It is with sorrow that we
report that Dr. Eduard Prugovecki passed away at his home in Lake
Chapala Mexico on October 13th
2003
Contents
Preface ix
List of Symbols xiii
Chapter 1. Survey of Principal Historical Developments 1
1.1. From Special to General Relativity
1
1.2. Geometry as Part of Physical
Theory 5
1.3. Quantum Theory and the Idea of
Fundamental Length 9
1.4. Localizability and Renormalizability
in Quantum Field Theory 12
1.5. Quantum Field Theory in Curved
Spacetime 15
1.6. From Canonical Quantum Gravity
to Superstrings 17
Chapter 2. Classical Frame Bundles in General Relativity
22
2.1. General Covariance under Coordinate
Transformations 22
2.2. General Covariance and Classical
Frame Bundles 26
2.3. Moving Frames in Principal
Frame Bundles 31
2.4. Gauge Invariance in Associated
Bundles 36
2.5. Connections and Gauge Transformations
39
2.6. Levi-Civita Connections and the
Strong Equivalence Principle 47
2.7. The Einstein Field Equations
and Canonical Gravity 53
Chapter 3. Quantum Frames and Spacetime Localizability 62
3.1. The Uncertainty Principle and
Representations of the Galilei Group 62
3.2. Quantum Mechanics and Informational
Completeness 68
3.3. Informationally Complete Nonrelativistic
Quantum Frames 76
3.4. Sharp-Point Limits of Nonrelativistic
Quantum Frames 82
3.5. Path Integration and Nonrelativistic
Quantum Frames 88
3.6. Poincaré Covariance and
Relativistic Quantum Localizability 95
3.7. Poincaré Covariance and
Quantum Lorentz Frames 99
3.8. Fundamental Special-Relativistic
Quantum Lorentz Frames 106
Chapter 4. Quantum Geometry over a Classical Base Spacetime
112
4.1. Quantum Frame Bundles and Associated
Bundles 112
4.2. The Internal Hilbert Structure
of Quantum Bundles 116
4.3. Connections on Affine Frame Bundles
and Associated Bundles 121
4.4. Connections and Parallel Transport
in Quantum Bundles 126
4.5. Quantum Tensorial Bundles and
Quantum Metrics 130
4.6. Quantum-Geometric Propagation
in Quantum Bundles 133
4.7. The Physical Meaning of Quantum-Geometric
Propagation 139
4.8. Relativistic Causality of Classical
and Quantum Propagation 145
Chapter 5. Massive Quantum-Geometric Boson Fields 153
5.1. Microcausality vs. Local Commutativity
of Quantum Fields 153
5.2. Quantum Frame Fields and Microcausality
158
5.3. Second-Quantized Boson Frames
in Fock Bundles 164
5.4. Propagators for Parallel Transport
in Fock Bundles 168
5.5. Geometric Localization of Boson
Field Exciton Modes 175
5.6. Quantum-Geometric Boson Field
Propagation in Free Fall 184
5.7. Quantum Field Interactions and
Relativistic Locality 193
5.8. Quantum-Geometric Propagation
of Interacting Boson Fields 200
7.1. Hilbert Space Representations
of the Poincaré Group 233
7.2. Krein Space Representations
of the Poincaré Group 236
7.3. Typical Krein Fibres
for Spin-1 Bosons 241
7.4. Gupta-Bleuler Bundles for Multi-Photon
States 245
7.5. Quantum-Geometric Propagation
in Gupta-Bleuler Bundles 251
7.6. The Geometric Interpretation
of Yang-Mills Gauge Fields 254
7.7. Global Gauge Transformations
on Principal Bundles 257
7.8. BRST and Anti-BRST Operators
on Spaces of Connections 260
7.9. The Quantum-Geometric Framework
for Yang-Mills Fields 265
Chapter 8. Geometric Quantum Gravity 271
8.1. Diffeomorphism and Poincaré
Gauge Invariance in CGR 271
8.2. Basic Aspects of Quantum-Geometric
Gravity 277
8.3. Superlocal Graviton States and
Internal Gauges 281
8.4. Linearly and Circularly Polarized
Graviton States 285
8.5. Quantum Gravitational and Berezin-Faddeev-Popov
Frames 291
8.6. Quantum Gravitational Gauge Supergroups
and Quantum Spacetime 295
8.7. Quantum Gravitational Connections
and BRST Symmetries 303
8.8. Superlocal Quantum Gravitational
and Matter Fields 307
8.9. Quantum-Geometric Evolution
of Gravitational and Matter Fields 312
8.10. Summary and Conclusion 320
References 323
Index 341
Preface
The primary goal of the present monograph is to explain and analyze, in
a manner accessible to readers with a graduate-student level of knowledge
and understanding of general relativity and quantum theory, the principles
of a quantum-geometric unification of these two disciplines, formulated in
a manner that can be justifiably said to supply the foundations for a quantum
counterpart of classical general relativity.
In view of the many attempts, carried out in the course
of the past several decades, at a consistent unification of general relativity
with quantum theory, no contemporary theoretical framework can lay the claim,
without qualifications, to the title “quantum general relativity.” On the
other hand, what the main principles underlying such a framework should
be emerges very clearly from the work of the founders of general relativity
and quantum theory. Indeed, the principles underlying the classical theory
of general relativity were enunciated by Einstein from the outset, and were
further clarified in his later writings; whereas those of quantum theory
were formulated, often explicitly, but sometimes only implicitly, by Bohr,
Born, Dirac, Heisenberg, and other well-known founders of quantum mechanics
in many of their early writings. Therefore, what the basic principles of
quantum general relativity should be can be discerned by examining
and analyzing those writings.
It is the intent of the various chapters in the
present monograph to study all the main physical and mathematical implications
of the principles that emerge from those writings, examine their mutual
compatibility at the conceptual level, and describe a geometric framework
capable of incorporating them in a consistent manner, by taking advantage
of recent advances in certain areas of mathematics and theoretical physics.
Clearly, such a framework should also be capable of incorporating theoretical
models whose numerical predictions would be in agreement with those of experimentally
well-supported conventional models in relativistic quantum theory. It will
be shown that such a task can be achieved by taking advantage of key geometric
aspects and results in the theory of gauge fields and of coherent states
developed during the last two decades, by incorporating into quantum mechanics
and quantum field theory the idea of fundamental length along the epistemic
lines first suggested by Born and Heisenberg, and by extrapolating the classical
concept of local frame of reference into a concept of quantum local
frame of reference, into which such a fundamental length is embedded from
the outset.
The introduction of a fundamental length in quantum
physics was advocated already in the 1930s by several of the best known
founders of quantum mechanics. However, for various historical reasons (which
will be discussed in the first chapter of this monograph), it was paid scant
attention by the subsequent generation of theoretical physicists. It was
only during the last decade that string theory revived the interest in this
idea by proposing the Planck length as a basic parameter related to the
extension of superstrings.
On the other hand, as opposed to the historical development
of general relativity as well as of quantum theory, string theory itself
emerged from phenomenological rather than from foundational considerations.
As a consequence, its underlying physical principles were not clearly
enunciated from the outset, and some are still obscure. Hence, the present
monograph will concentrate on a purely geometric formulation of quantum general
relativity into which only well-established gauge groups are incorporated.
In other words, the considered quantum geometries are based only on those
principles of general relativity and quantum theory that have survived the
test of time. On the other hand, these quantum geometries incorporate into
their very structure a fundamental length in such a manner that, in the non-relativistic
and the special relativistic regime, conventional results can be recovered
in the sharp-point limit that is obtained as that length is allowed to tend
to zero.
A very general quantum-geometric framework for general
relativity was expounded in a recent monograph of the present author. In
contradistinction, the present volume sets emphasis on the implementation
of fundamental physical principles rather than on mathematical generality,
as it strives to bring out to the fore the conceptual simplicity of the ideas
underlying those principles. These fundamental physical principles, and their
method of implementation, will be explained at a level accessible to students
who have already taken introductory courses in quantum mechanics and general
relativity, but have not yet necessarily acquired the background knowledge
in fibre and superfibre bundle theory, Krein spaces, gauge groups and gauge
fields, coherent states, and other physically or mathematically advanced
topics required in the formulation of quantum geometries. This means that
the basic concepts of these mathemat-ical disciplines will be explained in
this monograph in a physics context, by first describing the physical
motivation for their introduction, and then elaborating only on those mathematical
aspects that are of an immediate physical relevance. Those readers interested
in their more advanced mathematical aspects will be directed, in the main
text or in footnotes, to suitable publications.
This does not mean, however, that mathematical rigor
has been in any way compromised. Rather, the mathematical material that is
actually presented in this monograph is formulated in the mathematically sound
manner advocated by Dirac as a methodological guide to the formulation of
a “new relativistic quantum mechanics,” and with the use of precise notation
meant to bring out in strong relief the underlying physical and mathematical
nuances. Although this kind of notation is standard in advanced mathematics
textbooks, at first glance some of it might be unfamiliar to those readers
primarily used to the typical textbooks in relativistic quantum physics. To
assist that kind of reader, an extensive list of symbols has been provided.
With its help, a more careful inspection should reveal that the additional
information contained in the mathematically precise notation used in those
formulae, as compared with their often sketchier formulation that is customary
in typical textbooks on relativistic quantum mechanics and quantum field theory,
clarifies not only their mathematical but also their physical meaning. In
general, considerable emphasis is set on the kind of precise notation and
rigorous argument that is long overdue in some of the fields with which this
monograph is dealing.
Great emphasis is set also on the principle of mathematical
beauty and simplicity. This principle was championed by Einstein throughout
his life. As a methodological guide, it had its advocates also in Poincaré
and Weyl, but its strongest champion was again Dirac. Indeed, as stated
in a recent scientific biography of Dirac: “For Dirac this principle of
mathematical beauty was partly a methodological moral and partly a postulate
about nature's qualities. It was clearly inspired by the theory of relativity,
the general theory in particular, and also by the development of quantum
mechanics.” (Kragh, 1990, p. 277.)
These twin methodological principles of mathematical
soundness and beauty, advocated by Dirac, have extensively guided the approach
to quantum general relativity presented in this monograph. Of course, the
application of the second of these principles obviously required a discriminating
attitude, geared to taking into account the predictions of those conventional
quantum field theoretical models that have received indubitable experimental
support. With that stipulation in mind, it became possible to apply Dirac's
principle of mathematical beauty and simplicity to narrow down to unique
choices the wide range of alternatives to the various key features in the
formulation of quantum general relativity, which in previous publications
of this author were presented from a perspective that emphasized mathematical
generality. In particular, the adoption of this methodological principle
influenced the choice of quantum-geometric propagators, and the formulation
of quantum-geometric causality. This has a profound effect on the mathematical
form of the implementation of the concept of quantum-geometric propagation
of fields, which is central to the present formulation of quantum general
relativity. As a consequence, in the special relativistic context the formal
agreement with conventional quantum field theory becomes very straightforward
at the perturbative level, since it involves primarily the removal of finite
normalization factors that become infinite in the sharp-point limit in which
the adopted fundamental length l is allowed to tend to zero.
For the benefit of the reader who desires a concise
overview of the final outcome of the present formulation of quantum general
relativity, the principal ideas of the framework for geometric quantum gravity
that emerges by the end of the last chapter of the present monograph can
be summarized as follows:
1. The central concept of this framework is that of
quantum frame and superframe bundle. The frames and superframes in such
bundles take over the role played by the local Lorentz frames of classical
general relativity. Due to their informational completeness, these local
quantum frames and superlocal quantum superframes are capable of taking
over the role played by complete sets of (compatible) observables in orthodox
quantum mechanics.
2. In the quantum geometric regime all the counterparts
of the constraints that emerge from classical Yang-Mills field theories
and from classical general relativity are geometrized by means of gauge
groups, which inject them into Maurer-Cartan structural equations that govern
the construction of connection forms in principal quantum frame and superframe
bundles.
3. The elements of all massive quantum frame bundles
possess an operational interpretation at the microlevel, obtained by replacing
the test particles of classical relativity with quantum test bodies (i.e.,
geometro-stochastic excitons).
4. The quantum-geometric evolution of fields in mutual
local interactions within a quantum spacetime supermanifold is described
by a perpetually ongoing process of creation and annihilation of the geometro-stochastic
excitons associated with them.
5. In the resulting framework for geometric quantum
gravity, matter and radiation quantum fields in free fall propagate by parallel
transport along stochastic paths; those paths are the limits of broken paths
corresponding to time-ordered segmentations of a quantum gravitational spacetime
supermanifold.
6. The causal time-ordering in a quantum gravitational
supermanifold is intrinsic, since it is implicit in the (local) proper time
marked by the massive constituents of the quantum frames. This proper
time emerges from an adaptation to geometro-stochastic excitons of de Broglie's
(1923, 1924) original idea that a natural time is inscribed in all
matter in existence, since each elementary quantum object of rest-mass m
can be viewed as a natural clock with mean period T = 2(pi)/m
in Planck natural units.
7. The quantum general relativistic covariance principle
is embedded in a quantum gravitational supergroup, which incorporates the
semidirect product of two types of subgroups of gauge transformations: one
type pertains to the metric equivalence classes of quantum superframes that
are interrelated by supergauge transformations originating from superoperator
representations of the diffeomorphism group, and gives rise to equivalence
classes of mean Lorentzian metrics; the other type describes changes of
quantum superframes within the equivalence class for each of these mean
metrics, and is provided by superunitary representations of the Poincaré
group.
8. The strong equivalence principle is embedded in the
above type of Poincaré gauge invariance, as well as in the mode of
the quantum-geometric propagation of quantum fields, which takes place by
parallel transport along the arcs of the broken paths that are the horizontal
lifts of geodesics of the Levi-Civita connection in each of the metric equivalence
classes. These geodesics lie in the base manifold resulting from the natural
fibration of the quantum spacetime supermanifold into superfibres of quantum
superframes lying above the various points in that base manifold – which
can be viewed as a “classical” spacetime manifold that labels the mean stochastic
locations of geometro-stochastic excitons and of the fields producing or
annihilating them.
9. The quantum superposition principle is embedded in
the path integrals that describe the outcome of the quantum-geometric propagation
of all quantum superfields along all possible stochastic causal paths. The
causality of these paths is embodied in the above-described features of
quantum propagation, which reflect a microcausal time-ordering and an adaptation
of the geodesic postulate to quantum general relativistic propagation. Their
stochasticity is not due to the presence of probability measures over paths,
as in classical stochastic processes; rather, it is due to the fact that,
in constructing the limits leading to the quantum-geometric propagators
of quantum fields and superfields, the superpositions of propagators for
parallel transport are taken, with purely geometric weighting factors, over
all possible broken causal paths consisting of geodesic arcs.
The author would like to thank the organizers of the
XIIIth Workshop on Geometric Methods in Physics, in Bialowieza, Poland,
for the opportunity to present the main ideas in this monograph for professional
consideration and discussion prior to its publication. He also thanks Professors
J. A. Brooke and W. Drechsler for discussions and correspondence on some
of the topics treated in it. Furthermore, he would like to express his appreciation
to M. Clayton, E. Reyes and M. Sangani-Monfared for comments and valuable
assistance with the proof-reading of parts of the manuscript. The comprehensive
help with its over-all preparation which he re-ceived from his wife, Margaret
R. Prugovecki, and the moral support she extended to him throughout the
many years of research leading to the results presented in this monograph,
were truly invaluable.
Principles of Quantum
General Relativity
by Eduard Prugovecki
The present monograph explains, in a manner accessible to readers with
a knowledge of general relativity and quantum theory acquired from introductory
graduate-level courses, the principles of a quantum-geometric unification
of these two disciplines. These principles have their roots in the writings
of Einstein, Bohr, Born, Dirac, Heisenberg, and other founders of these
disciplines. Although their formulation and development has sometimes followed
a tortuous path, the origins of this path are clearly visible in those writings,
as shown in historical notes provided in appropriate contexts. By taking
advantage of recent advances in such areas of modern theoretical physics
and mathematics as the fibre bundle theory, gauge fields, group-theoretical
methods, coherent states, etc., these principles can be incorporated into
quantum geometries that provide the foundations of an extrapolation of general
relativity to the quantum regime. The main goal of this volume is to present
the resulting geometry of quantum general relativity in a manner which sets
emphasis on fundamental physical ideas and their precise mathematical implementation,
and which conforms to the methodologies championed in classical general
relativity and in relativistic quantum theory by Einstein and by Dirac,
as they founded these respective disciplines.