It is with sorrow that we report that Dr. Eduard Prugovecki passed away at his home in Lake Chapala Mexico on October 13th 2003

Table of Contents

Preface               xi     

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
        Notes 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 Hopf Bundles and Line Bundles    99
    3.9.     Quantum Frames and Quantum Metrics in Typical Quantum Fibres    105     
        Notes to Chapter 3    109

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 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

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-0 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-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    

Chapter 8. Relativistic Quantum Geometries for Spin-1/2 Massive Fields    245

    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

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 Electrodynamics    289
        Notes to Chapter 9    300

Chapter 10. Classical and Quantum Geometries for Yang-Mills Fields    307

    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

Chapter 11. Geometro-Stochastic Quantum Gravity    337

    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    

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         
    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 foundational measurement-theoretical considerations. These considerations have led to a formulation of nonrelativistic and special-relativistic quantum theory on phase space, whose pre-1984 results have been comprehensively summarized in a preceding monograph [P], and whose subsequent developments can be found in some of the more recent works cited in the main text of the present monograph. These developments reflect the possibility of resolving long-standing quantum paradoxes by a careful analysis of old as well as of new quantum-measurement schemes and experimental procedures.
    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 geometro-stochastic propagation, developed by the present author as a mathematically and epistemologically sound extrapolation of the standard path-integral formalism. Fundamentally, this idea proposes a modification of the concept of quantum point particle (which according to the orthodox interpretation displays the behavior of either a wave or a particle) into that of geometro-stochastic exciton, whose micro-dynamics simultaneously embodies the classical attributes of “waves” as well as of “particles”. Thus, geometro-stochastic excitons possess proper state vectors belonging to the fibres of the quantum bundles constituting quantum geometries, and are localized in relation to quantum Lorentz frames, which take over in the geometro-stochastic framework the role played by their classical counterparts in classical general relativity. These proper state vectors propagate externally (i.e., in the classical spacetimes which constitute the base manifolds of those quantum bundles) along stochastic paths in a manner which is formally analogous to that of classical particles in diffusion processes. On the other hand, they are superimposed at each location in the base manifold in the manner which in classical physics is associated only with the behavior of waves. Thus, the wave-particle dichotomy present in the orthodox approach to quantum mechanics is replaced, at the micro-level, with a unified physical picture of quantum behavior, which has no counterpart in classical physics. It is only at the macro-level that, in complete accord with Bohr's ideas, geometro-stochastic exciton behavior manifests itself, in the context of certain experimental arrangements, as that of a “particle”; whereas, in the context of some other experimental arrangements, it manifests itself as that of a “wave”. Furthermore, in the nonrelativistic context, the mathematical apparatus of quantum geometry allows the transition to a sharp-point limit, which mathematically corre-sponds to the transition to proper wave functions which are delta-like. In that limit the conventional quantum mechanics of pointlike particles is recovered, thus demonstrating that geometro-stochastic dynamics is a natural outgrowth of conventional quantum theory (cf. Chapters 3 and 4).
    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 framework of ideas, rather than as a finished theory, that the results on quantum geometries are presented in this monograph.
    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 truth in physical theories, which manifested itself in mathematical beauty – a beauty totally at odds with the ungainliness inherent in the ad hoc “working rules” of conventional renormalization theory. Indeed, some of his characteristic comments, such as the one that “people are ... too complacent in accepting a theory which contains basic imperfections” (Dirac, 1978a, p. 20), reveal that, already by that time, he was all too painfully aware that his goals and ideals had become decidedly “unfashionable” in the world of perpetually changing fashions in elementary particle physics.
    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.

        Toronto, October 1991

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