Mathematical Physics and Stochastic Analysis Essays in Honour of Ludwig Streit
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Mathematical Physics
and Stochastic Analysis Essays in Honour of Ludwig Streit
Editors
S. Albeverio, Ph. Blanchard, L Ferreira, T. Hida, Y. Kondratiev and R. Vilela Mendes
tt>
fe World Scientific Singapore • New Jersey • London • Hong Kong
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MATHEMATICAL PHYSICS AND STOCHASTIC ANALYSIS: Essays In Honour of Ludwlg Streit Copyright ® 2000 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means. electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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V
Preface In October 1998 a conference took place in Lisbon to celebrate Ludwig's Streit 60th birthday. This book assembles some of the papers presented at the conference as well as other essays contributed by the many friends and collaborators that wanted to honor Ludwig Streit's scientific career and personality. Ludwig Streit was born June 26, 1938 in Leipzig. He attended the High School "Wilhelmschule" in Kassel, in an experimental course with emphasis on social studies and graduated in 1957 as the best student of the year (with an additional exam with "Grosses Latinum" in 1958). From the Spring semester 1957 he was a student of mathematics and physics at the Faculty of Natural Sciences of Gottingen University. From the Spring 1960 he continued his studies at the University of Graz (Austria), at the Faculty of Philosophy, taking his Diploma Thesis in theoretical physics, with mathematics, philosophy and psychology as minors. In the Fall term 1961-62 he was assistant to Prof. H. Lehmann at the Institute of Theoretical Physics, University of Hamburg. He received his PhD "mit Auszeichung" at the University of Graz in 1962 with the disser tation "Analytische Eigenschaften der exakten Streuamplitude" which he wrote under the direction of Prof. W. Zimmermann at Hamburg. After two years, 1963-65, with the position of "Research Assistant" at the University of Hamburg, he was from May 1, 1965 to July 31, 1967 at the Seminar for Theoretical Physics, ETH, Zurich (Prof. M. Fierz and Prof. R. Jost). In 1967-68, resp. 1968-69, he was a Research Associate, resp. an As sistant Professor, at the Physics Department of Syracuse University. After a period as Member of the Technical Staff at Bell Telephone Laborato ries, Murray Hill, N. J., he was, in 1970, nominated Associate Professor at Syracuse University. He left Syracuse in 1972, to take over a position of Professor at the Physics Department, University of Bielefeld, which he has held ever since. Since 1973 he is also Honorarprofessor at the University of Graz. In 1980, for professional reasons, he declined the offer of a chair in theoretical physics at the University of Minister. From 1989 to 1991 he was titular of the chair for Mathematical Physics at the Universidade do Minho, Braga (Portugal) and since 1991 he holds a chair at the Mathematics Department of the Universidade da Madeira, Funchal (Portugal). In 1980-91 he was given an "Akademie" -Fellowship by the Volkswagen Foundation. He is the author of more than 150 research papers, on topics
vi ranging from mathematical and theoretical physics, to mathematics, espe cially the theory of stochastic processes, infinite dimensional analysis, the theory of dynamical systems and their applications (in natural sciences and engineering). He is also author of a very influential book on white noise calculus and its applications and editor of 16 proceedings volumes. His outstanding achievements in science are described more closely in two articles (by S. Albeverio and T. Hida) in this volume. Let us stress here that besides his research achievements Ludwig Streit has been very successful in giving stimulus and furthering new research directions, creat ing new research centres, organizing research workshops, and international conferences. We mention the creation of the Research Centre BiBoS at the University of Bielefeld (1984 - ), Bochum (1984 - 1997) and Bonn (1997 - ), the organization of a Series of Conferences "Bielefeld Encounters in Mathematics and Physics" at Bielefeld University (started around 1978, of which numerous Proceedings have appeared), the Research Years 1975-76, "Mathematical Problems in Quantum Dynamics", 1983-84, "Project No. 2 in Mathematics and Physics" and "The sciences of complexity: from math ematics and technology to a sustainable world", 2000-01 at the Center of Interdisciplinary Research (ZiF) of the University of Bielefeld. He has been Director of ZiF (1981-1985), International Adviser for the International Institute for Advanced Studies at Kansai Science City (Japan), has held executive positions in the International Association for Mathematical Physics (1979-1985), IUPAP Commission for Mathematical Physics (1981-1987), and he is member of several editorial boards of jour nals and books. He has given invited lectures at Universities in all continents, with re peated long stays in Bulgaria, China, France, Italy, Japan, Philippines, Russia, USA. Ludwig's "engagement" for science in Portugal deserves a special mention. Starting with his nominations in Braga and Madeira, he has developed an impressive activity of research and teaching, with an out standing impact on the country's development of scientific research. Let us mention here his creation of a school of Mathematical Physics in Madeira with several PhD students at international level, his influence on other stu dents in Portugal who came to Bielefeld and other Central European places to do research work, his creation of the Research Centre "Centro de Ciencias Matematicas", his participation in the Committee of the Portuguese Secretary of State for Research and Development for a national program in astronomy, his organization of several workshops and international work shops, conferences and Summer Schools in Madeira and Lisbon. The contributions of Ludwig Streit for the development of science in
vii Portugal were officially acknowledged and praised, at the time of the con ference, by the Minister of Science of the Portuguese Government. Ludwig Streit is an instant source of energy and inspiration in a broad area of mathematics and physics and their many apphcations in natural and social sciences. The contributions presented in this volume by stu dents, friends, and coworkers reflect, at least partially, the broad interests of Ludwig. The two first contributions by S. Albeverio and T. Hida are particular in the sense that they have as a topic specific aspects of the scientific work of Ludwig and the interaction with the authors over many years. The other contributions, arranged alphabetically by authors, also testify, in one way or another, the great influence of Ludwig in many areas of mathematics, physics and their apphcations. In the name of all friends and contributors, we wish Ludwig many more years of good health and of successful pursuing of his outstanding activities, in science and for the community.
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Contents
Preface Ludwig Streit's list of publications
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Some themes of the scientific work of Ludwig Streit S. Albeverio
1
Mathematics, Physics and Streit since 1975 T. Hida
8
Stochastic limit and interacting commutation relations L. Accardi, I. Ya. Aref'eva and I. V. Volovich
18
De Rham complex over product manifolds: Dirichlet forms and stochastic dynamics S. Albeverio, A. Daletskii and Y. Kondratiev
37
Real time random walks on p-adic numbers S. Albeverio and W. Karwowski
54
Characterization of test functions in CKS-space JV. Asai, I. Kubo and H.-H. Kuo
68
Nonlinear Lie algebras in quantum physics and their interest in quantum field theory J. Beckers
79
Continuous percolation: tree approximation and most probable clusters Ph. Blanchard, G. F. Dell'Antonio, D. Gandolfo and M. Sirugue-Collin
84
X
Rigged Hilbert space resonances and time asymmetric quantum mechanics A. Bohm and H. Kaldass
99
Q-Ising neural network dynamics: A comparative review of various architectures D. Bolle, G. Jongen and G. M. Shim
114
The relativistic Aharonov-Bohm-Coulomb problem: A path integral solution J. Bornales, C. C. Bernido and M. V. Carpio-Bemido
130
Statistical manifolds: The natural affine-metric structure of probability theory G. Burdet, Ph. Combe and H. Nencka
138
Brydges' operator in renormalization theory P. Cartier and C. DeWitt-Morette
165
Fluctuation of the Bose-Einstein condensate in a trap H. Ezawa, K. Nakamura, K. Watanabe and Y. Yamanaka
169
Time dependent and nonlinear point interactions R. Figari
184
The Cole-Hopf and Miura transformations revisited F. Gesztesy and H. Holden
198
Some models of nonideal Bose gas with Bose-Einstein condensate R. Gielerak
215
Configuration spaces for self-similar random processes and measures quasi-invariant under diffeomorphism groups G. A. Goldin
228
Stochastic processes and the Feynman integral Z. Haba
243
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Non-Markovian random flows and hedging stocks in manufacturing processes M.-O. Hongler
255
Optimal portfolio in a fractional Black and Scholes market Y. Hu, B. Oksendal and A. Sulem
267
Nonrenormalizability and nontriviality J. R. Klauder
280
On the spectrum of lattice Dirac operators C. B. Lang
285
Integrals of motion and quantum fluctuations V. I. Man'ko
302
Distributions Gaussiennes et applications aux equations aux derivees partielles stochastiques H. Ouerdiane
318
On Donsker's delta function in white noise analysis J. Potthoff and E. Smajlovic
332
L^uniqueness on measurable state spaces: A class of examples M. Rockner
350
A Glauber dynamics for the Sherrington-Kirkpatrick and Hopneld models E. Scacciatelli
359
External and internal geometry on configuration spaces J. L. Silva
366
Circular invariance of the Weyl form of the canonical commutation relation J. Stochel and F. H. Szafraniec
379
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Numerical solution of the nonlinear Schrodinger equation with Newton-type methods L. Vazquez, D. N. Kozakevich and S. Jimenez
384
Spinor description of a general spin-J system V. R. Vieira and P. D. Sacramento
400
Gluon condensate and a vacuum structure for nonabelian gauge theory R. Vilela Mendes
412
On the invariance principle and the law of iterated logarithm for stationary processes D. Volny and P. Samek
424
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Ludwig Streit's list of publications 1. "Die Gewichtsfunktion der Mandelstamdarstellung am Rande ihres Tragers" - // Nuovo Cimento 23, 934 (1962). 2. " Eine Landausingularitat der exakten Streuamplitude" - Ada Physica Austriaca 16, 12 (1963). 3. "S-Matrix Theory and Current Operator in Quantum Field Theory" - // Nuovo Cimento 34, 198 (1964). 4. "An Introduction to Theories of Integration over Function Spaces " Ada Physica Austriaca, Suppl. II, 2 (1966). 5. " Generalized and Reducible Fields in Fock Space" - Helvetica Physica Ada 39, 65 (1966). 6. "Test Function Spaces for Direct Product Representations of the Canonical Commutation Relations" - Comm. math. Phys. 4, 22 (1967). 7. "Unitarity of Pauli Transformations" - II Nuovo Cimento 57, 140 (1968). 8. "A Generalization of Haag's Theorem" - II Nuovo Cimento 62, 673 (1969). 9. "Properties of Quadratic Representations of the Canonical Commu tation Relations" (with J. R. Klauder) - J. Math. Phys. 10, 1661 (1969). 10. "Quantum Field Theory on Light-like Slabs" (with J. R. Klauder and H. Leutwyler) - // Nuovo Cimento 66, 536 (1970). 11. "The Construction of Physical States in Quantum Field Theory" Ada Physica Austriaca Suppl. VII, 335 (1970). 12. "Generalized Free Fields as Cyclic Representations of the Canonical Commutation Relations" - Ada Physica Austriaca 32, 107 (1970) 13. "Tachyon Quantization" (with J. R. Klauder ) - Boulder Lectures in Theoretical Physics, vol. XIV B (1973), p.539. 14. "Dilation and Conformal Invariance on Null Planes (with F. Rohrlich) - II Nuovo Cimento 7 B, 166 (1973).
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15. "Exact Solution of the Quadratic Interaction Hamiltonian" (with W. J. Eachus) - Rep. Math. Phys. 4, 161 (1973). 16. "Model Calculation of the Vacuum Energy Density for a Self-Coupled Bose Field" (with L. Pittner) - Ada Physica Austriaca 38, 361 (1973). 17. "Optical Equivalence Theorem for Unbounded Observables" (with J. R. Klauder) - J. Math.Phys. 15, 760 (1974). 18. " On the Interference between Coulomb and Short Range Interactions" (with W. Plessas, H. Zingl) - Ada Physica Austriaca 40, 274 (1974). 19. "On Interactions which cannot be Turned Off' - Ada Physica Aus triaca 42, 9 (1975). 20. "Gaussian Processes and Simple Model Field Theories" - in Proc. XII. Winter School for Theoretical Physics, Karpacz 1975. 21. "Mathematik und Physik - Interdisziplinaritat als Voraussetzung" in Interdisciplinary Systems Research, vol. 24 , Birkhaeuser, Basel, 1976. 22. "The Construction of Quantum Field Theories" - in Uncertainty Prin ciple and Foundations of Quantum Mechanics - A Fifty Years Survey (S.S. Chissick, ed.), London 1976. 23. "Lightlike Initial Data for Quantum Field Theory" - in Proc. XIII. Winter School for Theoretical Physics, Karpacz 1975. 24. "The Wondrous Things. A Review of Probabilistic Concepts in Quan tum Dynamics" - Ada Physica Austriaca Suppl. XV, 7 (1976). 25. "Stability of Scattering Phase Shifts" (with J. Froehlich) - Acta Phys ica Austriaca 47, 125 (1977). 26. "On Quantum Theory in Terms of White Noise" (with T. Hida) Nagoya Math. J. 68, 21 (1977). 27. "Energy Forms, Hamiltonians, and Distorted Brownian Paths" (with S. Albeverio, R. Hoegh-Krohn) - J. Math. Phys. 18, 907 (1977). 28. "A Correct Formulation for Non-Relativistic Point Interactions of N > 2 Particles" (with J. Froehlich, H. Zankel, H. Zingl) - in Proc. 4eme Session Biennale de Physique Nucleaire, tome 2, Lyon 1977.
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29. " Coulomb Correction in Proton-Proton Scattering" (withJ. Froehlich, H. Zankel, H. Zingl) - in Proc. 4eme Session Biennale de Physique Nucleaire, tome 2, Lyon 1977." A Renormalization Group model with Non-Gaussian Fixed Point" (with W. Karwowski) - Rep. Math. Phys. 13, 1 (1978). 30. "Null Plane Fields and Automodel Processes" (with R. Gielerak, W. Karwowski) - in Feynman Path Integrals (S. Albeverio et al. eds.), Springer Lecture Notes in Physics no. 106. 31. "Construction of a Class of Characteristic Functionals" (with R. Giel erak, W. Karwowski) - in Feynman Path Integrals (S. Albeverio et al. eds.), Springer Lecture Notes in Physics no. 106. 32. "The Coulomb Contribution to p-p Scattering Phase Shifts" (with J. Froehlich, H. Zankel, H. Zingl) - Invited contribution, 8th Intl. Conf. on Few Body Systems and Nuclear Forces, Springer Lecture Notes in Physics 82, p.109, 1978. 33. "Regularization of Hamiltonians and Processes" (with S. Albeverio, R. Hoegh-Krohn) - J. Math. Phys. 21, 1636 (1980). 34. "Coulomb Corrections in N-N Coupled States" (with J. Froehlich, H. Zankel) - Phys. Lett. 92B, 8 (1980). 35. "White Noise Analysis and the Feynman Integral" - in Functional In tegration, Theory and Application (J.P. Antoine, E. Tirapegui, eds.), p.43, Plenum 1980. 36. "Simple Coulomb Correction for Phase Shifts and Application to the p-p System" (with J. Froehlich, H. Zankel, H. Zingl) - Journ. Phys. G 6, 841 (1980). 37. " Capacity and Quantum Mechanical Tunneling" (with S. Albeverio, M. Fukushima, W. Karwowski) - Comm. math. Phys. 81, 501 (1981). 38. "Energy Forms, Schroedinger Theory, Processes" - Phys. Reports 77, 363 (1981). 39. " Generalized Brownian Functionals and the Feynman Integral" (with T. Hida) - Stock. Proc. Appl. 16, 55 (1983). 40. "An Improvement of Coulomb Corrections in the Phase Shift Analysis of Elastic 7r-0 16 Scattering and Predictions of the Cross Sections for 7r~-0 16 Scattering at Low Energies (with J. Froehlich, H. G. Schlaile, H. Zingl) - Z. Phys. A 302, 89 (1981).
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41. "Prediction of n-alpha Phase Shifts from p-alpha Data between 20 and 35 MeV" (with J. Frohlich, H. Zankel) - Nucl. Phys. A 384, 97 (1982). 42. "A Separable Approach to the B-S Equation and its Application to Nucleon-Nucleon Scattering" (with K. Schwarz, J. Froehlich, H. Zingl) - Acta Physica Austriaca 53, 191 (1981). 43. "Charged Particles with Short Range Interactions" (with S. Albeverio, F. Gesztesy, R. Hoegh-Krohn) - Ann. Inst. H. Poincare 37, 263 (1983). 44. "Exact Solution of a Quasipotential Equation for Charged Spinless Particles" (with I. Todorov) - Rep. Math.Phys. 24, 95 (1986). 45. "Generalized Brownian Functionals" (withT. Hida), VI. Conf. Math. Phys. Berlin 1981, Springer Lecture Notes in Physics, no. 153, 285287, 1982. 46. "White Noise Analysis and its Application to Feynman Integral" (with T. Hida) - in Measure Theory and its Applications (J. M. Bellay et al., eds.) Springer Lecture Notes in Mathematics no. 1033, 1983 47. "Separable Bethe-Salpeter Equation Kernels for Pion-Nucleon Scat tering (with J. Froehlich, K. Schwarz, H. Zingl) - Phys. Rev. C 25, 2591 (1982) 48. "Model Dependence of Coulomb-Corrected Scattering Lengths" (with S. Albeverio, L. S. Ferreira, F. Gesztesy, R. Hoegh-Krohn) - Phys. Rev. C 29, 680 (1984). 49. "Low Energy Parameters in Nonrelativistic Scattering Theory" (with S. Albeverio, D. Bolle, F. Gesztesy, R. Hoegh-Krohn) - Ann. Phys. 148, 308 (1983). 50. "Tabakin-Like Separable Kernels for the NN-Bethe-Salpeter Equa tion" (with G. Rupp, J. A. Tjon) - Proc. "Few-Body X" Karlsruhe 1983, vol.1, p.149. 51. "Stochastic Processes - Quantum Physics" - Acta Physica Austriaca Suppl. XXVI, 3 (1984). 52. "Quantum Mechanical Low Energy Scattering in Terms of Diffusion Processes" (with S. Albeverio, Ph. Blanchard, F. Gesztesy) - Springer Lecture Notes in Mathematics no. 1109, p.207, 1985.
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53. "Separable Potentials from Gamow States" (with M. Baldo. L. S. Ferreira) - Phys. Rev. C 32, 685 (1985). 54. "On the Connection between Schrodinger and Dirichlet Forms" (with S. Albeverio, F. Gesztesy, W. Karwowski) - J. Math. Phys. 26, 2546 (1985) 55. "Charged Particles with a Short Range Force: Perturbation Theory with Respect to the Range and other Small Effects" (with F. Gesztesy, G. Karner) - J. Math. Phys. 27, 249 (1986). 56. " Convergence of Dirichlet Forms and Associated Schrodinger Oper ators" (with S. Albeverio, S. Kusuoka) - J. Fund. Anal. 68, 130 (1986). 57. "Capacity, Green's Functions and Schroedinger equation" (with S. Albeverio, W. Karwowski, M. Roeckner) - in Infinite Dimensional Analysis and Stochastic Processes, Pitman 1985. 58. "An Exactly Solvable Periodic Schrodinger Operator" (with F. Gesztesy, C, Macedo) - J. Phys. A: Math. Gen. L503 (1985). 59. " Quantum Theory and Stochastic Processes - Some Contact Points" Invited Lecture, XV. Conf. Stoch. Proc. Appl., Springer Lecture Notes in Mathematics no.1203, p.197, 1985. 60. "Three Nucleon Bound State Collapse with Tabakin Potentials" (with G. Rupp, J. A. Tjon) - Phys. Rev. C 3 1 , 2285 (1985). 61. "Gamow Separable approximations for Realistic N-N Interactions: Single Channel Case" (with M. Baldo, L. S. Ferreira, T. Vertse) Phys. Rev. C 33, 1587 (1986). 62. "Scattering Theory for General, non-Local Interactions: Threshold Behaviour and Sum Rules" (with D. Bolle, F. Gesztesy, C. Nessmann) - Rep. Math. Phys. 63. " Gamow Vectors as Solutions of a Non-Hermitian eigenvalue Prob lem" (with M. Baldo, L. S. Ferreira) - Nucl. Phys. A 467, 44 (1987). 64. "Eigenvalue Problem for Gamow Vectors and a Separable Approxi mation for the NN Interaction (with M. Baldo, L. S. Ferreira) - Phys. Rev. C 36, 1743 (1987).
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65. "On the Origin of Chaos in Gearbox Models" (with M.O. Hongler) Physica 29D, 402 (1988). 66. "Map Dynamics in Gearbox Models" (with M. de Faria, R. Vilela Mendes) - in "Mathematics in Industry" (H. Neunzert, ed.), Teubner, Stuttgart 1988. 67. "Dirichlet Forms and White Noise Analysis" (with T. Hida, J. Potthoff) - Comm. math. Phys. 116, 235 (1988). 68. "White Noise Analysis and Applications" (with T. Hida, J. Potthoff) - in Mathematics+Physics, vol.3, World Scientific, Singapore 1989. 69. "An Inverse Problem for Stochastic Differential Equations" (with S. Albeverio, Ph. Blanchard, S. Kusuoka) - J. Stat. Phys. 57, 347 (1989). 70. "A Remark on the Second Virial Coefficient in the Quantum Lorentz Model (with C. Carvalho) - J. Phys. A: Math. Gen. 21, 3739 (1988). 71. "Analytical Estimates of Level Spacing for the Hydrogen Atom in a Strong Magnetic Field" (with L. Vazquez, R. Vilela Mendes) - in "Essays on classical and Quantum Dynamics", Gordon & Breach, 1991. 72. "Quasi-Energies, Loss-Energies and Stochasticity" (with G. Karner, V. I. M'anko) - Rep. Math. Phys. 29, 177 (1990). 73. "Quantum Kicks and Stochasticity" (with G. Karner, V. I. Man'ko) - Lebedev Physical Institute Proceedings 208, 226 (1991), Nauka, Moscow. English translation: Nova Science Publ., Commack, N.Y., 1992. 74. " The Vacuum of the Hoegh-Krohn model as a generalized White Noise Functional" (with S. Albeverio, T. Hida, J. Potthoff) - Phys. Lett. B 217, 511 (1989). 75. "Dirichlet Forms in Terms of White Noise Analysis I - Construction and QFT Examples" (with S. Albeverio, T. Hida, J. Potthoff, M. Roeckner) - Rev. Math. Phys. 1, 291 (1990) 76. "Dirichlet Forms in Terms of White Noise Analysis II - Construction of Infinite Dimensional Diffusions and QFT Examples" (with S. Al beverio, T. Hida, J. Potthoff, M. Roeckner) - Rev. Math. Phys. 1, 313 (1990).
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77. " Generalized Master Equations and the Telegrapher's Equation" (with M.-O. Hongler) - Physica A165, 196 (1990). 78. "Energy Forms in Terms of White Noise" in "Algebra and Analysis in Classical and Quantum Dynamics", (S. Albeverio et al, eds.) p.255, Reidel 1990 79. "A Characterization of Hida Distributions" (with J. Potthoff) - J. Fund. Analysis 10, 212 (1991) . 80. "A Characterization of White Noise Test Functionals" (with J. Pot thoff, H. H. Kuo) - Nagoya Math. J. 121, 185 (1991). 81. "A Probabilistic Connection between the Burger and a Discrete Boltzmann Equation" (with M.-O. Hongler) - Europhys. Lett. 12, 193 (1990). 82. "Generalized Radon-Nikodym Derivatives and Cameron-Martin The ory" (with J. Potthoff) - in "Gaussian Random Fields" (K. Ito and T. Hida, eds.) World Scientific, Singapore, 1991 83. "The Feynman Integrand as a Hida Distribution" (with M. de Faria, J. Potthoff) - J. Math. Phys. 32, 2123 (1991). 84. "White Noise Analysis and what it can do for Physics" (with J. Pot thoff) - in "Gaussian Random Fields" (K. Ito and T. Hida, eds.) World Scientific, Singapore, 1991. 85. "Spaces of White Noise Distributions: Constructions, Descriptions, Applications" (with Yu. G. Kondratiev) -. Rep. Math. Phys. 33, 341-366 (1993) 86. "A Remark about a Norm Estimate for White Noise Distributions" (with Yu. Kondratiev) - Ukrainean Math. J. (1992) no.7. 87. "The Characterization Theorem for Hida Distributions. Generaliza tions and Applications." - UMa-Mat 3/91, to appear in Proc. III. Int. Conf. Stoch. Proc, Physics and Geometry. 88. "A Generalization of the Characterization Theorem for Generalized Functions of White Noise" (with W. Westerkamp) - In: " Dynamics of Complex and Irregular Systems" Ph. Blanchard et al. eds., World Scientific, Singapore, 1993.
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89. "White Noise Analysis - Theory and Applications" - Qu. Prob. Rel. Topics VII, 337 (1992). 90. "Constructing the Feynman Integrand" (with D. C. Khandekar) Annalen d. Physik 1, 49 (1992). 91. "How to Generalize White Noise Analysis to non-Gaussian Measures" (with S. Albeverio, Yu. G. Kondratiev) - In: "Dynamics of Complex and Irregular Systems" Ph. Blanchard et al. eds., World Scientific, Singapore, 1993. 92. "The Feynman Integral -Recent Results" - In: "Dynamics of Complex and Irregular Systems" Ph. Blanchard et al. eds., World Scientific, Singapore, 1993. 93. "Quantum Mechanical Propagators in Terms of Hida Distributions", (A. Lascheck, P. Leukert, L. Streit, W. Westerkamp), Rep. Math. Phys. 33, 221-232 (1993). 94. "A New Look at Functional Integration", to appear in Proc. Capri Conf., World Scientific. 95. "White Noise Analysis and Functional Integrals", in "Mathematical Approach to Fluctuations" T. Hida, ed., World Scientific. 96. "Invariant states on random and quantum fields: <j> bounds and white noise analysis." (with J. Potthoff) J. Fund. Anal. I l l , 295-311 (1993) 97. "More about Donsker's delta function" (with A. Lascheck, P. Leukert, W. Westerkamp), Soochow Math. J. 20, 401-418 fl994). 98. "Generalized Functionals in Gaussian Spaces - the Characterization Theorem Revisited" (with Y. G. Kondratiev, P. Leukert, J. Potthoff, W. Westerkamp), J. Fund. Analysis 141, 301-318 (1996). 99. "An Introduction to White Noise Analysis" in "Stochastic Analysis and Applications in Physics" (A. I. Cardoso, M. de Faria, J. Potthoff, R. Seneor, L. Streit, eds.), Kluwer, Dordrecht 1994. 100. "Wick Calculus in Gaussian Analysis" (with Y. G. Kondratiev, P. Leukert) Ada Appl. Math. 44, 269-294 (1996) 101. "Some Recent Results in White Noise Analysis" (with M. de Faria) in "Stochastic analysis on infinite dimensional spaces" , Kunita and Kuo, eds., Longman, 1994.
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102. "Hida Distributions and More" in "Mathematical Approach to Fluc tuations" vol. 2, T. Hida, ed., World Scientific. 103. "Non-Gaussian infinite dimensional analysis" (with S. Albeverio, Y. Daletzky, Yu. Kondratiev) J. Fund. Analysis 138, 311-350 (1996). 104. "Intersection Local Times as Generalized White Noise Functionals" (with M. de Faria, T. Hida, H. Watanabe) Ada Appl. Math. 46, 351-362 (1997) 105. "A note on positive distributions in Gaussian analysis" (with Yu. Kondratiev, W. Westerkamp) Ukrainian Math. J. 47, no. 5 (1995) 106. "The Burgers Equation with a Non-Gaussian Random Force" (with F. Benth), In "Stochastic Analysis and Related Topics" (Decreusefond, L. Gjerde, J., 0ksendal, B., Ustiinel, A. S. eds.), Stochastic Monographs vol. 6, Birkhauser, p. 187-210, 1998. 107. "Generalized Functions in Infinite Dimensional Analysis" (with Yu. Kondratiev, W. Westerkamp, Y.A. Yan) Hiroshima Math. Journal 28, 213-260 (1998). 108. "Stochastic Differential Equations. A Pedagogical Random Walk." Proc. Euroconf. "Non-linear Klein-Gordon and Schrodinger Equa tions", ElEscorial, 1995. 109. "The Feynman Integral - Answers and Questions", in Proc. 1st Jagna Intl. Workshop on Adv. in Theor. Physics, p. 188 -199, Central Visayan Inst., Bohol, 1996. 110. "Generalized Appell Systems" (with Yu. Kondratiev, J. L. Silva) Meth. Fund. Anal. Top. 3, 28-61 (1997). 111. " The Feynman Integral for time-dependent anharmonic oscillators" (with M. Grothaus, D. C. Khandekar, J. L. Silva) J. Math. Phys. 38(6), 3278-3299 (1997). 112. "Feynman Integrals for a Class of Exponentially Growing Potentials" (with T. Kuna, W. Westerkamp) Journ. Math. Phys. 39, 4476-4491 (1998). 113. " Complex Gaussian Analysis and the Bargmann-Segal Space". (with M. Grothaus, Yu. Kondratiev) UMa-CCM 20/97, to appear in Meth. Fund. Anal. Topoloav.
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114. "Nonlinear Evolution Equations with Gradient Coupled Noise" (with F. Benth, Th. Deck, J. Potthoff) Lett. Math. Phys. 43, 267-278 (1998). 115. "Analysis on Poisson and Gamma spaces" (with Yu. Kondratiev, J. L. Silva, G. Us) Inf. Dim. Anal. Qu. Prob.1,91-117 (1998) 116. "Regular generalized functions in Gaussian analysis" (with M. Grothaus, Yu. Kondratiev) Inf. Dim. Anal. Qu. Prob 2, 1-25 (1999) 117. "Representations of diffeomorphisms on compound Poisson spaces". In H. Heyer, J. Marion (eds.): Analysis on infinite dimensional Lie algebras and groups. World Scientific, 1998. 118. "Scaling limits for the solution of Wick type Burgers equation, (with Grothaus, M. and Kondratiev, Yu.G), BiBoS preprint no.824/6/98 119. "The Renormalization of Self Intersection Local Times I: The Chaos Expansion." (with M. de Faria, C. Drumond). U. Lisbon preprint IFM 1/98, to appear in Inf. Dim. Anal. Qu. Prob 120. "A Generalized Clark-Ocone Formula" (with M. de Faria, M. J. Oliveira). U. Lisbon preprint IFM 2/98, to appear. xxii
121. "Construction of Relativistic Quantum Fields in the Framework of White Noise analysis" (with M. Grothaus) J. Math. Phys. 40, 53875405 (1999). 122. "Knots, Feynman Diagrams and Matrix Models" (with M. Grothaus, I. V. Volovich) BiBoS preprint no.826/11/98 123. "Multiple Intersection Local Times in Terms of White Noise" (with S. Mendonca) Universidade da Madeira preprint CCM 31/1998. 124. "Some Observations on stochastic vs. quantum dynamics - following Ezawa, Kauder, and Shepp" to appear in Proc. 2nd Jagna Intl. Work shop Mathematical Methods of Quantum Physics, Gordon+Breach. 125. "On regular Generalized Functions in White Noise Analysis and their Applications" (with M. Grothaus) BiBoS preprint, 1999. 126. "Quadratic Actions, Semi-Classical Approximation, and Delta Se quences in Gaussian analysis" (with M. Grothaus) Rep.Math. Phys. 44, 381-405 (1999).
xxiii
Books: • "White Noise. An infinite dimensional calculus." T. Hida, H. H. Kuo, J. Potthoff, L. Streit, Kluwer Academic, 1993. Books edited: • "Quantum Fields, Algebras, Processes" - Springer, Vienna 1980. • "Dynamics and Processes" (with Ph. Blanchard) - Springer Lecture Notes in Mathematics no.211, Berlin 1984. • "Resonances- Models and Phenomena" (with S. Albeverio, L. S. Ferreira) - Springer Lecture Notes in Physics no. 211, Berlin 1984. • "Mathematics+Physics. Lectures on Recent Results" 3 vols., World Scientific, Singapore 1985 - 1989. • "Path Integrals from meV to MeV" (with M. C. Gutzwiller, A. Inomata, J. R. Klauder) - World Scientific, Singapore 1985 • "Stochastic Processes - Mathematics and Physics" (with S. Albeverio, Ph. Blanchard) - Springer Lecture Notes in Mathematics no.1158, Berlin 1986. • "Models and Methods in Few-Body Physics" (with L. S. Ferreira, A. Fonseca) - Springer Lecture Notes in Physics no. 273, Berlin 1987. • "Stochastic Processes - Mathematics and Physics II" (with S. Albeve rio, Ph. Blanchard) - Springer Lecture Notes in Mathematics no. 1250, Berlin 1987. • "Taktische Kernwaffen - Die fragmentierte Abschreckung" (with Ph. Blanchard, R. Koselleck) - Suhrkamp, Frankfurt 1987. • "Stochastic Processes in Physics and Egineering" (with S. Albeverio, Ph. Blanchard) - Reidel 1988. • "Dynamics and Stochastic Processes" (with R. Lima, R. Vilela Mendes) - Springer Lecture Notes in Physics no. 355, Berlin 1990. • "White Noise Analysis - Mathematics and Applications" (with T. Hida, H. H. Kuo, J. Potthoff), World Scientific, Singapore 1990.
XXIV
• "Stochastic Processes and their Applications in Mathematics and Physics" (with S. Albeverio, Ph. Blanchard) - Kluwer 1990. • "Dynamics of Complex and Irregular Systems" (with Ph. Blanchard et al.), World Scientific, Singapore, 1993. • "Stochastic Analysis and Applications in Physics" (with A. I. Car doso, M. de Faria, J. Potthoff, R. Seneor), Kluwer, Dordrecht 1994. • "Non-Linear Klein-Gordon and Schroedinger Systems: Theory and Applications" (with L. Vazquez, V. Perez-Garcia) World Scientific, 1996.
1
SOME THEMES OF T H E SCIENTIFIC W O R K OF LUDWIG STREIT SERGIO ALBEVERJO Inst. Ang. Mathematik, Universitt Bonn Wegderstr. 6 D 53115 Bonn (Germany 1) (SFB 237; SFB 256; BiBoS (Bielefeld-Bonn); CERFIM (Locarno); Ace. Arch., USI (Mendrisio)) A short review of the scientific work of Ludwig Streit is given. Some recent de velopments in infinite dimensional analysis, stochastic analysis and mathematical physics stimulated by Streit's work are indicated.
1
Introduction
Ludwig Streit's work has strongly and deeply influenced and shaped contem porary work on mathematical and theoretical physics as well as on infinite dimensional analysis and the theory of stochastic processes and its applica tions. I have the honour to present some of his work. It is also a great pleasure for me since Ludwig strongly influenced my own scientific life, and I am grateful to him for this. My lecture will be a mixture of reminiscences and historical remarks on Ludwig's work, with a mentioning of some recent developments stimulated by his work. As exposed in more details in the preface of this volume, Ludwig Streit's research work in theoretical and mathematical physics was started in the early sixties in Hamburg (a leading milieu for quantum field theory, with H. Lehmann and W. Zimmermann), continued in Graz (a University to which Ludwig remained attached ever since) and at the ETH Zrich (where M. Fierz and R. Jost had brilliantly taken over the School of W. Pauli, since the late 50's). I first met Ludwig at the Institute of Theoretical Physics of the ETH in 65-66. At that Institute situated in the top floors of a small villa at the Hochstrae (the lowest being occupied by hydraulic engineers) there were at that time, besides Fierz and Jost, K. Hepp, W. Hunziker and a group of younger assistants and coworkers including J. Roberts, W. R. Schneider, R. Schrader, R. Seiler, W. Wyss, E. Zehnder... Ludwig impressed me by the maturity of his formation, especially in quan tum field theory (an area where I was still fighting with the first elements), by his working energy, and by his multiform and vivid cultural interests. Ludwig had already at that time a more constructive approach than the one which
2
was then fashionable, namely axiomatic quantum field theory. He was writ ing papers on functional integration methods in quantum field theory and quantum mechanics which were going to play a major role in my own forma tion, later on. (At that time such methods were generally considered rather " 'strange"' by leading mathematical physicists, a sort of mathematical game without deep consequences. The work of K. O. Friedrichs and I. Segal was quite isolated in the latter community). By favoring such methods Ludwig was in advance with respect to other mathematically oriented quantum field theorists, and this put himself in a position to anticipate the very beginning and then participate actively in future developments in the construction of quantum field models, which were to take over the stage towards the end of the 60's. After the Zrich's period came a period at Syracuse and the Bell Laboratories (68-72) where Ludwig met among others J. Klauder, with whom he was going to have an intense collaboration. In 71 I had the pleasure to meet Ludwig again at a Summer School in Boulder, Colorado. There I saw clearly what strucks me most of Ludwig: his capacity and art to quickly grasp the essential point of a complex scientific problem and to formulate it with a few pertinent words. This, coupled with his great technical skills, hard work and wisdom, allows him to sparkle new developments and be a source of continuous inspiration for his many students and coworkers. From 72 he was active at the University of Bielefeld, keeping however many close tights with Universities and Research Centers abroad (in fact in all continents). He is a passionate traveller, a real master in combining strong scientific interests with cultural enjoyments. As described in the preface, he has a particular attachment to Portugal, where he admirably managed to create, often under diflBcult circumstances, a very good and active school of mathematical physics. In Bielefeld, Madeira and elsewhere he has developed an impressive activity of research and teaching, stressing mathematics and physics as essential motors in many interdisciplinary contexts (see e.g. [21] and edited books in his list of references).. He has also been an attractor for many scientists' orbits including my own (I came first to Bielefeld from Norway through Ludwig's ZiF-project 76-77, and he was very influential in my final settling in Germany). It is impossible to give in a short paper a full overview of Ludwig's work, which comprises around 150 papers and books, in several areas of physics and mathematics and their applications. I am forced to limit myself here to describe main areas, with some characterization of topics through " 'Stichwrter"'. I will then pick up a few examples and indicate briefly how
3
they developed as active areas of contemporary research.
2
Research work of Ludwig Streit and some connected developments
For convenience I shall divide the research areas of Ludwig Streit into six groups: 1) Quantum field theory 2) Non relativistic quantum theory 3) Nuclear physics 4) Dynamical systems 5) Stochastic processes and applications 6) Infinite dimensional analysis and applications I shall say something on Ludwig's contribution to these areas. I would like to stress that my presentation is only meant to give an idea of the depth and extent of Ludwig's contributions and interest, it is however far from being complete. I apologize in advance for possibly forgetting some important con nections. I hope other contributions in this volume might help alleviating this.
2.1
Quantum field theory
In the years 62-64 Ludwig Streit's work concerned S-matrix theory (Mandelstan representation; study of Landau singularities; current operator...) [1-3]°. From 65 he was already introducing probabilistic methods in the study of quantum fields. His masterly written paper [4] was very influential (I have a personal depth towards it, for opening up my own perspective in the constructive tools for quantum fields). L. Streit was soon going to develop this study also in connection with the theory of stochastic processes
"The numbers refer to the publication list of Ludwig Streit.
4
and infinite dimensional analysis, especially since 76-77, when he started a very productive line of research centered around Dirichlet forms on one hand and the theory of white noise calculus on the other hand (see sect.2.2 below). In the years 66-70 Ludwig's work concerned the representation theory of canonical commutation relations ([5],[6],[9]), the study of Pauli transformations ([7]), of the phenomena connected with the Haag theorem which he extended ([8]), the study of quantum fields in "'light - like slabs'" ([10]). He also was a pioneer in the constructive approach in quantum field theory ([11],[12]). He was going to continue this line in 73 ([15], [16]) and in 75 ([19],[20], [22]). In 73 he also turned to the study of tachyons quantization ([13]), conformal invariance ([14]), renormalization group analysis ([20]) and optics ([17]). Quantum fields with light-line initial data were studied by Ludwig in 75 ([23]). In the period 77-78 he started pioneering work on random fields with scaling properties (non Gaussian fixed points) ([24], [30],[31]). Also he initiated, with R. Gielerak and W. Karwowski, work on the construction of Euclidean quantum fields via characteristic functional ([32]). Such methods were going to play later on an important role in the construction of a class of interacting local relativistic quantum fields ([122], [AGW] and references therein).
2.2
Non relativistic quantum theory
From 75 Ludwig introduced (in work with R. H0egh-Krohn and myself ([27], [34]), see also [39], [60]), the theory of Dirichlet forms and associated processes for the study of quantum systems with interactions which can be very singular. This was also developed subsequently in joint work with Ph. Blanchard, M. Fukushima, F. Gesztesy, W. Karwowski, S. Kusuoka and M. Rckner ([38], [53], [55], [57], [58]). Ludwig contributed in an outstanding way to the development of Dirichlet form theory and connected recent developments in stochastic analysis. His work was subsequently further developed in the frame of white noise analysis ([26], [52], [68], [69], [75]-[77], [79], [109]) (joint work with T. Hida, J. Potthoff and many coworkers, cf. Hida's contribution to this volume). In particular Ludwig's work permitted to define lower bounded Hamiltonians with point interactions in the many body problem via Dirichlet forms ([27]). Ludwig applied then such methods to the study of the quantum Lorentz gas (with C. Carvalho, see [71]). Other work of Ludwig in the area of quantum theory concerns the study of quantum chaos (with G. Karner, M'anko, L. Vazquez and R. Vilela
5
Mendes) ([72]-[74]), the study of periodic interactions ([59]) and the represen tation of quantum dynamic by Feynman paths integrals (see below under 2.6).
2.3
Nuclear physics
Ludwig Streit has given in this area important contributions concerning the study of Coulomb corrections and low energy phenomena. This work was done in collaboration with various groups in Graz (J. Frhlich, F. Gesztesy, K. Schwarz, H. G. Schlaile, H. Zankel, H. Zingl...) ([18], [25], [28], [29], [33], [37], [41], [42], [43], [48]), Lisbon (L. Ferreira, C. Macedo) ([49]), Bari (M. Baldo), Leuven (D. Bolte) ([50]). This work concerns in particular phase shift analysis in various systems, nucleon-nucleon and pion-nucleon low energy scattering, the study of charged particles and of low energy parameters. There are also important contribution with T. Vertse, C. Nessmann, G. Rupp, I. A. Tjon ([51], [54], [61]-[65]), and contributions concerning the study of systems with Coulomb and point interactions as suitable models of nuclear systems ([44], [56]). Other work concerns the study of resonances (Gamow potentials, sum rules...), see [54], [62], and of a quasi potential equation for charged spinless particles ([45]).
2.4
Dynamical systems
Various important contributions of Ludwig in this area concern e.g. the study of noise in deterministic gear boxes ([66], [67]), the study of the telegrapher equation ([78]) and its relations with the discrete Boltzmann equation ([82]).
2.5
Theory and applications of stochastic processes
As mentioned already in 2.1, 2.2, the theory of Dirichlet forms was introduced as an important tool in quantum theory in 75-77, with two main aims: the first is to study Hamiltonians with singular potentials, with applications in non relativistic quantum theory, and the second is to define and study quantum field models. Ludwig's contributions also concern the development of the theory of stochastic processes associated with Dirichlet forms, e.g. in work concerning stochastic differential equations with singular drift ([26],
6
[27], [34], [38], [39], [52], [109]), the characterization of hitting times and its relations with irreducibility and tunneling/no tunneling phenomena ([38]), the construction of infinite dimensional diffusions via white noise calculus ([68], [75]-[77], [79], [97], [115]). For the latter, where Ludwig was going to become one of the main actors worldwide, see his fundamental book with T. Hida, H. H. Kuo, J. Potthoff, which has become the standard reference for white noise calculus, and T.Hida's contribution to this volume. Let me stress here the momentum given by Ludwig to the study of certain stochastic partial differential equations (see [107], [109] and 0ksendahl's contribution in this volume). An important inverse problem for stochastic differential equations was also formulated and solved by Ludwig (in collaboration with Ph. Blanchard, S. Kusuoka, and myself, cf. [70]). It concerns the determination of the drift inside a spatial region, knowing the behaviour of the process outside the region. In recent years Ludwig has done outstanding work in the study of Donskers function and the intersection functionals of Brownian motions ([98], [105], [108], [120], [124], [127]).
2.6 Infinite dimensional analysis and applications Ludwig was an initiator and an essential motor in the development of a dis tribution theory on infinite dimensional spaces (Gaussian and non Gaussian white noise calculus), and its applications to differential and variational calcu lus on infinite dimensional spaces, see [36], [40], [46], [47], [81], [83], [85], [90], [92], [99]-[104], [106], [111]-[118], [121], [126] (and T. Hida's contribution). In particular Potthoff-Streits characterization theorem ([80]) has become a truly fundamental tool of Gaussian white noise calculus for Hida distributions and his formulation of Feynman path integrals as white noise functionals has lead to the determination and study of solutions of Schrdinger equations with a large class of non necessary bounded potentials ([84], [91], [93]-[96], [110], [112], [113]), and a rigorous formulation of Feynman's variational principle, cf. [110]. Moreover the Chern-Simons functional integrals arising in topological quantum field theory have found a rigorous realization as white noise functionals, via Potthoff-Streits characterization theorem, in work by Leukert, Schfer and myself (abelian case), see [ASch], [LS], and Sengupta and myself [ASe] (non abelian case). Further related recent work concerns the reformulation of the perturbation series by white noise functionals in [123] and the extension of work in [AGW] for the construction of a class of local relativistic quantum fields with indefinite metric and non trivial scattering ([122]).
7
Acknowledgements It is a special pleasure to express here my deep gratitude to Ludwig Streit for the strong influence he has had on my work and for the scientific and human advise he has given me during many years of friendship. I wish Ludwig from the bottom of my heart many more years of good health and success in his scientific and personal life.
References [LS] P. Leukert and J. Schfer. A rigorous construction of Abelian ChernSimons path integrals using white noise analysis. Rev. Math. Phys., 8: 445-456, 1996. [ASch] S. Albeverio and J. Schfer. Abelian Chern-Simons theory and linking numbers via oscillatory integrals. Journal Math. Phys., 5: 2157-2169, 1995. [ASe] S. Albeverio and A. Sengupta. A mathematical construction of the nonAbelian Chern-Simons functional. Comm. Math. Phys., 186: 563-579, 1997. [AGW] S. Albeverio, H. Gottschalk, and J.-L. Wu. Models of local relativistic quantum fields with indefinite metric. Rep. Math. Phys., 40: 385-394, 1997. Numbered references refer to L. Streits list of publications.
8
MATHEMATICS, PHYSICS A N D STREIT SINCE 1975 TAKEYUKI HIDA Department of Mathematics, Faculty of Science and Technology, Meijo University, Nagoya, 458-8502, Japan
Dedicated to Professor Ludwig Streit on the occasion of his sixtieth birthday The original interest in the subjects on Mathematical Physics has been entirely inspired by Professor Ludwig Streit since the time when I first met him at RIMS (Research Institute for Mathematical Sciences) of Kyoto University, 1975. Since then, we have been more than a best friend in Mathematics, Physics and every thing. Now it seems to be a good opportunity to have a quick review what we have done together and to remind what be has been proposing for the future directions on White Noise Analysis and Physics.
§0. Prelude. It is my great pleasure to write this note on the occasion of the sixtieth birthday of Professor Ludwig Streit. He has been my teacher of physiscs and the best collaborator on mathematics and physics. Without his suggestion I would have wandered over an immense continent of scientific terrain carrying no map. Now, I want to take this opportunity to offer him my con gratulations upon this anniversary and express my deep thanks to him for the very best friendship. Thus, let me start this note with an overview of our footstep in a chronological order. While doing so, we shall see his contribution to the science and his stimulating idea for scientific research. §1. Historical note. 1) Having had an exciting encounter at RIMS we are ready to carry on our joint work in the field of mathematical physics. In late 1975 Professor Streit kindly invited me to ZiF (Zentrum fuer interdisciplinaere Forschung) at Bielefeld, where the project "Quantendynamik: Mathematik und Modelle" was going on.
9
As is well known, stochastic analysis, in particular, white noise analysis has profound connection with quantum dynamics. The first joint study, starting on this occasion, with Professor Streit was concerned with the Euclidean fields. Although many inter esting results have already been obtained by several authors using stochastic analysis until that time, we did extend the known results and obtained new results by appealing to the theory of white noise analysys, which was rather new approach to theoretical physics. For example, causality in the sense of stochastic process can be involved there. It was until 1977 the first joint paper [3] appeared. To continue our study I have visited ZiF almost every year since then, being supported by institutions such as ZiF, BiBoS, JSPS (Japan Society for Promotion of Science), and the visits have always been quite fruitful. This may be seen in our several joint papers and partly found in the ZiF bulletin (Jahresbericht). Another successful period was the time when he visited Nagoya University for three months supported by JSPS in 1977. I was given strong influence from him; for one thing my approach to generalized white noise functionals was very much encouraged by him on this occasion. At that time generalized Brownian functionals (now called gen eralized white noise functionals) was gradually becoming popular, and we reported an important application to the Feynman path integrals in 1981 at Berlin Conference on Mathematical Physics (see [5] that appeared in 1982). 2) Next significant epoch was the HAS (International Institute for Advanced Studies, located south of Kyoto) time. It was 1992 when I started to organize a basic research project at HAS under the title "Mathematical approach to fluctuations". He was a core member among the invited speakers. Preceding this, he was ap pointed as an advisor of the HAS, because of his career that he directed ZiF project so successful. The fluctuation project continued until 1995. The results can be seen in the publications; Proceedings consisting of two volumes
10
[14]. His great contribution to the success of the project is highly appreciated. 3) Madeira time. He has been holding an honour to have joint appointmet at Bielefeld University and University of Madeira. He has frequently organized international conferences at Madeira on not only mathematical physics, but also on wavelet, complexity and others that are in interdisciplinary fields. The Madeira Uni versity publications tell us the research has always been very active there. I visited Madeira often to participate in those international conferences and workshops and we, of course, have had very ex citing time. In addition, I used to be impressed at his idea to encourage young scientists. 4) Academic Frontier project at Meijo University. In 1997 a new academic frontier project started at Meijo Uni versity ; the title of the project is "Quantum Information Theo retical Approach to Life Science". On the occasion of the opening of theoretical section he gave an opening lecture, in which he sug gested our future directions. His continuous collaboration with us at Nagoya is certainly indispensable and we are grateful to him for his extremely valuable contribution. §2. Complex random systems and applications. We are interested in complex random systems with some structure like stochastic processes depending on time or space-time parameter and further random fields with more general parameter, say a man ifold in Euclidean space, for which good applications to quantum fields would be expected. The basic idea of analyzing such complex random systems comes from the notion of the innovation; innovation of a stochastic process and that of a random field. To give an intuitive meaning of this concept we refer to LeVy's infinitesimal equation of a stochas tic process X(t), and its generalization to the case of a random field X(C), where the parameter C is taken to be a closed, smooth and covex manifold moving around in a Euclidean space.
11
For a stochastic process X(t) the infinitesimal equation is of the form: 8X(t) = $(X(s), s < t\ Y(t), t, dt), where $ is a non-random function and where Y(t) is the inno vation which is an infinitesimal random variable such that it is independent of the X(s),s < t, and that it contains the infor mation as much as the X(t) gains during the infinitesimal time interval [t, t + dt). Although this equation has only a formal sig nificance, it does give us good profound meaning concerning the probabilistic structure of X(t). If the infinitesimal equation is given, and if the innovation is actually formed, then it would be fine that the given process X(t) is expressed as a function of the Y(s), s < t,. By doing so, we are ready to analyze the function of independent random variables and the analysis is causal. There are many possibilities of taking the probability distribution of the independent system {y(t)}. Among others is the most important and typical one, namely a white noise. It is 1) independent at every t, 2) stationary in t, 3) standard Gaussian in distribution and satisfies 4) some analytic properties of sample functions (like general ized functions of t of rank 1). It is a generalized stochastic process with characteristic func tional
C(t) =
eM-\jm2dt},
where f runs through a nuclear space E, say the Schwartz space S. A concrete expression of a white noise is obtained by taking the time derivative of a Brownian motion B(t), that is B(t) is a white noise.
12
Having been suggested by Professor John R. Klauder we call a collection of infinitesimal random variables satisfying 1), 2) and 4) a system of idealized elementary random variables, abbr. i.e.r.v. As is mentioned above, a white noise is the most important exam ple of a system of i.e.r.v.'s. Our idea is the following. Once a system of i.e.r.v.'s is formed from a given random phenomenon, we can find a function (actually a functional) of i.e.r.v.'s which are viewed as the variables. Then, we wish to analyze in a similar manner to the case of non-random functional by introducing a nice representation of the random func tional. Thus, the complexity of the random system in question can be investigated by the analytic properties of the function, which is non-random, and by the probabilistic properties of the basic i.e.r.v. 's. If a complex random system is a stochastic process or a ran dom field, then the innovation plays the role of i.e.r.v.'s. In the case where i.e.r.v. is taken to be a white noise, the well estab lished white noise analysis can immediately be applied. It should be noted that for the white noise B(t), the time variable t ap pears explicitly (without smearing B(i)), which means that the white noise analysis is fiiting for the so-called causal calculus, as was mentioned before, where the time development is explicitly expressed (see [3],[14] and so forth). Once a system of variables (that are i.e.r.v.'s) is given, we shall be in a position to introduce a suitable class of functions of the given variables. The class of course depends on the phenomena to be expressed in terms of the given i.e.r.v.'s, and from another standpoint we claim that the first choice of the class should be a system of elementary functions like polynomials in those variables. As for the latter, we are led to introduce a class of generalized functions (in fact, generalized functionals) of white noise. Note that sample functions of the B(t) are generalized function, so that we are requested to introduce nonlinear functionals of generalized functions. There is naturally requested advanced theory of func-
13
tional analysis. To be lucky there is an advantage of our white noise analysis; namely a suitable class of generalized functionals of white noise can naturally be introduced with the background of probability theory and of the classical theory of functional anal ysis. The space that has been introduced is wide enough for our purpose and it is usually written as (S)*. On this space various basic operators are acting to establish the advanced white noise analysis. With this framework we can carry on calculus and even harmonic analysis, where everything is infinite dimensional. To be extremely important, there is a famous charcterization theorem for a generalized white noise functional obtained by Potthoff and Streit [11]. When a white noise calculus is performed, we are always asking if the functionals appearing in the computations are living in our favorite space (S)*. Together with other reasons we realize a significance of the characterization theorem. Before the theorem is stated, it is necessary to introduce the Stransform to have a representation of an (S)'-functional. Let
T°(S).
The second type is described by the translation invariant Hamiltonian
V = Y.VI,J
= *£ Iv[Pl,...pI\q1,...qj)
l,J
(
I
l,J
J
\
(2.25)
J
I
J
9 r aa trubled and there a Cleary the delta function causes are singular terms r Vito in (2.25). This singularity is called the volume singularity. To give a Cleary the delta function causes a(2.25) trubleone andhasthere are singular terms meaning to hamiltonian with interaction to introduce a volume V/,0 inthen (2.25). Thisthe singularity called the volume singularity. To give cut-off perform vacuum is renormalization and vacuum dressing and a meaning hamiltonian interaction (2.25) one has to introduce a volume only after to that to removewith cut-off. This procedure defines the hamiltonian perform the this vacuum renormalization and vacuum and incut-off a new then space. To avoid difficulty we will assume that for dressing translation only afterinteraction that to remove cut-off. This procedure defines the hamiltonian invariant there are no pure creation and annigilation terms. In in a new space. To avoid we will assume for translation higher dimensions there are this otherdifficulty singularities caused by that the slow decrease invariant interaction there are no pure creation and annigilation of the kernel ii at infinity. These singularities give rise to the wellterms. knownIn higher dimensions there are other singularities caused by the slow decrease ultraviolet divergencies in quamtum field theory. In this paper we assume also of the kernel at infinity.cut-off. These singularities give rise to the well known that there is an vultraviolet ultraviolet divergencies in quamtum theory. In this we assume For fermions the interaction has field the form (2.22) withpaper a* —¥ b* and also the that there is an ultraviolet cut-off. hermitian condition means now For fermions the interaction has the form (2.22) with a* —> b* and the hermitian condition means now v(pi--pi\qi--.qj) =%j...ft|p/--.pi) (2.26)
^2PI - x! ) n *(pi) Pi n (Qr)dq
24
If one assumes that «(pi...p/|gi...gj)G5(^J+J))
(2.27)
then operator V is a bounded operator with V = V and the total hamiltonian H is selfadjoind on D(H0). 3
The Stochastic Limit
The stochastic limit is now widely used in the consideration of the large time/weak coupling behaviour of quantum dynamical systems, see for example 5
Let be given a quantum system described by the Hamiltonian H = H0 + XV where A is the coupling constant. The starting point of the stochastic limit is the equation for the evolution operator in interaction picture
where V(t) =
e«"oVe-itH0
The main idea is that there exists a new quantum field (master field) and a new evolution operator U{t) (they both live in a new Hilbert space) which approximates the old one U(x){t)KU{\2t) and the approximation is meant in the sense of appropriately chosen matrix elements. The above approximation suggests a natural interpretation of the van Hove rescaling 5 A ->• 0, t ->• oo so that X2t ~ constant = r (new time scale): it means that we measure time in units of 1/A2 where A measures the strength of the self-interaction. By putting T = 1 we see that the van Hove rescaling is equivalent to the time rescaling t ->• t/A 2 , and therefore the limit A -¥ 0 will capture the dominating contributions to the dynamics in the new time scale (the error can be estimated to be of order A2). It is remarkable that in this limit the dominating contributions can be explicitly resummed giving rise to a new unitary operator. A simple change of variables shows that the time rescaling t —V t/X2 is equivalent to the following rescaling of the Schrodinger equation for the evolu tion operator: dC/(A) t/A2)
i at
= -i\v{t/X2)uW{tlX2) X
(3.28)
25
The unitary operator U{t) is then obtained by taking the limit A —► 0: W(0=limtfW(. Here e< = ±. 4
Matrix elements of composite operators
Let us consider the Hamiltonian HX = H0 + XV
where the free Hamiltonian H0=
f e(p)c+(p)c(p)d3p+
{c(p),c+(p')} = S(P~P'),
fu>(k)a+{k)a(k)d3k, [a(k),a+(k')] = S(k - k')
26 and the interaction Hamiltonian: V=
f d3kd3pg(k,p)(c+(P)c(p
- k)a(k) + h.c.)
(4.30)
Here g(k,p) is a test function and e(p) and w(k) are one-particle dispersion laws, for example e(p) = p 2 /2, u(k) = \k\. The rescaled collective fields in this case have the form Ax{p,k,t)
= |e^ac+(p)a(fc)c(p-ik)e-^ = \ A
c+(p)a(k)c(p-k)eitEWx3
A
(4.31) A$(j>,k,t) = ^ ^ c + { p - f c ) a + ( f c ) c ( p ) e - ^ = i C + (p-*)a + (fc)c(p)e- a E ^*^ A a (4.32) where £(p, k) = e(p) - w(Jfc) - e(p - A;) (4.33) is the corresponding energy. Let us consider the matrix (n~ ,n+) elements
0. Definition of these diagrams will be given below. Next we prove that such diagrams consist of connected parts each of those contains one line coming in and one line coming out and moreover these connected parts are in fact noncrossing or half-planar diagrams. And finally we show that diagrams with N connected components being half-planar (non-crossing) diagrams are described by N copies of operators satisfying the entangled commutation relations. Generically, the sets of momenta corresponding to different vertices are different. However, it may happens that the same set of momenta corresponds to two different vertices. More precisely, momenta which come in the first vertex come out from the second one and viceversa. DEFINITION 2. We say that two incident vertices of a given connected diagram are conjugated if the momenta coming in the first vertex come out
28
from the second vertex, i.e. the vertices have the same momenta but with the opposite orientation. If the i-vertex has a conjugated vertex then we denote the latter by i. A typical example of diagrams containing at least one pair of conjugated vertices is a diagram with the mass insertion such that this insertion contains a line that does not cross other lines of the diagram (see Fig.2).
'
\
'
\
I
\ i.
J.
Figure 2: Diagram with a pair of conjugated vertices
Another simple example of a diagram containing only conjugated vertices is a disconnected diagram with two connected parts each of which is the second order mass insertion (Fig.3).
-i
1
' '
\ \
_i
Figure 3: Disconnected diagram with two pairs of conjugated vertices
On fig. 1 we present a diagram with a pair of conjugated vertices. Let us prove the main LEMMA 2. // a connected diagram doesn't consist only from pairs of conju gated vertices then it vanishes in the limit A -> 0 (in the sense of distributions).
Proof. A diagram, representing a matrix element (4.34) being integrated over t\,..., tv with test functions, can be schematically represented as
29
Figure 4: Diagram with a pair of conjugated vertices
A+B-l
h
e'EL.^/^p,?)
L
v
IJ dP*I[*l'i[ 0. In the case when n = n0 the exponent in (4.43) vanishes and generically we get the non-zero answer:
/ ( n ^ ) ) *({**,}, (M.^) 11*11*
(4-44)
31
Suppose now that v is an odd number, v — In — 1. In this case the given diagram cannot be divided into pairs and therefore its contribution goes to zero with A — ► 0. More explicitly. For the case of odd v, v = 2n — 1, we once again select n vertices and make the following change of variables
( t i , . . . , t 2 n - i ) ->• {Ti,.-.,Tn;ti.,j
= l , . . . , n 0 ; t i n + r , r = 1 , . . . ,n - 1 - n 0 ) (4.45)
with the T* as before (see (4.39)). U, = t\. +
\2TJ
Uj = ti„+j + A2r,-,
,
1 <j < n 0 ,
n 0 + 1 < j < n - 1,
tin = A 2 r n The difference with the case of even v is that we get an extra factor A in front of the integral . f , y \
j
g
l—lj = \
T,E,. iY^JEii+Ej.Vi./X2 '
'l g
i—l] = lK
'l
'j'
','
*E""' x,(£'i+£i.+i)k+yA! g
n
^)=no+r
J
»+J'
n0
0(« + AT, t,p, g) JJ dpdg J ] dtj JJ d'»-+i j=l
n+J'
( 446 )
_;=n0
Here we use the same schematical notations as in (4.41). When A 4 Owe neglect the A-dependence of
0. The lemma 2 is proved. 3. // o diagram with (n~,n+) external lines consists only from pairs of conjugated vertices then n~ = n+ and the diagram has on n+ connected parts. Each connected diagram has (1,1) external lines and is half-planar, i.e. it can be drown in the half-plane without self intersections. LEMMA
We will not present here the simple proof of this lemmata. Now the theorem follows from the above three lemma.
32
5
Entangled Commutation Relations
5.1
Stochastic limit for N point correlators in one-particle states
The following theorem has been proved in 1 3 . THEOREM
2. The stochastic limit
lim Ax(p,k,t)
A-»0
= B~{p,k,t),
lim A$(p,k,t)
X-yO
=
B+(p,k,t)
exists in the sense of the convergence of the matrix elements (ti = ±) lim < 0\c(q)Ax1{pi,kutl)...Ax-(Pn,kn,tn)c+(q')\0
>
= ((pukut1)...BTM) 3X = (XOkeA, A e n, X k € TCm{M. -+TMV). m
r
Let us remark that for u € TC {M) we have Vu € J C will use the notations < X(x),Y(x)
m_1
(4)
(M ->TM). We
>x= Y, (*k(*),*k(*))T„kM, divX = Yl div^X^, (5) kez* kez"4 dtVk meaning the divergence with respect to x k . The space M has a Banach manifold structure with the Banach space lb(Zd -*■ B.N) of bounded sequences Y = (yk)kez*,*k € KN, equipped with the norm imi„ := sup ||Yk||RW , (6) kez*
40
as the model. However, this norm being not smooth, one gets difficulties in using this manifold structure for the purposes of stochastic analysis. The way of overcoming of this difficulty was proposed in 3 , 4 . In these works, an analog of Riemannian structure on M was introduced. On a heuris tic level, the tangent space to M at the point x can be identified with the space x k 6 Z i T z k M . In order to define a differentiable structure on M , it is natural to consider some Hilbert subspace of x k 6 Z .iT x k M. Let h := h(Zd -* R^.) be the space of summable sequences p = (pk)kez,i of positive numbers. For a f i x e d p e l i let us define the space T P l , = I X 6 x kGZ< ,T xk Jlf : Y, P* H*kHr.kM < ° 4 , { kez TMp) of differentiable vector fields over M , were introduced. That is, / € C x (Mp -> TMp), iff the matrix V/(ar) := (V,/ k (x)),
(9)
of partial (covariant) derivatives generates a bounded operator in T X M P , con tinuous in x. We will use the notation T M :=2 1 M 1 , where 1 is the weight sequence with elements Ik = 1. The space M p possesses the metric pp defined by
Pp{x,y)2 = 5Z P(*k,*k)9Jfc, kez d
(10)
which makes it a complete metric space. Let /i be a probability measure on M differentiable in the sense that the following integration by parts formula holds true: for any u G TCX (M -> R 1 ) ,
41
and any vector field X e TC1 (M ->TM) / J2 (Vku(x),X k (a;))T„ k M **(*) = - / / ? > ( * ) **(*), ^ kez* ^
(H)
with some /?£ € L 2 ( M , / J ) . /?£ is called the logarithmic derivative of n in the direction X. We assume that /3£ is given by 0x(«) = E (^W-■Xk(s))T. k M + dfoXkfr)), kez^
(12)
where /3M(x) = (/3k 0*0) e C 1 ( M P -»■ TM P ) for some weight sequence p£ We will call 0^ the (vector) logarithmic derivative of \i. 3
h.
Differential forms and de R h a m complex
The simplest differential forms over M are the forms with both image and domain consisting of cylinder elements. The space of m— times differentiable forms of the order m of such type will be denoted by .Ffi™. We will use the notation 7Tln := U m ^n™. Each v € .Ffi„ has the form v x
() =
v
53 ki
*i
k-(*)i
(13)
knCZi
where Ukx k„(«) € T ^ M A ... A T ^ M, and the sum is finite. Actually, each w G T£ln can be regarded as a differential form on the manifold MA for some finite A C Zd. For such forms, the symbols V* resp. A* will denote the covariant derivative resp. the Bochner Laplacian (Afcw(a:) := TrV^w(x)), w.r.t. X*. We will use the notation Vw = (Vfcw)fc€Zd, Aw = X) AjfcW. For w e TQ.n we have Vw e , H } n . We will use the notation <w{x),v(x)
>x=
^ ki
(win
k„(a;),Uk,
k.Wk^AfA.-.Ar^M.
k„&Zd
(14) Let / x b e a differentiable measure on M with the logarithmic derivative p. We introduce spaces of integrable forms L* iln as the completions of F£ln w.r.t. the norm \\-\\s given by
IMC = /
E («ki,...,k„(a0>t'ki i k,,...,k„6Z'
^ ( x ) ) ^ ^ MA...AT,k, Af
d/x(a;). (15)
42
In order to define de Rham complex over M, we need spaces of smooth forms. Given a Hilbert space )C, we define the space A"AC which is the n—fold antisymmetric tensor power of the Hilbert space AC. We introduce in the stan dard way the exterior multiplication A : A"AC x A m AC -»• A n + m A C ,
(16) 0 A tp:=^±I^.ASn+m((l> n'.ml
VO,
ASn+m : ®n+mIC -* A n + m K. being the antisymmetrization operator, and creation resp. annihilation operators a*(Jfc):AnAC->An+1AC, k G K, (17) am(k):=kA
resp. a(k) : An+1AC -»• AnAC
(18)
(the adjoint operator). Let us define the bundle A T M with fibres AT Z M. Sections of this bundle can be considered as differential forms over M (we identify the space T^M with TmM). Each section w of A T M can be represented as a sum of sections with cylinder images, w(*) =
5Z k,
«ki.....k.0O. (a:))i )kl) ... )k „ eZ , ,
(20)
where VtWk, k n (i) e C(TX\M,Tx]Si MA...AT lk> M) (if exists). We can now define the space C 1 (M P -+ AnTM) of differentiable forms, requiring that Vw(s) G C(TXMP, A n T x M),
(21)
and is bounded uniformly in x G M. Similarly we define the spaces C m (M p -¥ ATM). We wiU use the notation ft™(M„) := Cm(Mp -¥ A T M ) .
(22)
43
We introduce the exterior differential dn^CMp^n^CMp),
(23)
dnwix) = (n + l)ASn+1 (Vw(z)),
(24)
setting
where ASn+i : (£) n+1 TXM -»• A"+1 TXM is the antisymmetrization opera tor, and Vw(x) € C(TXMP -> A"TXM) is identified with the element of (AnTsM) € .Ffi„+i we have d*w € flJ,(Mp). In what follows, we will also use the spaces L"(M -* A"TM,,/j) of integrable forms with values in A n TM„ with the norm given by (JIHIA-T.M.
*(x) = Rn(x)-[V0"(x)]An. (33) Formula (32) implies that R£(x) does not depend on the choice of the basis in (30). Both Hjf resp. H * are non-negative self-adjoint operators in L2ft„ and generate therefore strongly continuous contraction semigroups Tff(t) resp. T*(t) in it. The following result characterizes the properties of these semi groups. Theorem 4.2 Let us assume that 0" 6 C4(MP -► TMP). Then : 1) both Hjf and H* are essentially self-adjoint operators on TCPQA); 2) for any weight sequence q the semigroup T^(t) leaves invariant the space C{M -¥ A"TM,), and for v G C(M -¥ ATM,) we have II T J?(*)«(*)|IA-T.M 1 = T "W ll«HA-TM, (*)!
(34)
3) for the weight sequence q = p or q = p _ 1 the semigroup T*{t) leaves invariant the space C(M -t ATM,), and for v G C(M -> ATM,) tue /w«e ||T«(t) V (x)|| A „ T>Mq < e ^ T ^ t ) HHIA.TM, (*).
(35)
where r is such that r • (h,h)A„TmM < (R£(x)h, /I)A»T,M, for each h e AT*M, and x G M; 4) let us assume that there exists a constant ro is such that ro ■ (h, fe)A,TiM < (RJJ(X)/I,/I) A „ T M M for each h G AT^M, and x G M; then the semigroup
46 T£(*) leaves invariant the space C(M -+ ATM), and for v G C(M -4 A"TM) we have ||T?(*M*)|| A ,. T . M < e-tr" T„(t) ||t,|| A „™ (*).
(36)
The prooffollowsfrom a probabilistic representations of the semigroups Tjf(i) and T*(i), which will be given in Section 5. Let us recall that a semigroup T(t) (acting on functions on M) is called hypercontractive, if for all 1 < s < a < oo T{t) : L'(M,M) -*• L"(M,n),
(37)
and T(t) is a contraction when ^
> e" 2tA
(38)
for all t > 0 and some A > 0. Formula (34) emphasizes a certain Markov property of Tjf (t) and implies its hypercontractivity provided TM is hypercontractive. An analogous state ment for T*(£) requires positivity of R£. The following result is the corollary of Theorem 4.2. Theorem 4.3 Let us assume that in the framework of Theorem 4-2 r > 0 resp. r0 > 0. Then: 1) the semigroup T*(i) is Markov in the sense that for q = p orq = p~l resp. g= l
IW<Mz)| A „ T . M , < T„(«) IMU-™, (*);
(39)
2) the semigroup T*( A" TM.g,fi), where q = p - 1 resp. q = 1; 5^ «/TM(t) is Aj/peram£racttve, Men T%(t) is also hypercontractive (with the same X), in the sense that T*(t) : L'(M -► AnTMq,(i) -> L a (M -+ AnTM,,/x), where q=p~l 5
(40)
resp. q = 1, ond T^(t) is a contraction when (38) holds.
Probabilistic representations of semigroups
The aim of this section is to obtain probabilistic representations of the semi groups Tjf(t) and T*(i), and to prove with their aid Theorem 4.2. The construction of the diffusion process in M, which gives the stochastic dynamics associated with the classical Dirichlet form £M, is given in 2 - 4 . This process was constructed as the strong solution to an infinite system of SDE
47
d(k(t) = &«(*))# + P(&(t)) o dufcft), k € Z d ,
(41)
in the Stratonovich form on M, where w^, k € Zd, are independent Wiener processes in a given Euclidean space R n such that M c R " isometrically, and P(m) : R n -¥ TmM is the orthoprojector. We will also write this system in the form of one SDE #(*) = J ^ K W ) * + P(«*)) o dw{t)
(42) d
on M, where w is the cylinder Wiener process in /C := /2(Z -► R"), and P(a;) is the block-diagonal operator K. -> r x M with diagonal blocks Pkkfa) = P(xk). It is easy to see that A(x) £ HS(K.,TXMP) for each p eh. Theorem 5.1 (3, 4) Let &> € Cl(Mp -»> TM P ). TAen fAe SDE (42) has a unique solution £x(t) for any initial data x € M. £x{t) depends continuously (in the mean square sense) on the initial data x. The corresponding generator coincides on TC1 (M) with the Dirichlet operator of measure /*. Our next goal is to construct the parallel translation of differential forms along solutions of (42). For this, we need an analog of Levi-Civita connection onM. Let OM be the orthonormal frame bundle over M. The space OM has obviously the structure of a compact manifolds which fits into the framework of previous sections. We define the product manifold 0M:=x k 6 Z d0M.
(43)
This space will play the role of orthonormal frame bundle over M, with fibres O s M:=x k6Z 1 and a weight sequence q of positive numbers (not necessarily decreasing), we consider the bundle V := An TMq over M. Let v G V x and define by p(g)v the natural action of g G O {TXM) := xkeZ*0 (TX\M) on Vx, O (TmM) being the space of orthogonal linear transformations of TmM. We can now define the parallel translation p
W)--Vil0)-+Vm
(50)
along the solution £(t) of (42) as **«*)« = PWMO)- 1 )*, (51) where j(t) solves (49). The transformation P^ is obviously orthogonal. Let J be a continuous operator field on M, J(x) G C(VX). We define the mapping J : OM -> V€(0) by the formula J(*) := p ( « b ~ M - , ^ M * ) ) p ( * 0 . «b = 0(0), and consider the equation
(52)
= E < w(^{t)),PUt)Vv(t)
>,
(54)
where nv(t) is the solution to (53) with the initial condition y, and denote by Hi,J its generator. Proposition 5.1 Forut T(?$A -> V ) we have &<Ju{x) = l&u{x)+
< a ( i ) , V u ( i ) >x +Jm{x)u(x).
(55)
st
Proof. Let us first assume that a = 0 and J = 0. This case reduces obviously to the case of a single manifold, and the corresponding generator is equal to \ A by the finite dimensional theory 1T. In the general case, applying Ito formula to the function $(z,v) =< u(n(z)), P(ZZQX)V > on O M x V x , we obtain the additional first-order term V , * ( z , « ) [3(*)] + V.*(z,t>) [J(z)v] = < Vu(*(*)) [a(ir(x))],p(zz^1)v + < u(-w{z)),J(z)p(zZo1)v
> (56)
>,
which implies the result. The following statement follows in a straightforward way from the Ito formula and Gronwall inequality. Proposition 5.2 The semigroup T^'J(t) satisfies the following estimate: \\T*'J(t)v(x)\\{yl)m
< e tc T J(x) for each x, as operators in V,. Remark 5.1 Let us consider the bundle W := A n T M , i , where the weight sequence q1 is such that q£ < ?k for each k. Then we have V x C W x for each x e M . Let us assume that there exists a constant c\ such that Cyl > J(x) for each x, as operators in Wx. Similar arguments show that the process n(t) satisfies also the estimate
lln(tM0||W{(1)=e*Mb?(0)llw.0,
(58)
50
which implies that T €,J (f) leaves invariant the space C(M -► W ) , and for v € C(M -► W ) we have ||T € ' J (*)«(*)|| (W , ) . < e te ' I*(t) || V || W , (x).
(59)
Now we can construct probabilistic representations for Bochner and de Rham semigroups and to obtain with their aid a proof of Theorem 4.2. Let us observe that on TQ,n we have H« = -H«,
(60)
where H € is the generator of the parallel translation along the paths of the "stochastic dynamics" process f associated with /i, defined by SDE (42). If ( - H ^ , ^ n „ ) is essentially self-adjoint, the semigroup T^(i) has a simple probabilistic interpretation. Indeed, in this case Tjf(i) is the unique semigroup with the generator (—H^,^fi„) , and therefore for w € C(M -»• AnTM) we have T«(t)u;(x) = E(P t M& (*))).
(61)
In order to obtain a probabilistic representation for the semigroup T^ associated with H*, let us observe that for w € FSln we have H«'- R -w = -H*w.
(62)
The corresponding semigroup T*' -R S (t) leaves invariant the space C(M -»• An TMp-i), and coincides on this space with T^, provided H* is essentially selfadjoint on ^ n „ . The latter fact holds true if /?" € C 4 (M P -> TM,) (in this case both semigroups T*(f), T* , - R - (t) leave invariant the space C 2 (M -> An TMp-i), which obviously contains in the domain of Hjf and H*). This implies Theorem 4.2. 6
Stochastic dynamics for lattice models associated with Gibbs measures on product manifolds
Let us consider a family of potentials U = {UA)A€{I, U\ 6 (?(AfA). Let ft(k) be the family of all sets A G fi containing the point k G Zd. We will assume the following: (Ul) £
sup |l/ A (x)| < oo
Aen(k)* 6 M
(63)
51
for any k € Zd. Let n be a Gibbs measure on the Borel er-algebra B(M), associated with the family of potentials U. We denote by G{U) the family of all such Gibbs measures. G{U) is non-empty under the condition (63), see e.g. 19, 18. Let us now assume that the family of potentials U satisfies (in addition to (Ul) the following conditions: (U2) Us € C 1 (M A ) for each A, and sup V |||V k tf A |||™ < oo, k
(64)
6zdA6n
where |||Vk[/A|||TM :=sup l6M ||V k £/ A (a;)|| TiikM ; (U3) U\ G C2 (A) for each A, and there exists C < oo such that SU
P E E IIIVjVkt/A|||TM®TM < C, (65) JezrM p ). We summarize our discussion in the following Theorem 6.2 Let the family of interactions U satisfies (Ul), (U2), (US) and (66), and \i e G{U). Then there exists a weight sequence p€ h such that the statements of Theorems 4-2 and 4-S hold true. Remark 6.2 An application of an approximation technique similar to the one developed in3,7 gives the possibility to prove the essential self -adjointness of operators Hjf and H * on FSln only under the conditions (Ul), (U2), (US), and therefore to avoid the additional condition (66) in Theorem 6.2. References 1. S. Albeverio: Some applications of infinite dimensional analysis in math ematical physics, Helv.Phys.Acta 70 (1997), 479-506. 2. S. Albeverio, A. Daletskii, Yu. Kondratiev: Infinite systems of stochastic differential equations and some lattice models on compact Riemannian manifolds, Ukr. Math. J. 49 (1997), 326-337. 3. S. Albeverio, A. Daletskii, Yu. Kondratiev: Stochastic analysis on prod uct manifolds, in the book: "Stochastic Dynamics", ed. H. Crauel and M. Gundlach, Springer 1999 (Proceedings of the conference "Random Dynamical Systems", Bremen, April 28 - May 2, 1997). 4. S. Albeverio, A. Daletskii, Yu. Kondratiev. Stochastic equations and Dirichlet operators on product manifolds, Univ. Bonn, Preprint No. 591 SFB 256 (1999). 5. S. Albeverio, A. Daletskii, Yu. Kondratiev: Stochastic analysis on prod uct manifolds: Dirichlet operators on differential forms, Univ. Bonn, Preprint No. 598 SFB 256 (1999). 6. S. Albeverio, Yu. Kondratiev: Supersymmetrie Dirichlet operators, Ukr.Math.J. 47 (1995), 583-592.
53
7. S. Albeverio, Yu. Kondratiev, M. Rockner: Uniqueness of the stochastic dynamics for continuous spin systems on a lattice, J.Func.Anal., 133, No.l (1995), 10-20. 8. S. Albeverio, Yu. Kondratiev, M. Rockner: Quantum fields, Markov fields and stochastic quantization, in the book: Stochastic Analysis: Mathematics and Physics, Nato ASI, Academic Press (1995) (A. Car doso et al.,eds). 9. S. Albeverio and M. Rockner: Dirichlet forms on topological vector spaceconstruction of an associated diffusion process, Probab. Th. Rel. Fields 83 (1989), 405-434. 10. A. Arai: Supereymmetric extension of quantum scalar field theories, in Quantum and non-commutative analysis, H.Araki et al. (eds.), 73-90, Kluwer Academic Publishers, Holland (1993). 11. A.Arai: Dirac operators in Boson-Fermion Fock spaces and supereym metric quantum field theory, Journal of Geometry and Physics 11 (1993), 465-490. 12. A.Arai, I.Mitoma: De Rham-Hodge-Kodaira decomposition in oo-dimensions, Math. Ann.,291 (1991), 51-73. 13. A. Bendikov, R. Leandre: Regularized Euler-Poincare number of the in finite dimensional torus, to appear. 14. Yu. M. Beresansky and Yu. G. Kondratiev: "Spectral Methods in Infinite Dimensional Analysis", NaukovaDumka, Kiev, 1988.[English translation: Kluwer Akademic, Dordrecht/Norwell, MA, 1995]. 15. L. Cycon, R. G. Froese, W. Kirsch, B. Simon: "Schrodinger Opera tors with Applications to Quantum Mechanics and Global Geometry", Springer, 1987. 16. Yu. L. Daletcky and S. V. Fomin: "Measures and Differential Equations in Infinite-Dimensional Space", Kluwer Academic, Dordrecht, Boston, London, 1991. 17. K. D. Elworthy: Geometric aspects of diffusions on manifolds, Lecture Notes in Math., 1362, Springer Verlag, Berlin and New York, 276-425 (1988). 18. V. Enter, R. Fernandez, D. Sokal: Regularity properties and Pathologies of Position-Space renormalization-group transformations, J. Stat. Phys. 2, Nos. 5/6 (1993), 879-1168. 19. H. O. Georgii: "Gibbs measures and phase transitions", Studies in Math ematics, Vol. 9, de Gruyter, Berlin, New York (1988). 20. L.Gross: Hypercontractivity and logarithmic Sobolev inequalities for Clifford-Dirichlet forms, Duke Math. J., 43 (1975), 383-386 .
54
REAL TIME R A N D O M WALKS ON P - A D I C N U M B E R S SERGIO ALBEVERIO Institut fur Angewandte Mathematik, Universitdt Bonn, Bonn, Germany WITOLD KARWOWSKI Institute of Theoretical Physics University of Wrocllaw, Wroclaw, Poland Dedicated to Ludwig Streit on the occasion of his 60 birthday The field of p-adic numbers is a complete metric space under a non-Archimedian metrics. For this reason it is a suitable mathematical framework for the description of hierarchical phenomena appearing in many disciplines. Here we summarize some investigation of the real time random walks on p-adic based on the properties of transition functions obtained by solving the Kolmogorov equations.
1
Introduction
The last two decades have seen a steady growing interest in p-adic analysis. A substantial part of mathematical investigations has received momentum from physical motivations. There have been two main physical ideas leading towards application of local fields and in particular the p-adic numbers 1 . One came from particle physics. It was based on the conjecture that the space time at Planck distances may have a structure better described by a field other than the one of the real numbers. This has given motivation to exten sive mathematical work which resulted in elaborating interesting structures going under the names of p-adic quantum mechanics and p-adic quantum field theory 2,3 . In this paper we will relate mainly to another idea coming from statisti cal physics, in particular in connectioin with models describing relaxation in glasses. The non exponential nature of those relaxations has been interpreted as a consequence of a hierarchical structure of the state space, which can in turn be put in connection with p-adic structures. In fact p-adic numbers can be interpreted as having a "tree structure". In the physical literature various models of "diffusion" or random walks on (finite or infinite) trees have been investigated 1,15 . In particular L. Brekke and M. Olson4 constructed a class of random walks on p-adic numbers and discussed the relation with relaxations in glasses. This and pure mathematical interest motivated systematic studies of ran-
55 dom walks indexed by continuous real time with the field of p-adic numbers as state space. This note is intended to present some methods and results ob tained in this direction. The Levy processes on p-adic numbers have been con structed by different methods see Evans 16 and Figa-Talamanca 5 , but we shall be mostly concerned with the approach developed by Albeverio, Karwowski7,8 and later extended by Karwowski, Vilela-Mendes 9 , Albeverio, Karwowski, Zhao 10 and Albeverio, Karwowski, Yasuda 13 . For the relation betrween Figa-Talamanca 5 and Albeverio, Karwowski 7|8 see Husssmann 6 . In Section 2 we give basic properties of p-adic numbers and show how the translation and rotation invariant Levy processes are obtained by solving the corresponding Kolmogorov equations. We also present spectral properties of the generator. In Section 3 we generalize this method to cover corresponding processes taking values in weighted state space. In Section 4 we discuss conditions for a process to be recurrent and also hitting and exit times for p-adic balls. In Section 5 we discuss a p-adic trace formula and its analogy with the Selberg trace formula. 2
Levy Processes on Qp
We begin with basic definitions and facts about p-adic numbers 14 . Let p > 1 be a prime number. A p-adic number a is associated with the formal power series oo
a=J2^iP^
(2-1)
i=N
where AT is an integer and a,- = 0 , 1 , . . . , p - 1. With addition and multiplica tion defined in the natural way for formal power series the set Qp of all p-adic numbers becomes a field. Given a G Qp, set i0 for the smallest value of i in the sum (2.1) for which at ^ 0. Then we put
NI P =p- i 0 .
(2-2)
The map a -+ ||o|| p defines a norm in Qp, which has the non Archimedian triangle property ||a + % ^ m a x { | | a | | p , | | 6 | | p } ,
(2.3)
56 and Qp with this norm is a complete separable locally compact totally dis connected space (with the cardinality of the continuum). The series (2.1) converges with respect to the || || p norm. Let a G Qp, Mel, then the set K(a>PM)
= {x € Q P ; \\a - x\\p ^ pM} M
is called a sphere (or a ball) of radius p consequences of (2.3) i) If x e K{a,pM)
then K{x,pM)
=
(2.4)
centered at a. We note the following K(a,pM)
ii) If at e Qp, 1=1,... ,p, \\a, - ak\\p = pM+\
k # I then ^
K{ahpM)
=
M+1
K(ak,p
)
M
in) Let /C be the family of all disjoint balls of radius pM. Then K.M is countable and writing K.M = {Ki}^ we have U Ki = Qp. iv) K(a,pM)
is open and compact.
Let £ denote the er-algebra generated by the family of all balls in Qp. Then the set function /z defined on the balls by M
(K{a,pM))
=pM,
a € Qp, M <E Z
(2.5)
can be uniquely extended to a measure on £ ) p also denoted by /*. It is a Haar measure for the additive group in Qp. Put G+ for Qp as an additive group and G. = {x € QP, \\x\\p = 1}
(2.6)
for Qp \ {0} as a multiplicative group. Then G. defines a group of automor phisms Qz, z E G* of G+ by Qza = za, a € G+. We put G for the semidirect product of C?» and G+ relative to 0 : G = G. x e G+
(2.7)
i.e. G = {ff€ [z,a]:zeG*,
a€G+}
(2.8)
and 5i52 = [z\z2,ziai
+02].
(2.9)
Defining action of G on Qp by 0Z = [z, a]x = zx + a,
x e Qp, g € G
(2.10)
57
we find that G is a doubly transitive group of isometries for Qp We also have (see 1 3 ): Proposition 2.1 /z is the Haar measure for the group G. There are different starting points possible for the construction of random processes. Before presenting the approach proposed in Albeverio et al7-8 we briefly mention two other a) Levy-Khinchine representation 12 ' 16 . Put {x} for the fractional part of x G Qp. The normalized additive character on Qp (G+) is defined by X{x)= exp(27ri{x}) and then the Fourier transform of a complex valued function <j> e L1 (Qp, fi) is defined by
m = f x&MzMdx) Any Levy process on Qp can be defined by the LeVy-Khinchine formula
*(t,fl = exp< tj\x(xO-lMdx)\ where v is a cr-finite measure on £
(2.11)
satisfying v({x € Qp; \\x\\p > pM})
0 defined on the space D(QP) of locally constant functions with compact support by the formula
Da
+w = g lp-C-i / w* - f) - *(*)] Wvw;0"1^) ■
(2-12)
Qy
R e m a r k 2.1 Recently H. Kaneko22 modified formula (2.12) and obtained new random walks which apparently do not belong to any class discussed in this note.
58
The construction proposed in Albeverio et al 7 ' 8 is based on the Kolmogorov equations. We begin by taking KM for the state space. Then the system of forward and backward Kolmogorov equations reads PKiKi (*) = -a{Kj)PKtKi
(*) + £
u(Kf, Ki)PKiK,
(t)
(2.13a)
resp.
PKtKi(t) = -a{Ki)PKiKi{t) + Y,^K^Kf)P^f^t>>
(2-13b)
/#* with t > 0, i,j € N and the initial condition P K ^ . (0) = dy. a(Kj) is interpreted as intensity of the state Kj and u{K^Kj) as the infinitesimal transition probability from the initial state K{ to the target state Kj. Till now K,M is just a countable set. Its non Archimedian structure must be imprinted in the coefficients u(Ki,Kj). We shall do it as follows We say that a sequence of real numbers A = {a(n)}„ 6 z belongs to the class A iff a) b)
a(n) ^ a(n + 1), lima(n)=0.
(2.14)
n—>oo
Let A G A. If for any M e Z and m € N put U(M) = a(M - 1) - a(M) and u{M,m) = (p - l ) - 1 ? - " " - 1 C/(M + m ) . MKif\Kj
= 0 then dist p (J^i, Kj) = pM+m for some m € N. Then we define «(lfi,iir i ) = «(M,m)
(2.15)
a(Kj) = Y,u(Ki,Kj)
(2.16)
Requiring further
we immediately obtain a(Kj) = a(M).
59
The equations (2.13) with the coefficients specified by (2.15) and (2.16) can be explicitly solved. Further by shrinking the initial ball to a point while keeping the target ball fixed one arrives at the formulas which can be extended to the transition function of a random walk with Qp as the state space 7 ' 8 . Namely oo
Pt(x, K(a,pM))
=p-1{p-
l)53p-»exp{-T M .H*}
(2.17a)
t=0
if x eK(a,pM)
and
M\\ Pt(x,K(a,pM))=
— p—m
P'1 (P - 1) 5 Z P~* exp{-TAf+m+< *} «=0
- exp{-T W + m _i t}
(2.17b)
if dist p (x, K(a,pM)) = pM+m. These formulas hold for all x,a € Qp, M G Z. We put here rn = ( p - l ) _ I [ p a ( n ) - a ( n + l)] ,
neZ.
(2.18)
The transition function Pt{x,B) (t > 0 , i £ Qp, B G £ ) determined by the formulas (2.17) can be seen as the integral kernel for a Markovian semigroup (Tt,t>0)mL2(Qp]ti). Set Tt — e~Ht. Then H is a positive self adjoint operator in this space. Its action on the indicator functions for the balls can be given explicitly, which provides sufficient information for the spectral analysis of H. It turns out that the spectrum of If is essential pure point
( 0 , 7 ) •
If — H0 is the
(5.1)
If the process in Qp is denned by a(n) = p~" then the right hand side can be expressed in terms of ||7|| p i.e. the p-adic distances between 0 and 7. The process in K(0,1) can be viewed as obtained from the process on Qp by identification of the points x € A"(0,1) and x + 7, 7 G T. Conversely if — Ho generates a process on K(0,1) such that T r e - " 0 * = ft(0,0) < 00 then there is a family of processes on Qp such that (5.1) holds. References 1. R. Rammal and G. Toulouse, Ultrametricity for Physicists, Rev. Modern Phys. 58, 765-78 (1986) 2. V.S. Vladimirov, I.V. Volovich, E.I. Zelenov, p-adic Numbers in Mathe matical Physics, World Scientific, Singapore (1993) 3. A. Khrennikov, p-adic Valued Distributions in Mathematical Physics. Kluver, Dordrecht (1994) 4. L. Brekke, M. Olson, p-adic Diffusion and Relaxation in Glasses, Preprint EFI Chicago (1989) 5. A. Figa - Talamanca, Diffusion on compact ultvametvic spaces. In: Noncompact Lie Gruops and Some of Their Applications, Ed. E.A. Tanner, R. Wilson, Kluwer, Dordrecht (1994) 6. S. Hussmann, Random walks on p-adic tree. SFB 237 Preprint No. 359 '97 (1997) 7. S. Albeverio, W. Karwowski, Diffusion on p-adic Numbers, pp. 86-99 in •.Gaussian Random Fields, Ed. K. Ito, H. Hida, World Scientific, Singa pore (1991) 8. S. Albeverio, W. Karwowski, A Random Walk onp-adics: The Generator and its Spectrum, Stochastic Processes and their application, 53, 1-22 (1994) 9. W. Karwowski, R. Vilela-Mendes, Hierarchical Structures and Asymmet ric Processes on p-adics and Adeles, J. Math. Phys. 35, 4637-4650 (1994) 10. S. Albeverio, W. Karwowski, X. Zhao, Asymptotic and Spectral Results for Random Walks on p-adics, Stochastic Processes and their Applica tions, 83,39-59 (1999)
67
11. R. Lima, R. Vilela-Mendes, Stochastic Processes for the Turbulent Cas cades, Phys. Rev. E 5 3 , 3536-3540 (1996) 12. K. Yasuda, Additive Processes on Local Fields, J. Math. Sci. Univ. Tokyo, 3, 629-654 (1996) 13. S. Albeverio, W. Karwowski, K. Yasuda, Trace Formula for p-adics, (to be published) 14. N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta-Function, Springer, New York, 2nd ed. (1984) 15. M. Schreckenberg, Long Range Diffusion in Ultrametric Spaces, Z. Phys. B60, 483^88 (1985) 16. S.N. Evans, Local Properties of LeVy Processes on a Totally Disconnected Group, J. Theoret. Probab. 2, 209-259 (1989) 17. V.S. Vladimirov, Generalized Functions over the Field of p-adic Numbers, Russian Math. Surveys 43, 19-64 (1988) 18. A.N. Kochubei, Parabolic Equations over the Field of p-adic Numbers, Math. USSR Izviestiya 39, 1263-1280 (1992) 19. M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes, De Gruyter, Berlin 1994 20. S. Albeverio, Z. Zhao, On the Relation between different Construction of Random Walks on p-adic, (to be published) 21. D.A. Hejhal, The Selberg trace Formula for PSL(2,R), Vol. 1,2, Lecture Notes in Mathematics 548 resp. 1001, Springer-Verlag (1976 resp. 1983) 22. H. Kaneko, A class of spatially inhomogeneous Dirichlet spaces on the p-adic number field. Preprint (1999)
68 CHARACTERIZATION OF TEST F U N C T I O N S IN CKS-SPACE NOBUHIRO ASM* Graduate School of Mathematics Nagoya University Nagoya 464-8602 JAPAN IZUMI KUBO Department of Mathematics Faculty of Science Hiroshima University Higashi-Hiroshima 739-8526 JAPAN HUI-HSIUNG KUO t Department of Mathematics Louisiana State University Baton Rouge LA 70803 USA We prove a characterization theorem for the test functions in a CKS-space. Some crucial ideas concerning the growth condition are given.
1
Introduction
Let £ be a real countably-Hilbert space with topology given by a sequence of norms {| • | } £ 1 0 (see n . ) Let £p be the completion of £ with respect t o t h e norm | • | p . Assume the following conditions: (a) There exists a constant 0 < p < 1 such t h a t | • | 0 < p\ ■ |i < • • • < pp\ ■ \p
0, there exists q > p such t h a t the inclusion m a p iq>p : £Q —► > then its [£p]*a-norm is given by / oo
II*H-»I/«=
,
\ V2
a
E^-i -P
•
^
70
This Gel'fand triple [£]a C (L 2 ) C [£\*a is called the CKS-space associated with a sequence {a(n)}£L 0 of numbers satisfying the above conditions (Al) and (A2). Several characterization theorems for generalized functions in [£]^ have been proved in the paper 4 . However, no characterization theorem for test functions in [£]a is given. The purpose of the present paper is to prove such a theorem. In addition we will mention some crucial ideas in order to get a complete description of the characterization theorems for test and general ized functions in our ongoing research collaboration project. We remark that similar results have been obtained by Gannoun et al. 5 . 2
Characterization theorems
For £ e £c (the complexification of £,) the renormalized exponential function :e(''t~*: is defined by oo
:e 0,
as n -> oo.
This shows that condition (Al) implies condition (A2). On the other we see t n a t hand, by applying Corollary 4.4 in 4 to the sequence {^y}^Lo 1)J condition (B2) implies condition (Bl). Moreover, it is easy to check that condition (B3) implies condition (B2). Now, we study the characterization theorem for test functions in [£]aFirst we prove a lemma. L e m m a 2.2 Assume that condition (Bl) holds and let F be a function on £c satisfying the conditions: (1) For any £, n in £c, the function F(z£ + 77) is an entire function of z G C. (2) There exist constants K,a,p>0
such that
\F(0\ 0: l(/n,6®"-®€n-^|