Principles of Quantum General Relativity

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


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
Chapter 6. Massive Quantum-Geometric Fermion Fields    212
    6.1.    Fock-Dirac Bundles and Bispinor Quantum Frame Fields      212    
    6.2.     Standard Berezin-Dirac Quantum Superfibres    216    
    6.3.     Quantum Berezin-Dirac Superfibre Bundles    221    
    6.4.    Parallel Transport in Fock-Dirac Bundles    224    
    6.5.    Geometrically Localized Fermion Field Supermodes    227
    6.6.    Quantum-Geometric Fermion Field Propagation    230
Chapter 7. Massless Quantum-Geometric Gauge Fields    233
    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        


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.

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