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
Prefacexi
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
*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
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.
Preface
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.