QUANTUM GEOMETRY

It is with sorrow that we
report that Dr. Eduard Prugovecki passed away at his home in Lake
Chapala Mexico on October 13 |

Table of Contents

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

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

Notes to Chapter 2 63

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 Hopf Bundles and Line Bundles 99

3.9. Quantum Frames and Quantum Metrics in Typical Quantum Fibres 105

Notes to Chapter 3 109

*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

5.1. Klein-Gordon Quantum Bundles 137

5.2. Parallel Transport in Klein-Gordon Bundles 141

*5.3. Quantum Torsion and the Klein-Gordon Quantum Connection 147

5.4. Geometro-Stochastic Propagation in Klein-Gordon Quantum Bundles 149

5.5. The Physical Interpretation of the GS Klein-Gordon Framework 155

*5.6. GS Propagation in Klein-Gordon Bundles and Quantum Diffusions 158

5.7. Relativistic Causality and Quantum Stochasticity 165

Notes to Chapter 5 171

*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

*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-0 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

8.1. Fock-Dirac Bundles for Spin-1/2 Charged Quantum Fields 246

8.2. Parallel Transport and Stress-Energy Tensors in Fock-Dirac Bundles 248

8.3. Second-Quantized Frames in Berezin-Dirac Superfibre Bundles 250

8.4. Geometro-Stochastic Propagation in Fock-Dirac Bundles 255

Notes to Chapter 8 258

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 Electrodynamics 289

Notes to Chapter 9 300

10.1. Basic Geometric Aspects of Classical Yang-Mills Fields 308

10.2. Gauge Groups of Global Gauge Transformations in Principal Bundles 311

*10.3. Graded Lie Algebras Generated by Connection Forms 316

10.4. BRST Transforms and Ghost Fields in Classical Yang-Mills Theories 320

*10.5. Lorenz and Transverse Gauges in Typical Weyl-Klein Fibres 324

*10.6. Geometro-Stochastic Quantization of Yang-Mills Fields 330

Notes to Chapter 10 334

11.1. Canonical 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-Stochastic 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

11.11. Principles of GS Propagation in Quantum Gravitational Bundles 405

11.12. Foundational Aspects of GS Quantum Cosmology 412

Notes to Chapter 11 421

Developments in Relativity and Quantum Theory 433

12.1. Positivism vs. Realism in Relativity Theory and Quantum Mechanics 435

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 sections marked with an asterisk can be omitted at a first reading.

The present monograph provides a systematic and basically self-contained introduction to a mathematical framework capable of incorporating those fundamental physical premises of general relativity and quantum mechanics which are not mutually inconsistent, and which can be therefore retained in the unification of these two fundamental areas of twentieth-century physics. Thus, its underlying thesis is that the equivalence principle of classical general relativity remains true at the quantum level, where it has to be reconciled, however, with the uncertainty principle. As will be discussed in the first as well as in the last chapter, conventional methods based on classical geometries and on single Hilbert space frame-works for quantum mechanics have failed to achieve such a reconcilia-tion. On the other hand, foundational arguments suggest that new types of geometries should be introduced.

The geometries proposed and studied in this monograph are referred to as quantum geometries, since basic quantum principles are incorporated into their structure 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 theory, group representation theory, and modern formulations of differential geometry. The developed physical concepts have their roots in nonrelativistic and relativistic quantum mechanics in Hilbert space, in classical general relativity and in quantum field theory for massive 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 mathematical structures that can serve as frameworks within which such theories can be developed in an epistemologically 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 perspectives on the world of elementary particles.

The essential ideas and techniques of the varied and rich disciplines treated in this monograph are explained in the appropriate sections 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 space, the reader is directed in a carefully guided manner to specific sections 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 accessible to most graduate students in physics and in mathematics. 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 mechanics (cf., e.g., Chapters 1-4 of [PQ]), that is not absolutely mandatory, since all the basic concepts and results are explained in the text, and for the details which are not covered, instructions are given as to where to find them in the key references. For readers at a more advanced level, detailed references to conference proceedings, lecture notes and contemporary papers published in professional physics and mathematics journals are provided in the notes at the end of each chapter. Consequently, this book can be used also as a reference manual and guide to literature for research in the areas it covers.

From the mathematical point of view, the quantum geometries presented in this monograph are infinite-dimensional fibre bundles associated with principal bundles [C,I] whose structure groups incorporate the Poincaré group – or its covering group ISL(2,C)). The base manifolds of these fibre bundles are Lorentzian manifolds, or their appropriate frame-bundle extensions; whereas, their typical fibres are infinite-dimensional (pseudo-) Hilbert spaces or superspaces. The study of connections on such fibre bundles poses interesting mathematical problems, which appear to have received scant attention thus far.

From the physics point of view, the principal areas of application of the present framework are to quantum field theory in curved spacetime and to quantum gravity. The ensuing methodology is distinct from that of other approaches to these disciplines in that it is derived from

From the point of view of the quantum mechanics on phase space presented in Part I of [P], the present quantum geometries enable an extrapolation of their special-relativistic frameworks to the general-relativistic regime. In this context it should be noted that such an extrapolation had been attempted in Part II of [P], but that it ran into the same main difficulty as the more conventional approaches to quantum field theory in curved spacetime – namely, it did not succeed in properly adapting the equivalence principle of classical general relativity to the quantum regime. The present fibre-theoretical framework has, however, succeeded in that task. In fact, although this framework displays many novel features which are of independent mathematical interest, the achievement of that goal represents its major motivation from the point of view of physics. Consequently, abundant corroborative quotations of well-known authorities in the field are provided, not only as vouchers of the fundamental need for a radical revision of many of the conventional ideas in relativistic quantum theory, but also as a guide to further independent study, that might lead some readers to new ideas of their own.

Central to the application of the present quantum geometry framework to quantum physics is the idea of

Thus, when taken in conjunction with the idea of geometro-stochastic quantum propagation, the quantum geometry framework provides a viable blueprint for the consistent unification of general relativity and quantum theory, which does not give rise to conflicts with conventional theory in those areas where that theory has received unquestionable experimental support. Moreover, as will be seen in the present monograph, quantum geometries can also incorporate many of the theoretical ideas which are in vogue at the present time. Hence, it can serve as a basic framework within which such ideas can be formulated with the mathematical clarity and rigor required for the understanding of their multifaceted physical implications. It is, therefore, primarily as a

In the course of the four decades which followed after the early numerical successes of renormalization theory in quantum electrodynamics, the world of elementary particle physics has witnessed a great variety of constantly changing fashions: in turn, there were vigorous promotions of dispersion relations, Regge poles, current algebras, quark models and QCD, supersymmetry, grand-unification, superstrings (the “Theory Of Everything”) – and new fashions are constantly emerging. On the other hand, as will be seen from the numerous quotations provided in the present text, during that same period, the still living grand masters of twentieth century physics repeatedly urged the pursuit of deeper mathematical as well as epistemological analyses. Although their pleas remained mostly unheeded in the world of very rapidly changing fashions in quantum theoretical physics, they kept warning against the prevailing profes-sional compla-cency in some of the most relevant areas. Most steadfast in his refusal to accept fashionable but fundamentally flawed developments in those areas of quantum theory which he had founded was P. A. M. Dirac.

When I was a graduate student, I had the privilege of seeing Dirac, and of listening to him, as I attended seminars at which he was present during an extended visit which he made to the Institute of Advanced Studies in Princeton in the early 1960s. By that time he was, of course, already a legendary figure – one of the revered, great physicists of this century, whose professional stature overshadowed that of all the other distinguished physicists who were in regular attendance at those weekly seminars. However, while many of those well-known luminaries liked to impress the audience with comments which confirmed their intellectual brilliance, as a rule Dirac kept his counsel, and limited himself to only an occasional question, which would be pertinent but always unprepossessing. It was only many years later, after I began to read his critical assessments (cf., e.g., pp. 8, 190, 244, 289, 481) of the very foundations of some of the theories presented at those seminars, that I came to appreciate the understated greatness of his genius, and the depth of his commitment to an ideal of

Considering that Dirac is the universally acknowledged founder of quantum field the-ory, the very title of his last paper (“The Inadequacies of Quantum Field Theory” – cf. p. 190), bespeaks of lofty professional ideals and ethical standards, maintained throughout his entire life with a steadfastness and uncompromising integrity that has almost no parallel in this century. Although his critical comments might not have reached the fashion-conscious amongst the theoretical and the mathematical physicists, his words and deeds have inspired and provided moral support to me, as I trust it did to many other researchers who share his ideals and basic values with regard to the principles and aims of theoretical physics. It is therefore with genuine reverence and overwhelming spiritual gratitude that I dedicate this work to the memory of Paul A. M. Dirac – a great physicist, as well as a truly great man.

I would like to express my thanks to all those colleagues who have contributed useful comments and information in the course of more than two decades, during which the programme recounted in this and my two preceding mono-graphs ([PQ] and [P]) was gradually developed. Professors James A. Brooke and Wolfgang Drechsler had the opportunity to examine the first draft of many of the chapters in the present monograph, and I hereby thank them for their valuable comments. Special thanks are due to Scott Warlow for his very careful proofreading of the entire first draft, and for his many insightful comments and suggestions. I also wish to thank the editor of the Fundamental Theories of Physics series, Professor Alwyn van der Merwe, for enabling me, for a second time in eight years, to expound the outcome of my research into the foundations of relativity and quantum the-ory in the cogent form of a basically self-contained monograph published in this series. Last, but certainly not least, I wish to express my gratitude to my wife, Margaret R. Prugovecki, for her assistance with the preparation of the manuscript, and her moral support during what proved to be a protracted and arduous journey into uncharted territories.

EDUARD PRUGOVECKI

Toronto, October 1991