Table Of ContentQuantum Geometry
Fundamental Theories of Physics
An International Book Series on The Fundamental Theories 0/ Physics:
Their Clarification, Development and Application
Editor: AL WYN VA N DER MERWE
University oj Denver, U.S.A.
Editorial Advisory Board:
ASIM BARUT, University oj Colorado, U.S.A.
HERMANN BONDI, University ojCambridge, U.K.
BRIAN D. JOSEPHSON, University ojCambridge, u.K.
CUVE KILMISTER, University 0/ London, U.K.
GÜNTER LUDWIG, Philipps-Universität, Marburg, Germany
NATHAN ROSEN, Israel Institute ojTechnology,/srael
MENDEL SACHS, State University ojN ew York at Buffalo, U.S.A.
ABDUS SALAM, International Centre/or Theoretical Physics, Tt·ieste, Italy
HANS-JÜRGEN TREDER, Zentralinstitut für Astrophysik der Akademie der
Wissenschaften, Germany
Volume48
Quantum Geometry
A Frameworkfor
Quantum General Relativity
by
Eduard Prugovecki
Department o/ Mathematics.
University o/Toronto.
Toronto. Canada
Springer-Science+Business Media, B.Y.
Library of Congress Cataloging-in-Publication Data
Prugoveck1. Eduard.
Ouantum geometry : a framework for quantum general relat1v1ty / by
Eduard Prugoveck1.
p. em. -- (Fundamental theor1es of phys1es ; v. 48)
Ineludes b1bl1ograph1eal referenees (p. ) and 1ndex.
1. Geometrie quant1zation. 2. General relat1v1ty (Phys1es)
I. T1tle. II. Series.
OC174.17.G46P78 1992
530' . 1--d e20 92-2540
ISBN 978-90-481-4134-0 ISBN 978-94-015-7971-1 (eBook)
DOI 10.1007/978-94-015-7971-1
Printed on acid-free paper
All Rights Reserved
© 1992 Springer Science+Business Media Dordrecht
Originally published by KIuwer Academic Publishers in 1992
Softcover reprint of the hardcover 1s t edition 1992
No part of the material protected by this copyright notice may be reproduced or
utilized in any fonn or by any means, electronic or mechanical,
including photocopying, recording or by any infonnation storage and
retrieval system, without written pennission from the copyright owner.
To the memory of a great physicist of penetrating vision and lasting achievements,
whose scientific legacy and uncompromising integrity continue to inspire:
Paul Adrien Mauriee Dirae
(1902-1984)
"One must seek a new relativistic quantum mechanics
and one's prime concern must be 10 base it on
sound malhematics."
P. A. M. Dirae (1978)
Table of Contents
Preface Xl
Chapter 1. Principles and Physical Interpretation of Quantum Geometries 1
1. 1. The Historical Background of Physical Geometries 2
1 .2. The Incompatibility of Quantum Theory with Classical Relativistic Geometries 6
1.3. Basic Principles of Quantum Geometry 10
1.4. Physical Features of Geometro-Stochastic Propagation 16
1.5. The Physical Nature of Geometro-Stochastic Excitons 22
Notes to Chapter 1 27
Chapter 2. The Fibre Bundle Framework for Classical General Relativity 30
2.1. Tensor Bundles over Four-Dimensional Differential Manifolds 31
2.2. General Linear Frame Bundles over Four-Dimensional Manifolds 35
2.3. Orthonormal Frame Bundles over Lorentzian Manifolds 40
*2.4. Parallel Transport and Connections in Principal and Associated Bundles 43
*2.5. Connection and Curvature Forms on Principal Bundles 48
2.6. Levi-Civita Connections and Riemannian Curvature Tensors 52
2.7. Einstein Field Equations and Principles of General Relativity 57
N otes to Chapter 2 63
Chapter 3. Stochastic Quantum Mechanics on Phase Space 66
3.1. Nonrelativistic Systems of Imprimitivity 67
3.2. Nonrelativistic Systems of Covariance 72
3.3. Relativistic Systems of Imprimitivity 77
3.4. Relativistic Systems of Covariance 80
3.5. Probability Currents and Sharp-Point Limits 85
3.6. Path Integrals in Stochastic Quantum Mechanics 88
3.7. Quantum Frames and Quantum Informational Completeness 94
*3.8. Kähler Metrics and Connections in HopfBundles and Line Bundles 99
3.9. Quantum Frames and Quantum Metrics in Typical Quantum Fibres 105
Notes to Chapter 3 109
vili Table ofContents
Chapter 4. Nonrelativistic Newton-Cartan Quantum Geometries 116
*4.1. Classical Newton-Cartan Geometries 117
*4.2. Newton-Cartan Connections in Bargmann Frame Bundles 121
*4.3. Quantum Newton-Cartan Bundles 123
*4.4. Geometro-Stochastic Propagation in Quantum Newton-Cartan Bundles 127
Notes to Chapter 4 134
Chapter 5. Relativistic Klein-Gordon Quantum Geometries 136
5.1. Klein-Gordon Quantum Bundles 137
5.2. Parallel Transport in K1ein-Gordon Bundles 141
*5.3. Quantum Torsion and the K1ein-Gordon Quantum Connection 147
5.4. Geometro-Stochastic Propagation in K1ein-Gordon Quantum Bundles 149
5.5. The Physical Interpretation of the GS Klein-Gordon Framework 155
*5.6. GS Propagation in K1ein-Gordon Bundles and Quantum Diffusions 158
5.7. Relativistic Causality and Quantum Stochasticity 165
N otes to Chapter 5 171
Chapter 6. Relativistic Dirac Quantum Geometries 176
*6.1. Spinorial Wave Functions in Wigner and Dirac Representations 177
6.2. Standard Fibres for Dirac Quantum Bundles 181
*6.3. Dirac Quantum Frame Bundles 183
*6.4. Dirac Quantum Bundles 186
Notes to Chapter 6 188
Chapter 7. Relativistic Quantum Geometries for Spin-O Massive Fields 190
*7.1. Canonical Second-Quantization in Curved Spacetime 192
*7.2. Spontaneous Rindler Particle Creation in Minkowski Spacetime 199
*7.3. Ambiguities in the Concept of Quantum Particle in Curved Spacetime 205
7.4. Fock Quantum Bundles for Spin-O Neutral Quantum Fields 211
7.5. Parallel Transport and Action Principles in Fock Quantum Bundles 216
7.6. Relativistic Microcausality and Geometro-Stochastic Field Locality 222
7.7. Strongly and Weakly Causal Geometro-Stochastic Field Propagation 230
7.8. Interacting Quantum Fields in Extended Fock Bundles 233
Notes to Chapter 7 240
Chapter 8. Relativistic Quantum Geometries for Spin-l/2 Massive Fields 245
8.1. Fock-Dirae Bundles for Spin-1!2 Charged Quantum Fields 246
8.2. Parallel Transport and Stress-Energy Tensors in Foek-Dirae Bundles 248
8.3. Second-Quantized Frames in Berezin-Dirae Superfibre Bundles 250
Table o/Contents ix
8.4. Geometro-Stochastic Propagation in Fock-Dirae Bundles 255
Notes to Chapter 8 258
Chapter 9. Quantum Geometries for Electromagnetic Fields 260
9.1. Krein Spaces for Momentum Space Representations of Photon States 261
9.2. The Typical Krein-Maxwell Fibre for Single Photon States 266
9.3. Gupta-Bleuler Quantum Bundles and Frames 271
*9.4. Parallel Transport in Gupta-Bleuler Quantum Bundles 278
9.5. Stress-Energy Tensors and GS Propagation in Gupta-Bleuler Bundles 284
*9.6. Geometro-Stochastic vs. Conventional Quantum Electrodynamies 289
Notes to Chapter 9 300
Chapter 10. Classical and Quantum Geometries for Yang-Mills Fields 307
10.1. Basic Geometrie Aspeets of Classical Y ang-Mills Fields 308
10.2. Gauge Groups of Global Gauge Transfonnations in Principal Bundles 311
*10.3. Graded Lie Algebras Generated by Conneetion Fonns 316
10.4. BRST Transfonns and Ghost Fields in Classieal Yang-Mills Theories 320
*10.5. Lorenz and Transverse Gauges in Typical Weyl-Klein Fibres 324
*10.6. Geometro-Stochastie Quantization of Y ang-Mills Fields 330
Notes to Chapter 10 334
Chapter 11. Geometro-Stochastic Quantum Gravity 337
11.1. Canonieal Gravity and the Initial-Value Problem in CGR 339
11.2. Contemporary Approaches to the Quantization of Gravity 346
11.3. Basic Epistemic Tenets of Geometro-Stochastie Quantum Gravity 353
11.4. Observables and Their Physical Interpretation in CGR and QGR 360
11.5. Quantum Pregeometries for GS Graviton States 371
11.6. Lorenz Quantum Gravitational Geometries 379
11.7. Internal Graviton Gauges and Linear Polarizations 384
11.8. Null Polarization Tetrads and Graviton Polarization Frames 388
11.9. Quantum Gravitational Faddeev-Popov Fields and Gauge Groups 392
11.10. Quantum Gravitational BRST Symmetries and Connections 397
11.1l. Principles of GS Propagation in Quantum Gravitational Bundles 405
11.12. Foundational Aspects of GS Quantum Cosmology 412
Notes to Chapter 11 421
Chapter 12. Historical and Epistemological Perspectives on
Developments in Relativity and Quantum Theory 433
12.1. Positivism VS. Realism in Relativity Theory and Quantum Mechanics 435
x Table ofContents
12.2. Conventionalistic Instrumentalism in Contemporary Quantum Physics 439
12.3. Inadequacies of Conventionalistic Instrumentalism in Quantum Physics 443
12.4. General Epistemological Aspects of Quantum Geometries 456
12.5. The Concept of Point and Form Factor in Quantum Geometry 461
12.6. The Physical Significance of Quantum Geometries 466
12.7. Summary and Conclusions 470
Notes to Chapter 12 474
References 486
Index 513
Note: The seetions marked with an asterisk can be omitted at afirst reading.
Preface
The present monograph provides a systematic and basicaIly self-eontained introduetion to a
mathematieal framework eapable of ineOIporating those fundamental physical premises of
general relativity and quantum meehanics which are not mutually ineonsistent, and which
ean be therefore retained in the unifieation of these two fundamental areas of twentieth
eentury physics. Thus, its underlying thesis is that the equivalenee principle of classical
general relativity remains true at the quantum level, where it has to be reeonciled, however,
with the uneertainty principle. As will be discussed in the first as weIl as in the last chapter,
eonventional methods based on classical geometries and on single Hilbert space frame
works for quantum meehanics have failed to aehieve such a reconciliation. On the other
hand, foundational arguments suggest that new types of geometries should be introdueed.
The geometries proposed and studied in this monograph are referred to as quantum
geometries, sinee basic quantum principles are ineorporated into their strueture from the
outset. The mathematical tools used in constructing these quantum geometries are drawn
from functional analysis and fibre bundle theory, and in particular from Hilbert space the
ory, group representation theory, and modern formulations of differential geometry. The
developed physical eoncepts have their roots in nonrelativistic and relativistic quantum me
chanics in Hilbert spaee, in classical general relativity and in quantum field theory for mas
sive and gauge fields. However, the principal aim of this monograph is to deal not with
specific physical theories, beyond QED and quantum gravity, but rather with general math
ematieal structures that ean serve as frameworks within which such theories can be devel
oped in an epistemologieally and mathematically sound manner. On the other hand, we
shall demonstrate that the novel features of these frameworks not only clarify some long
standing questions of quantum field theory in curved spacetime and of quantum gravity,
but also give rise to some new perspeetives on the world of elementary particles.
The essential ideas and teehniques of the varied and rieh disciplines treated in this
mono graph are explained in the appropriate seetions of its text. In order to deepen the
reader's understanding of those more technical aspects which could not be included due to
limitations on spaee, the reader is direeted in a carefully guided manner to specific seetions
of a score of key references, singled out from the list of references provided at the end of
this book. It is therefore hoped that despite the advanced nature of the presented material,
this monograph will be aecessible to most graduate students in physics and in mathematies.
Thus, although it is desirable that a student already have some understanding of the
mathematical foundations of classical general relativity (cf., e.g., Chapters 1-5 of [W] from
amongst the aforementioned key references) and of standard nonrelativistic quantum
meehanics (cf., e.g., Chapters 1-4 of [PQ]), that is not absolutely mandatory, since all the
