking Symmetry breaking is quite common in classical physics. For any potential with multiple minima, the particle can be stationary at any of the minima of the potential and consequently satisfy the equations of motion. In the case of the doublewell 2 potential given by V = g8 (x 2 − a 2 ), the particle can be classically at rest at either of the two points given by x = ±a, which breaks the x → −x parity symmetry of the potential. In quantum mechanics, the Mermin–Wagner theorem states that quantum mechanics cannot break continuous symmetries, but leaves open the possibility of breaking discrete symmetries. Symmetry in quantum mechanics means the following. Suppose H commutes with some operator R, that is, [H, R] = 0. The theory is said to have an unbroken symmetry, or a symmetric state space, if the ground state  is invariant under R, namely that it satisfies R = , symmetric theory. The theory has spontaneously broken symmetry if R = , spontaneously broken symmetry.
(16.70)
In particular, if R is the parity operator defined by R xR ˆ = −x, ˆ one has x ˆ = −R xR. ˆ It follows from the above equations that = 0, symmetric, x ˆ = 0, nonsymmetric.
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16.12.1 Infinite well Let us consider two infinitely deep potential wells. If the particle is in well I, it has an entire Hilbert space VI which is disjoint from the particle in well II; that is VI VI I = 0, and VI ∩ VI I = φ, with I , I I being the vacuum states. The discrete symmetry of the potential V (−x) = V (x) is broken by the vacuum since RI = I I = I , nonsymmetric vacuum.
16.12.2 Double well Let us now consider the doublewell potential Hamiltonian H =−
1 ∂2 g2 2 + (x − a 2 )2 . 2m ∂x 2 8
ˆ t (x) = ψt (−x). Let the Let R be the parity operator such that xRψt = Rψ “false” vacuum, centered around x = ±a, be denoted by ± . Then R± = ∓ , nonsymmetric vacuum. Hence ± spontaneously breaks the Rsymmetry. It will be shown that the true vacuum is invariant under R, namely that R = , symmetric vacuum.
(16.71)
16.13 Restoration of symmetry For the finite doublewell, it is shown how the symmetry of x → −x is restored by tunneling. Consider only the ground state sector of the Hilbert space of the double well. To any order in perturbation theory, one has for the lowest energy states of the double well the lowest order Hamiltonian given by ! E0 0 , H0  ± a = E0  ± a, H0 = 0 E0 with the degenerate eigenstates, as discussed in Section 16.7, given by ! ! 1 0 a , −a . 0 1
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In the limit of τ → ∞, all the states except the two lowest lying states decouple from the transition amplitude ae−τ H  − a. On this two dimensional subspace of V, the Hamiltonian H is a 2 × 2 matrix and hence yields the expansion ae−τ H  − a a(1 − τ H ) − a −τ H (a, −a), aH  − a ≡ H (a, −a). Equation 16.53 yields −τ H (a, −a) = τ NeSc . and hence (note the negative sign) H (a, −a) −Ne−c/g . 2
The full Hamiltonian is consequently given by $ # 2 E0 −Ne−c/g , H = 2 −Ne−c/g E0 with eigenstates and eigenvalues 1  = √ (a +  − a), 2 1 1 = √ (a −  − a), 2
E = E0 − Ne−c/g , symmetric, 2
E = E0 + Ne−c/g , 2
antisymmetric.
The vacuum state  was obtained earlier using the Ising approximation in Eq. 16.68. The true vacuum state for the double well is symmetric under the parity operator. The false vacua that to lowest order in perturbation theory start from either of the degenerate perturbative vacua, namely ±a, are corrected by the instantons, and symmetry is “restored” in the sense that a calculation for the vacuum state that includes the instanton contributions yields the actual symmetric and unique vacuum state. Instantons represent large quantum fluctuations that are represented by tunneling from one false vacuum to the other. The transition amplitudes could not have been obtained by considering only small fluctuations about the false vacua. It is the large fluctuations obtained by perturbing about the Ninstanton configurations that restore the symmetry of the false vacuum. A superposed state is completely nonclassical: the point particle is in an indeterminate state, existing simultaneously at two distinct points ±a. This is how quantum mechanics restores parity symmetry.
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16.14 Multiple wells By labeling the two wells as 1 and 2 one can write the Hamiltonian H as HN = E0 (11 + 22) − t (12 + 21),
(16.72)
with t = Ne(−c/g 2 ). The symmetry operator is then R1 = 2,
R2 = 1.
Consider the potential that has minima at sites na, with n = 0, 1, 2, . . . N. A calculation similar to the doublewell case gives to leading order that the effective low energy Hamiltonian is HN = E0
N
nn − t nn + 1 + n + 1n ,
(16.73)
n=1
where for simplicity we assume space is periodic with N +1 = 1. The symmetry of the Hamiltonian is given by the shift operator Rn = n + 1, [HN , R] = 0. One can verify that the eigenstates of HN are given by N 1 inθ θ = √ e n, HN θ = (E0 − 2t cos θ)θ . N n=1
with the true ground state given by the symmetric combination N 1 n, E = E0 − 2t.  = √ N n=1
The θ parameter is a measurable quantity and occurs in many theories, including the Yang–Mills Lagrangian. 16.15 Summary Nonlinear path integrals can have qualitative properties that cannot be obtained by perturbing about the linear part of the theory, which is defined by the quadratic piece of the Lagrangian. The doublewell potential was chosen to illustrate this aspect of nonlinear path integrals. Parity is a symmetry of the doublewell potential that is broken by the false vacuum of the linearized theory. The path integral was studied to ascertain whether the nonlinearities of the Lagrangian can restore parity symmetry. The domain of integration over which the path integral is defined is an infinite dimensional function space. The doublewell action has classical multiinstanton
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solutions that are configurations in function space far from the linear domain. To probe the nonlinearities of the theory, the path integral is expanded about the classical solutions and is evaluated using perturbation theory about the classical solution. The instantons have a zero mode, which is not constrained by the action, that arises from the instanton being free to tunnel from one false vacuum to the other at any time. The zero modes lead to a summation over all instants that the instanton can tunnel and produces a large effect. Examining the lowest energy states, it was seen that tunneling creates an offdiagonal term in the Hamiltonian which cannot be obtained by perturbing about the linearized theory. The offdiagonal term in turn leads to the Hamiltonian having a true vacuum (ground) state that preserves parity. The multiinstanton expansion can be approximated by a dilute gas of noninteracting instantons. This approximation scheme was used to evaluate the correlation function of the double well; the correlation length obtained shows that it depends on the coupling constants in a nonperturbative manner, further demonstrating the essential distinction between a linear and a nonlinear theory.
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17 Compact degrees of freedom
Quantum mechanical degrees of freedom come in many kinds and varieties and are variables that can take values in many different kinds of spaces (manifolds). Compact and noncompact degrees of freedom are two important types. In essence, a compact degree of freedom takes values in a finite range whereas the noncompact take values in an infinite range. The simplest compact degree of freedom is the Ising spin variable μ that takes two values, namely μ = ±1, and which was studied in Chapters 8 and 9. In Chapters 11 to 16, the noncompact degree of freedom x – taking values on the real line, that is x ∈ [−∞, +∞] – was studied for various models. Compact degrees of freedom have many specific properties not present for the noncompact case, and the focus of this chapter is on these properties. The degrees of freedom that take values in compact spaces for two different cases are discussed, with each case having its own specific features, as follows: • A degree of freedom taking values on a circle S 1 . This case has a nontrivial topological structure that occurs in a wide range of problems, with the simplest case being of a quantum mechanical particle moving on a circle S 1 . • A degree of freedom taking values on a two dimensional sphere S 2 , a compact space. This degree of freedom can represent a quantum particle moving on a sphere and is the simplest case of a quantum particle moving on a curved manifold. In Section 17.1 the degree of freedom taking values on a circle is introduced, and in 17.2 its multiple classical solutions are derived. The degree of freedom taking values on a sphere S 2 is discussed in Section 17.3 and its Lagrangian is derived in Section 17.4; a divergence arising from the curvature of the sphere is discussed in Section 17.5. Sections 17.6–17.8 apply the S 2 degree of freedom to study of the statistical mechanics of the DNA molecule.
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17.1 Degree of freedom: a circle The S 1 degree of freedom takes its values on a circle. The space S 1 is geometrically flat, similarly to the real line, but is a topologically nontrivial manifold: S 1 is not simply connected; what this means is that a loop on the circle cannot smoothly be contracted to a single point. One can wind around the circle n times, and all these windings are inequivalent since they cannot be mapped to, say, a single loop. The existence of the winding number is the single most important reflection of the nontrivial topology of S 1 . Consider a particle moving on a circle of radius R; a degree of freedom x then has the property that all points x + 2πnR, with n = 0, ±1, ±2, . . . ± ∞ are equivalent. In particular, the state function ψ(x) has the symmetry1 ψ(x + 2πnR) = ψ(x),
n = 0, ±1, ±2, . . . ± ∞.
The Hamiltonian is given by 1 ∂2 , x ∈ [−πR, +πR]. 2m ∂x 2 The state function has the normalization +π R 1= dxψ(x)2 . H =−
−π R
The normalized eigenfunctions of the Hamiltonian are given by H ψn = En ψn , einx/R , ψn = √ 2πR
En =
n2 , 2mR 2
n ∈ Z.
(17.1)
It is convenient to define a new angular variable θ such that x θ= ∈ [−π, +π]. R Then 1 ∂2 , 2mR 2 ∂θ 2 with the orthonormal state functions given by +π R +π dxψn∗ (x)ψm (x) = R dθψn∗ (θ)ψm (θ) = δn−m . H =−
−π R
−π
1 This condition can be relaxed to
ψ(x + 2π nR) = eiφ ψ(x),
n = 0, ±1, ±2, . . . ± ∞.
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The evolution kernel, for Euclidean time τ , is given by ˆ K(θ , θ; τ ) = θ e−τ H θ = e−τ En ψn (θ )ψn∗ (θ) 1 = 2πR
∞
e
n
− 12
τ mR 2
n2 in(θ −θ)
e
.
(17.2)
n=−∞
17.1.1 Poisson summation formula The Poisson summation formula is a useful result in the study of a quantum particle moving on a circle, and is given by the identity ∞
δ(ξ − n) =
n=−∞
∞
e2π iξ .
(17.3)
=−∞
To prove the Poisson summation formula, let f (ξ ) =
∞
δ(ξ − n) ⇒ f (ξ + m) = f (ξ ),
m = integer.
n=−∞
Let ξ = 0 and m = 1; then f (1) = f (0). Hence f (ξ ) is a periodic function on the interval 0 ≤ ξ < 1; note that the point ξ = 1 has been excluded, since due to periodicity, it is identical to the point ξ = 0. Consider the following Fourier expansion of f (ξ ): ∞
f (ξ ) =
e2π iξ f ,
=−∞
⇒ f =
1
0
1
dξ e−2π iξ δ(ξ − n) = 1,
n=−∞ 0
∞
⇒ f (ξ ) =
∞
dξ e−2π iξ f (ξ ) = e2π iξ ,
=−∞
which proves Eq. 17.3. The identity yields ∞ n=−∞
fn =
∞
dξ −∞
∞ n=−∞
δ(ξ − n)f (ξ ) =
∞ =−∞
∞
dξ e2π iξ f (ξ ),
(17.4)
−∞
where f (ξ ) is the extension to a continuous argument ξ of the function fn defined on a lattice.
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17.1.2 The S1 Lagrangian The Lagrangian is given by the Dirac–Feynman formula
K(θ , θ; τ ) = θ e−H θ = N ()e L(θ,θ ) .
(17.5)
Taking the limit → 0 in Eq. 17.2 is hopeless since higher and higher order terms are necessary to evaluate the sum. To take the limit one needs to invert τ → τ1 in Eq. 17.2. The Poisson summation formula given in Eq. 17.4 yields
θ e
−τ H
∞ 1 − τ 2 n2 in(θ −θ) θ = e 2mR e 2πR n=−∞ ∞ 1 − τ ξ2 dξ = e2π iξ e 2mR2 eiξ(θ −θ) 2πR =−∞ ∞ m − mR2 (θ −θ−2π )2 = e 2τ . 2πτ =−∞
Hence, taking the limit of τ = yields 1 m − 1 mR2 (θ −θ)2 −H θ + O e− θ e e 2 2π = N ()e L ,
(17.6)
(17.7)
and the Lagrangian is given by 1 θ − θ L = − mR 2 2 m with N () = . 2π
!2
dθ 1 = − mR 2 2 dt
!2 ,
θ ∈ [0, 2π ],
17.2 Multiple classical solutions The evolution kernel is defined by
K(θ , θ; τ ) =
Dθe
"τ 0
L
.
(17.8)
The path integral measure, for τn = n , N = τ , is given by
&
+N/2
Dθ = lim
N→∞
n=−N/2
+π π

m dθn . 2π
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The action for the particle on a circle is given by 1 S = − mR 2 2
τ
dt 0
dθ(t) dt
!2 ,
θ ∈ [0, 2π ].
The action is similar to that of a free particle, except that now θ is a compact variable, taking values on a circle. To evaluate the evolution kernel, a semiclassical expansion is carried out about all the classical solutions. The equations of motion are given by θ¨ = 0 ⇒ θcl = (a + bt) mod 2π, B.C. θc (0) = θ, θc (τ ) = θ + 2π,
: winding number.
(17.9)
The multiple classical solutions are shown in Figure 17.1 by displaying θ as a noncompact variable. The winding number, shown in Figure 17.2, reflects the fact that θ(t) is a compact variable that can satisfy the boundary conditions by winding
Figure 17.1 Classical solutions satisfy the boundary conditions, shown for zero, one and two windings around the compact direction.
Figure 17.2 Winding number; the unbroken line is the classical solution with no winding and the dashed line is the classical solution with one winding around the compact direction.
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around the compact direction. Eq. 17.9 shows that the final position is equal to θ modulo 2π, since the compact quantum variables θ(t) are periodic. Hence (θ − θ) 1 t t + 2π , θ˙cl = θ − θ + 2πl . τ τ τ The classical action for the th winding number is given by 2 mR 2 τ 2 mR 2 θ − θ + 2πl θ˙cl dt = − · . Sc () = − 2 2 τ 0 θc (t) =θ +
As expected, the more the classical solution winds, the larger is the velocity it requires; hence, classical solutions with higher and higher winding numbers have more and more negative values for the classical action, as given by Sc (). A semiclassical expansion of the path integral about all the classical solutions, similar to the one carried in Section 16.6 for the double well potential, yields ∞ ∞ mR 2 2 eSc = N(τ ) e− 2τ (θ −θ+2π ) . Z = DθeS = N(τ ) =−∞
=−∞
To obtain the prefactor "N (τ ) one needs to analyze the path integral over the compact variable, namely DθeS ; consider the change of variables θ(t) = θc (t) + z(t), B.C. z(0) = z(τ ) = 0.
(17.10)
This yields, from Eq. 17.9, the action τ 1 1 1 2 2 2 2 2 ˙ ˙ S = − mR (θcl + z˙ ) dt = − mR z˙ 2 dt θcl dt − mR 2 2 2 0 2 θ − θ + 2πl 1 1 2 (17.11) = − mR − mR 2 z˙ 2 dt. 2 τ 2 The quantum variable z(t) is noncompact, taking values of −∞ ≤ z(t) ≤ +∞; the fluctuations about the classical solutions are noncompact, making the evaluation of the prefactor an exact Gaussian path integral. The prefactor N (τ ), in fact, is the same as that for a free particle on the real line and it has been evaluated earlier in Section 5.8. Since z(t) does not depend on the classical solutions, it completely factorizes from the sum over the classical solutions and yields the final result " − 12 mR 2 z˙ 2 Dze eScl () = N (τ ) eScl () K(θ , θ; τ ) = =
m 2πτ
∞
e
2 2 − mR 2τ (θ −θ+2π )
.
(17.12)
=−∞
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The result obtained in Eq. 17.12 was obtained earlier, in Eq. 17.6, using the Poisson summation formula. The summation n in Eq. 17.2 is a summation over all the discrete momentum states of the particle whereas in Eq. 17.12 is a summation over all the winding numbers. For a compact variable, the momentum and winding number are “dual” to each other. For noncompact variables, the concept of winding numbers does not exist and hence there is no duality.
17.2.1 Large radius limit The limit of the radius of S R → ∞ takes the circle into the real line. We recall from Eq. 17.2 1
K(θ , θ; τ ) =
∞ 1 − 12 mτ n22 i n ·R(θ −θ) R e R e . 2πR n=−∞
(17.13)
Let p = n/R; for R → ∞ we have −∞ ≤ p ≤ +∞,
∞
=R·
dp.
n=−∞
Let us define Rθ = x, Rθ = x . Hence ∞ m − m (x −x)2 1 τ 2 − 2m z iz(x−x ) K(x , x; τ ) = dze e = . R e 2τ 2πR 2πτ −∞ Furthermore, from Eq. 17.12 ∞ ∞ m − 1 mR2 (θ −θ−2π )2 m − m (x −x−2π R)2 e 2 τ = e 2τ lim K = R→∞ 2πτ =−∞ 2πτ =−∞ m − m (x −x)2 ⇒K= . (17.14) e 2τ 2πτ We have recovered the result for the evolution kernel of a free particle moving on the real line, given in Section 5.8.
17.3 Degree of freedom: a sphere The main new feature of a degree of freedom that is a circle, for example a quantum particle moving on a circle S, is the winding number arising from the fact that the space is not simply connected. The degree of freedom taking values on a sphere is exemplified by a quantum particle moving on a two dimensional sphere S 2 . The two dimensional sphere is simply connected in that any closed loop on
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Figure 17.3 Particle moving on a sphere.
S 2 can be continuously contracted to a point, with every intermediate loop in the mapping being points in S 2 . Hence S 2 has a trivial topological structure and has no winding number; in particular, unlike the S 1 case, the S 2 degree of freedom does not have multiple classical solutions. Although S 2 is topologically simple, it has another important geometrical feature, namely that of curvature: the two dimensional sphere is a Riemannian manifold with constant curvature; for a sphere of radius R the Ricci scalar curvature is given by 2/R 2 . A particle moving on a sphere is shown in Figure 17.3; the position of the particle on the sphere is specified by two angles, which in spherical coordinates shown in Figure 17.4 are given by the polar and azimuthal angles θ, φ respectively. The threedimensional Laplacian, in spherical polar coordinates r, θ, φ, is given by 2 2 = ∂ + ∂ + ∂ = 1 ∂ r − L , ∇ ∂x 2 ∂y 2 ∂z2 r ∂r 2 r2
where L2 =
1 ∂ ∂ ∂ + 2 + cot θ , 2 ∂ϕ 2 ∂θ ∂θ sin θ
0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π.
The position of the particle on the sphere is given by R (cos θ sin ϕ, cos θ cos ϕ, sin θ) . A particle moving on a sphere of fixed radius r = R has ∂r∂ = 0. Two particles at the ends of a rigid rod undergoing rotations form a (linear) rigid rotor. A particle confined to move on the surface of a sphere is equivalent to the motion of a rigid
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17.4 Lagrangian for the rigid rotor
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Figure 17.4 Spherical coordinates.
Figure 17.5 A quantum mechanical rigid rotor.
rotor (for example, a diatomic molecule), where I is the moment of inertia of the rigid rotor, which is shown in Figure 17.5. The Hamiltonian of the rigid rotor is given by H =−
2 2 2 2 L = − L, 2mR 2 2I
I = mR 2 .
(17.15)
The eigenfunctions are the spherical harmonics Ylm (θ, ϕ), given by Gottfried and Yan (2003), and satisfy H Ylm =
2 l(l + 1)Ylm (θ, φ) , 2I
l = 0, 1, 2, . . . ∞, − l ≤ m ≤ l.
17.4 Lagrangian for the rigid rotor The rigid rotor is described by two angular degrees of freedom, namely θ, φ; the Hilbert space has coordinate basis states θ, φ and the dual (“momentum”) coordinates are l, m with
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θ, ϕl, m = Ylm (θ, ϕ). Note that
π
π
dθ 0
−π
dϕ sin θ θ, ϕ θ, ϕ = I.
The Lagrangian is given by the Dirac–Feynman formula
N () e L = θ , ϕ e− H θ, ϕ. It is more convenient to use the momentum basis p, q since, for infinitesimal time , it is equivalent to the spherical harmonic basis l, m. The completeness equation is given by +∞ dpdq p, qp, q = I, θ, ϕp, q = eipθ+iqφ . 4π 2 −∞ The completeness equation yields, from Eq. 17.15 ! 2 − 2I q2 +p2 −ip cot θ dpdq sin θ N () e L = eip(θ −θ )+iq (ϕ −ϕ ) e 4π 2 ! q2 −ϕ − +p2 dpdq ip θ −θ+ cot θ +iq ϕ 2 2I ( ) sin θ 2I = e e 4π 2 ! 2 2θ I 2 2I − sin ϕ −ϕ ) − 2 θ −θ+ 2I cot θ ( 2 sin θe = e . Hence for → 0, the Lagrangian of the rigid rotor is given by !2 I I 2 2 ˙ θ+ L = − sin θ ϕ˙ − cot θ . 2 2 2I The normalization N () =
! 2I sin θ
(17.16)
(17.17)
depends on θ and is not a constant. This is a typical case for all path integrals on curved manifolds. The prefactor N () cannot be obtained from the Lagrangian and is a result of the state space of the degree of freedom taking values on a curved manifold. The classical Lagrangian is given by the velocity of the particle on a sphere and is I L = − sin2 θ ϕ˙ 2 + θ˙ 2 , I = mR 2 . 2
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17.5 Cancellation of divergence
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Compared to the classical Lagrangian above, the extra term 2I cot θ in Eq. 17.16 is a quantum correction to L. The path integral is2 K θ , ϕ ; θ, ϕ; t = θ , ϕ e−τ H θ, ϕ N−1 & 0 & % % 1 N−1 θn+1 , ϕn+1 %e−H % θn , ϕn dθn dϕn sin θn =N n=0
=N
N−1 &
n=1
dθn dϕn sin θn eS ,
(17.18)
n=1
S=
N−1
L (θn+1 , ϕn+1 , θn , ϕn , ),
n=0
where N is a normalization constant.
17.5 Cancellation of divergence The path integral measure, from Eq. 17.18, is given by N−1 &
sin θn = e
(N−1 n=1
ln(sin θn )
.
n=1
Note that there is a problem with the measure term as it is apparently divergent since, in the limit → 0, N−1
1 1 ln (sin θn ) → · ln (sin θn ) = n n=1
τ
dt ln (sin θ (t)) → ∞.
0
The only way to obtain a finite result from the path integral is if all the divergent terms that appear in the path integral exactly cancel. To show the cancellation of the 1/ term, a perturbation expansion is carried out for the path integral in powers " of θ."The functional integral Dφ is performed while " keeping the functional integral Dθ fixed. Performing the functional integral Dφ" generates a term for the θ variable that precisely cancels the divergent term − 21 θ 2 that comes from the path integral measure. Note that 0 ≤ θ ≤ π ; the minimum value of the action S is around θ = π/2. Hence, the angle θ is shifted to θ˜ as shown in Figure 17.6, so that the shifted angle, in the path integral, is constrained to be near zero due to the action. 2 Henceforth, unless required, we set = 1.
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Figure 17.6 Spherical coordinates with angle θ measured from the “forward” yaxis.
Hence, the angle θ is shifted to θ˜ = θ − π/2, with −π/2 ≤ θ˜ ≤ π/2; dropping the tilde yields, from Eq. 17.16 I I L = − cos2 θ φ˙ 2 − θ˙ + tan θ 2 2 2I Let τ → ∞; the path integral is given by $ # & cos θt exp Z = DθDφ
∞
!2 .
' dtL(t) .
−∞
t
Consider the case of the moment of inertia I 1 and the action is expanded in powers of 1/I . Hence θ2 I 1− + ··· L− 2 2
!2
I φ˙ 2 − θ˙ 2 + O(1). 2
Note that cos θ 1 −
" & θ2 1 τ 2 ⇒ cos θt e− 2 0 dtθ . 2 t
Using the notation
∞
−∞
dt ≡
,
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17.6 Conformation of DNA
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the partition function is given by ' ∞ " 2 I I I 1 θ − 2 2 2 2 ˙2 ˙ ˙ Z θ φ + ··· . DθDφe exp − φ − θ + 2 2 2 −∞ (17.19) The generating function for a free particle, namely ' ' I 1 1 2 Dφ exp − jt Dt−t jt , φ˙ + j φ = exp Z 2 2I has been evaluated in Eq. 12.1. The correlation function is given by
G(t − t ) = E[φt φ ] = t
and yields E[φ˙ t φ˙ t ] =
1 1 ∂t ∂t φt φt = I I
dω eiω(t−t ) , 2π ω2
(17.20)
dω iω(t−t ) 1 = δ t − t . e 2π I
Hence, for discrete time E[φ˙ t2 ] =
1 , I
(17.21)
and from Eqs. 17.19 and 17.21 " 2 I I 1 − 2 2 1 2 θ ˙ θ 1+ Z Dθe − θ 2 I 2 ' 1 I Dθ exp − θ˙ 2 , terms exactly cancel! 2 The divergent piece of the measure term is canceled by a term generated from the kinetic term of the ϕ variable. It can be shown that the cancellation holds to all orders in powers of 1/ [ZinnJustin (1993)]. In summary, all the singular terms in the path integral cancel, yielding perfectly finite results for all computations.
17.6 Conformation of DNA A polymer is a linear chain of molecules and can be considered to be a string laid out in space. The conformation (shape) of a polymer is statistically determined by the likelihood of it taking the various allowed shapes. The statistical mechanics of the system – in equilibrium at temperature T – is described by the competition between the entropy of the chain of length L and the energy required for bending it.
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t2 t1 y
x
Figure 17.7 The trajectory of a particle specified by its tangent vectors.
A polymer in space is shown in Figure 17.7. If one does not need to know the location of the polymer in threedimensional space, then as shown in Figure 17.7, the polymer’s configuration is completely determined by specifying its three dimensional tangent vector along its length. Let us consider a DNA molecule, with total length L, that is free to move in a solute, discussed by Phillips et al. (2008). The curve of the DNA in three dimensional space is parameterized by a parameter s; the vector t(s) that is tangential to the shape of the DNA specifies the shape of the DNA, as shown in Figure 17.8. The parameterization is chosen so that, for every s, the tangential vector has unit length, namely t(s) · t(s) = 1, and takes values on a twodimensional sphere S 2 . In terms of the spherical polar angles given in Figure 17.4, the tangent vector has the coordinates t = (sin θ cos φ, sin θ sin φ, cos θ) . The degree of freedom for the DNA is taken to be the tangent vector t, which takes all possible allowed values along the curve occupied by the polymer (DNA); the random configurations of the DNA can be modeled by assigning a probability distribution for the different configurations of t(s). The statistical mechanics of the DNA molecule is modeled by the following “action” and Lagrangian which, to leading order in ξ , are given by ! L ξ dt 2 ξ ξ dsL, L = − = − sin2 θ φ˙ 2 − θ˙ 2 . S= 2 ds 2 2 0 The role of time is played by the parameter s that runs along the curve.
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17.7 DNA extension
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Figure 17.8 DNA polymer in threedimensional space.
The energy of bending is given by H = −kB T S = kB T × and the partition function Z=
ξ 2
H Dt exp − kB T
L
ds 0
'
dt ds
!2 ,
=
DteS .
Since the action is a nonlinear functional of the degrees of freedom, the partition function can be evaluated perturbatively by expanding the action about θ = φ = 0. The correlation function of the zcomponent of the tangent vectors is given by G s, s = E[tz (s)tz (s )]. In order to calculate the force required to stretch the DNA" we need to include a L force term applied along zˆ in the action S that is given by f 0 tz ds. Hence !2 ξ ξ d t + f tz = − sin2 θ φ˙ 2 + θ˙ 2 + f cos θ. L=− 2 ds 2 17.7 DNA extension "L The chain’s extension is given by P = 0 cos θ(s)ds. Note that if all the tangent vectors are parallel, then θ = 0 and z = L leading to the full extension of the DNA. The average extension is given by
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1 P¯ = E[P] = Z
Dt
T
cos θdseS =
0
∂ ln Z . ∂f
For the small force limit f 1, which yields the expansion L f2 L Z(f ) = DteS 1 + f cos θ(s)ds + dsds cos θ(s) cos(s ) + · · · 2 0 0 = 1 + G1 + G2 + O(f 3 ). For ξ 1 the Lagrangian is approximately given by !2 1 ξ ξ ξ ξ L − φ˙ 2 1 − θ 2 − θ˙ 2 − φ˙ 2 − θ˙ 2 + O(θ 4 ). 2 2 2 2 2 For making the computation well defined we introduce a regulator ω and consider ξ ξ 2 L = − φ˙ 2 − θ˙ + ω2 θ 2 . 2 2 The limit ω → 0 will be taken at the end of the calculation. The propagator, from Eq. 11.9, is approximately given by e−ωs−s  E[θ(s)θ(s )] . 2ωξ
For the first term note that L G1 = f E[cos θ(s)]ds, 0 iθ(s) S 1 1 −iθ(s) 0 Dθ e Dθeiθ(s) eS0 . e = +e E[cos θ(s)] = 2Z Z Hence, for jl = δ (s − l), we have from Eqs. 12.1 and 17.20 1
cos θ = e− 2
"
jl Dll jl
1
= e− 4ωξ ⇒
1
lim cos θ = e− 4ωξ → 0
ω→0
⇒ G1 = 0. The second term is given by f2 G2 = 2
L
dsds G s, s , G s, s = E[cos θ(s) cos(s )].
0
The path integral is approximated by 1 DθDφeS cos θ(s) cos θ(s ) G s, s Z
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17.8 DNA persistence length
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"L ξ 2 1 2 2 ˙ Dθe− 0 2 (θ +ω θ )ds cos θ(s) cos θ(s ) Z 1 = E ei{θ(s)+θ(s )} + ei{θ(s)−θ(s )} . 2
Let us define the external current for the two terms, respectively, by dlji (l)θ(l), i = 1, 2 ⇒ j1 (l) = δ (l − s) + δ l − s , j2 (l) = δ (l − s) − δ l − s . The first term for G s, s is zero since ! 1 dldl (δl−s + δl−s ) e−ωl−l  (δl −s + δl −s ) lim exp − ω→0 4ωξ ! 1 −ωs−s  → 0. = lim exp − 1+e ω→0 2ωξ Hence, the correlator is given by ! 1 1 −ωl−l  dldl (δl−s − δl−s ) e G s, s = lim exp − (δl −s − δl −s ) ω→0 2 4ωξ # %$ % ! 1 %s − s % 1 1 = lim exp − 1 − e−ωs−s  . = exp − ω→0 2 2ωξ 2 2ξ The correlation length is 2ξ , where ξ is called the persistence.
17.8 DNA persistence length DNA can be thought of as a jointed chain having rigid links of persistence length ξ , as shown in Figure 17.9. Typically, ξ 50 nm. The distance between base pairs is approximately 0.33 nm, hence ξ is about 150 base pairs. Total length of the DNA is L 16 μm and f 0.1 pN [Phillips et al. (2008)]. ξ
ξ
ξ
DNA Freely Jointed Chain
Figure 17.9 A DNA polymer with finite correlation length is equivalent to a freely jointed rigid chain.
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For f 0, the correlator yields the partition function s−s  f2 L f2 L dsds G s, s = 1 + dsds e− 2ξ Z(f ) = 1 + 2 0 4 0 1 ∼ = 1 + f 2 Lξ for L ξ. 2 Hence, from the equation above f 0 ⇒
E[P] 1 ∂ ln Z(f ) = = f ξ, DNA extension per unit length. L L ∂f
The other limit is that of a large force f 1, in which case the action is approximated by ! ' 1 2 ξ 2 ξ f 2 ξ ˙ 2 ξ ˙2 2 ˙ ˙ θ + θ , L − φ − θ + f 1 − θ + ··· = − φ − 2 2 2 2 2 ξ L L 1 ds cos θ L − ds θ 2 . P= 2 0 0 In this approximation, using the result of the simple harmonic oscillator given in Eq. 11.10 (with m = ξ and ω2 = f/ξ ), E[θ(s)θ(s )] = yields the result 1 E[P] = Z
√ √ 1 √ e−( f / ξ )s−s  2ξ f
L
1 DθDφ ds cos(θ(s))e L − 2 0 L 1 1 =L− ds √ 2 0 2ξ f E[P] 1 ⇒ =1− √ . L 4ξ f
L
ds θ 2 (s)eS
S
0
For intermediate f , the following equation interpolates between small f and large f : fξ
1 1 E[P] − . + 2 L 4 4 (1 − E[P]/L)
The graph of force versus extension for the DNA is shown in Figure 17.10. The path integral for the DNA’s conformation can answer more complicated questions such as what is the likelihood of the DNA looping and interacting. Intersections are sometimes crucial for obtaining the full information encoded in the base pairs as the interaction. When the DNA loops, there are special proteins sitting on the DNA that lock the intersections and hence bringing otherwise distant base pairs into close proximity [Phillips et al. (2008)].
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17.9 Summary
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E[P] / L
Figure 17.10 Extension of the DNA.
17.9 Summary The two cases of compact degrees of freedom that were studied, namely S 1 and S 2 , have qualitative features that occur widely in nonlinear theories. The S 1 degree of freedom is periodic and hence is a topologically nontrivial theory that has multiple classical solutions, classified by the number of times the classical path winds around S 1 . Furthermore, the momentum is discretized due to the periodicity of the degree of freedom. The path integral was defined using the Hamiltonian. The semiclassical expansion of the path integral yielded the exact result; the S 1 theory has the simplifying feature that the semiclassical expansion yields the exact path integral. The Poisson summation formula allowed the semiclassical expansion to be exactly resummed and leads to the Lagrangian description of the degree of freedom. In effect, the Poisson summation formula interchanges the sum over the winding number of the classical solutions with the discrete momentum of the periodic degree of freedom. The conformational properties of the DNA molecule – in equilibrium at finite temperature – were modeled by an S 2 degree of freedom. It was seen that the mathematics for describing the statistical mechanics of a system by a path integral is identical to that used for describing a quantum system, with the difference lying only in the interpretation of the results. The key new feature of the S 2 degree of freedom is that S 2 is a manifold with constant nonzero curvature. This leads to a nontrivial measure for the path integral,
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which, in turn, gives rise to apparent divergences in the path integral. The divergence arising from the nontrivial measure is a generic feature of all path integrals defined for degrees of freedom that take values in a curved manifold. An expansion in powers of the inverse of the moment of inertia I showed that, to lowest order, all the divergences exactly cancel; it can be shown that this cancellation takes place to all orders [ZinnJustin (1993)]. The path integral yields finite and welldefined results for all physical quantities. Various experimentally observable properties of the DNA were derived to illustrate the flexibility and utility of modeling the system using a path integral.
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Index
Qoperator matrix elements, 298, 300 similarity transformation, 297, 299, 301, 303 S1 classical solution multiple, 387, 388, 390 winding number, 389, 390, 392 Lagrangian, 387, 388, 390 S2 cancellation divergence, 394–397, 399 correlator, 398, 399, 402 DNA polymer, 396, 397, 399 acceleration evolution kernel symmetry, 292, 294, 309, 311 Hamiltonian classical, 283, 285 Lagrangian coordinate, 274, 276 domains, 278, 280 path integral coordinate, 274, 276 propagator equal frequency, 331, 333 action, 105 acceleration completeness equation, 286, 288 equations of motion, 276, 278 antifermion, 204 conjugation, 205 Hilbert space, 206 state space, 204 Bohr, 7 boundary conditions acceleration, 289, 291, 293 canonical equations, 80 circle
Lagrangian, 387, 388, 390 classical action acceleration, 277, 279 magnetic field, 268 oscillator, 233 classical equation of motion doublewell, 356, 357, 359 classical solution acceleration, 275, 277 vacuum, 312, 314 instanton, 358, 359, 361 oscillator, 231 coherent states path integral, 99 commutation equation Euclidean, 89 Minkowski, 89 completeness equation, 32, 74 Jordan block 3×3, 345, 347 coherent states, 32, 35, 39, 98 coordinate basis, 31 dual eigenstate, 324, 326 eigenstates, 37 fermion, 203 fermion–antifermion, 207 Ising, 163 magnetic field, 181 Jordan block, 336–339 Jordan block 2×2 , 342, 344 matrix elements, 32 momentum basis, 42 pseudoHermitian, 319, 321 conditionality probability oscillator, 239 conservation laws energy, 82 symmetry, 82 Copenhagen interpretation, 7 correlator S 2 , 398, 399, 402
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“9781107009790AR” — 2013/11/5 — 21:26 — page 410 — #428
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410 correlator (con’t) linear regression, 185 creation operator, 226 degree of freedom, 10 S 1 , 385, 386, 388 S 2 , 390, 391, 393 circle, 385, 386, 388 compact, 384, 385, 387 continuous, 30 periodic, 39 sphere, 390, 391, 393 density matrix mixed states, 50 pure states, 50 destruction operator, 226 Dirac bracket, 90 bracket notation, 12 constraint commutation equation, 91, 92 two constraints, 91 Dirac brackets action acceleration, 280, 282 Dirac delta function, 33 Dirac–Feynman formula, 67, 74 continuous paths, 69 discrete paths, 69 DNA correlation length, 400, 401, 404 extension, 398, 399, 402 force extension, 401, 402, 405 jointed chain, 401 persistence length, 400, 401, 404 statistical mechanics, 397, 398, 401 doublewell kink, 358, 359, 361 multikink, 361, 362, 364 doublewell potential Ising model, 375, 376, 378 eigenfunctions evolution kernel, 249 eigenfunctions pseudoHermitian left, 306, 308 right, 306, 308 pseudoHermitian H , 305, 307 pseudoHermitianH † , 305, 307 evolution kernel, 63 S 1 , 390, 391, 393 large radius, 390, 391, 393 indeterminate final position, 256 circle, 386, 387, 389 constant source, 260 doublewell
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Index singularity, 360, 361, 363 eigenfunctions, 249 free particle, 93 magnetic field, 267 oscillator, 230 Faddeev–Popov analysis instanton zero mode, 363, 364, 366 Fermi pseudopotential, 59 fermion, 198 antifermion eigenstates, 214 calculus, 198 complex, 207 Gaussian, 209 Gaussian integration, 207 generation function, 209 Hamiltonian, 214 Hilbert space, 201 integration, 200 Lagrangian, 217 normal ordering, 212 path integral, 211 real, 207 variables, 199 fermion–antifermion conjugation, 206 Feynman path integral, 61, 70 evolution kernel, 72 see path integral, 61 Fokker–Planck path integral, 156 free energy oscillator, 243 functional differentiation, 115 chain rule, 115 Gaussian path integrals, 251 Gaussian integration, 129 Nvariables, 131 fermions, 207 Gaussian random variable, 130 generating function oscillator, 234 Hamiltonian, 20, 22, 64 acceleration operator, 285, 287 eigenfunctions, 51 Euclidean, 84, 85 fermionic, 214 Fokker–Planck ground state, 156 pseudoHermitian, 155 Jordan block, 330, 332 2×2 , 342, 344 mechanics, 80 oscillator, 226
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“9781107009790AR” — 2013/11/5 — 21:26 — page 411 — #429
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Index path integral, 75 phase space quantization, 94 pseudoHermitian, 155, 295–298 critical, 331, 333 eigenfunctions, 304–307 equivalent, 297, 299 excited states, 313, 315 similarity transformation, 296, 298 quadratic momentum, 87 quasiHermitian, 296, 298 rigid rotor, 392, 393, 395 selfadjoint extension, 55 Hamiltonian: Fokker–Planck, 151 pseudoHermitian, 153 harmonic oscillator forced coherent states, 102 coherent states, 101 Heisenberg, 7 Heisenberg algebra unitary representation, 49 Heisenberg commutation equation, 47 Heisenberg equations pseudoHermitian Hamiltonian, 321, 323 Hilbert space, 14 fermionic, 201 indeterminate paths, 23 instanton, 354, 355, 357 classical solution, 358, 359, 361 coefficient, 367, 368, 370 correlation function, 372, 373, 375 dilute gas, 373, 374, 376 expansion spectral representation, 363, 364, 366 multi, 369, 370, 372 transition amplitude, 370, 371, 373 zero mode doublewell, 361, 362, 364 Faddeev–Popov analysis, 363, 364, 366 Ising 2 × N lattice, 176 block spin, 191 correlator open chain, 167 periodic chain, 169 degree of freedom, 161 magnetic field, 180 correlator, 184 evolution kernel, 182 magnetization, 190 partition function, 189 transfer matrix, 181 magnetization, 183 model, 161
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411
nonlocal, 376, 377, 379 partition function, 172 path integral, 171 periodic lattice, 168 renormalization, 191 spin, 161 state space binary, 163 Ising model, 161 doublewell potential, 375, 376, 378 magnetic field, 180 Ising spin Bloch sphere, 164 Heisenberg operator, 167 Schrödinger operator, 167 Ito chain rule, 137 discretization, 135 Ito calculus, 136 Jordan block 2×2 Schrödinger equation, 343, 345 2×2 , 340, 342 completeness, 336–339 Hamiltonian, 339, 341 propagator, 337, 339, 344, 346–349 kink doublewell, 358, 359, 361 Kolomogorov, 127 Lagrangian, 69, 105 S 1 , 387, 388, 390 acceleration, 273, 275 Euclidean, 84 fermionic, 217 path integral, 75 rigid rotor, 393, 394, 396 Langevin equation linear, 140 nonlinear, 145 potential, 143 Laplacian, 391, 392, 394 Legendre transformation, 81 magnetic field path integral, 267 measurement, 18 momentum Euclidean, 84 momentum basis path integral, 243 multikink doublewell, 361, 362, 364 multiinstantons, 369, 370, 372 normal ordering, 98 normal random variable, 129, 130
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“9781107009790AR” — 2013/11/5 — 21:26 — page 412 — #430
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412 normalization rigid rotor, 393, 394, 396 objective reality, 15, 23, 127 operator Hamiltonian, 43 acceleration, 285, 287 momentum domain, 52 selfadjoint domain, 52, 54 Weyl, 43 multiplication, 44 shift, 44 operators, 14, 30 exponential, 252 fermionic, 211 position momentum, 50 selfadjoint, 51 oscillator classical action, 233 classical solution, 231 source, 234 conditionality probability, 239 eigenstates, 226 evolution kernel, 230 finite lattice, 241 generating function, 234 Hamiltonian, 226 infinite time, 230 normalization, 233, 254 simple harmonic, 225 transfer matrix lattice, 246 overcomplete basis coherent states, 98 fermion, 203 path integral acceleration, 286, 288 coherent states, 99 continuum limit, 76 Euclidean, 86 evolution kernel, 73 fermionic, 211 free particle, 240 Gaussian, 251 Hamiltonian, 106 indeterminate positions, 261 Lagrangian, 106 Minkowski, 85 momentum basis, 243 periodic, 253 phase space, 85 quantization, 105 time lattice, 75 pfaffian, 208 phase space
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Index path integral acceleration, 283, 285 Poisson bracket, 87 Euclidean, 88 Jacobi identity, 88 Poisson summation formula, 386, 387, 389 polymer tangent vector, 397, 398, 400 potential acceleration, 274, 276 quadratic, 274, 276 delta function, 57 doublewell Ising model, 375, 376, 378 quartic, 350, 351, 353 probability conditional, 63, 128 joint, 128 marginal, 128 probability amplitude, 10, 24 composition rule, 76, 291, 293 time evolution, 61 probability theory classical, 127 propagator acceleration path integral, 279, 281 Jordan block, 337, 339, 344, 346–349 lattice oscillator, 245 oscillator finite time, 227 pseudoHermitian operators, 322, 324 state space, 324, 326 pseudoHermitian dual eigenstates, 307, 309 quantum entity, 26 definition, 27 quantum mechanics operator formulation, 22 three formulations, 25 quantum numbers, 46 quantum paths infinite divisibility, 70 quantum state, 11 quantum superstructure, 8 quartic potential, 350, 351, 353 random paths, 142 random variable, 127 renormalization recursion Ising, 193 block spin magnetic field, 195
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“9781107009790AR” — 2013/11/5 — 21:26 — page 413 — #431
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Index flow Ising, 194 Ising spin, 191 sample space, 127 Schrödinger equation, 19, 64 Jordan block 2×2, 343, 345 measurement, 8 properties, 21 semiclassical approximation, 351, 352, 354 semiclassical expansion doublewell, 362, 363, 365 integral, 352, 353, 355 simple harmonic oscillator see oscillator, 225 spectral representation instanton expansion, 363, 364, 366 spherical harmonics, 392, 393, 395 spin decimation Ising, 175 spontaneous symmetry breaking, 379, 380, 382 state space basis states, 35 continuous degree of freedom, 30 degree of freedom, 11 pseudoHermitian Hamiltonian, 318, 320 state vector statistical, 8 state vector collapse, 19 stochastic, 125
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stochastic quantization, 148 stock price, 137 geometric mean, 138 superposition indeterminate paths, 65 quantum, 21 symbol, 11 symmetry breaking multiple wells, 382, 383, 385 symmetry restoration, 380, 381, 383 ground state, 382, 383, 385 tangent vector polymer, 397, 398, 400 tensor product state space, 17 time Euclidean, 84 transfer matrix Ising spin, 165 oscillator, 246 transition amplitude, 63 fermionic, 219 vacuum state Hamiltonian pseudoHermitian, 309, 311 white noise, 132 Wilson expansion, 139 Ito calculus, 138
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“9781107009790AR” — 2013/11/5 — 21:26 — page 414 — #432
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