For my record, I list here some interesting papers by Salisbury et al. which cover fundamental questions of relevance on, should I say, “pre-” quantum gravity matters.
The lines of research are summarized as follows:
- preservation of general coordinate transformation and additional gauge symmetries in the transition from a Lagrangian to a Hamiltonian description;
- the nature of observables in classical general relativity, and their potential usefulness in the construction of an eventual quantum theory of gravity;
- construction of diffeomorphism invariants (observables) in general relativity;
- history of constrained hamiltonians.
The papers of relevance are the following:
- Realization in phase space of general coordinate transformations [Phys. Rev. D 27, 740, 1983];
- Gauge transformations in the Lagrangian and Hamiltonian formalisms of generally covariant theories [gr-qc/9612037 = PRDvol55,no2,658,1997]: establishes the general framework in which gauge variables are retained as canonical variables;
- Reduced phase space: quotienting procedure for gauge theories [math-ph/9811029]: describes an alternative algorithm to the Dirac-Bergmann constraint procedure for constructing a self-consistent Hamiltonian model;
- Gauge group and reality conditions in Ashtekar’s complex formulation of canonical gravity [gr-qc/9912085]: discusses Ashtekar’s complex connection approach to gravity;
- Gauge Transformations in Einstein-Yang-Mills Theories [gr-qc/9912086]: discusses gauge symmetries in Einstein-Yang-Mills models;
- The gauge group in the real triad formulation of general relativity [gr-qc/9912087]: discusses a real triad version of canonical gravity;
- Gauge symmetries in Ashtekar’s formulation of general relativity [gr-qc/0004013]: proposes a gauge averaging procedure modeled after an approach of Rovelli’s, though retaining gauge variables and recognizing the essential distinction between time evolution and realizeable canonical gauge symmetries;
- Quantum general invariance and loop gravity [gr-qc/0105097]: preliminary exploration into the construction of diffeomorphism invariants using dynamical field-dependent finite gauge transformations;
- Quantum General Invariance [Proceedings of the Ninth Marcel Grossmann Meeting held in Rome in 2000 ("Quantum general invariance", in Proceedings of the Ninth Marcel Grossmann Meeting, edited by V.G. Gurzadyan, R. T. Jantzen and R. Ruffini, (World Scientific, New Jersey, 2002), 1300-1301)]: continued the exploration of finite gauge transformations;
- The issue of time in generally covariant theories and the Komar-Bergmann approach to observables in general relativity [ gr-qc/0503013 = Phys.Rev. D71 (2005) 124012]: constructs local invariants through the use of intrinsic coordinates. This can be accomplished in the canonical framework in general relativity using Weyl curvature scalars, as was first suggested by Komar and Bergmann. One essential new observation in this work is the recognition that gauge variables become functionals of the non-gauge variables, and consequently in the quantum theory they become subject to fluctuations. In particular, in canonical quantum gravity the light cone is itself fluctuating [the authors supposedly show that there is no conceptual problem for the canonical formulation of generally covariant theories because the mathematical identification of the Hamiltonian as a gauge generator is erroneous (resolution of the time evolution versus gauge problem)];
- Rosenfeld, Bergmann, Dirac and the Invention of Constrained Hamiltonian Dynamics [physics/0701299]: In a paper appearing in Annalen der Physik in 1930 Leon Rosenfeld invented the first procedure for producing Hamiltonian constraints. He displayed and correctly distinguished the vanishing Hamiltonian generator of time evolution, and the vanishing generator of gauge transformations for general relativity with Dirac electron and electrodynamic field sources. Though he did not do so, had he chosen one of his tetrad fields to be normal to his spacetime foliation, he would have anticipated by almost thirty years the general relativisitic Hamiltonian first published by Paul Dirac.
Links will be added later [Edit: almost all included now]. It would be interesting to follow these matters in light of recent advances in canonical quantum gravity. Marcus over at Physics Forums have prepared a selection (actually, an invitation to a poll) of papers published in the arxiv in 2008 in canonical quantum gravity.