We propose a solution to the problem of time for systems with a single global Hamiltonian constraint. Our solution stems from the observation that, for these theories, conventional gauge theory methods fail to capture the full classical dynamics of the system and must therefore be deemed inappropriate. We propose a new strategy for consistently quantizing systems with a relational notion of time that does capture the full classical dynamics of the system and allows for evolution parametrized by an equitable internal clock. This proposal contains the minimal temporal structure necessary to retain the ordering of events required to describe classical evolution. In the context of shape dynamics (an equivalent formulation of general relativity that is locally scale invariant and free of the local problem of time) our proposal can be shown to constitute a natural methodology for describing dynamical evolution in quantum gravity and to lead to a quantum theory analogous to the Dirac quantization of unimodular gravity.
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For an analysis of approaches broadly constituted along these lines, we refer the reader to the recent reviews of Anderson [13, 14]. For a comprehensive modern text on canonical quantum gravity see . The canonical formulation of general relativity was originally developed by Dirac  and Arnowitt, Deser and Misner (‘ADM’) [17, 18].
By conventional gauge theory methods we will understand: (i) Dirac quantization, (ii) reduced phase space quantization, and (iii) Faddeev–Popov gauge fixing, as outline in Sect. 2.
N.B. What we are proposing is only claimed to be a consistent methodology for the quantization of what is, under our definition, a relational theory. There are, of course, alternative types of quantization, just as there are alternative definitions of relationalism.
We are indebted to Tim Koslowski for helping to clarify this key point.
Here, and below, by equitable we will mean dependent upon the contributions of all the dynamical variables/subsystems of the system in question.
We are again indebted to Tim Koslowski for his valuable insight in regards to this extension procedure.
Second class constraints can always be made first class by a suitable redefinition of Ω following Dirac .
There is also a global requirement that Σ gf only intersects the gauge orbits once. We will assume that this requirement can be satisfied.
Crucially, this is the step that can be performed in shape dynamics that is highly non-trivial in the ADM formulation of general relativity.
This distinction constitutes a temporal orientation rather than a temporal direction, which would imply an arrow of time.
See [28, p. 280] for a analogous case.
In this section we will sometimes write the coordinates of p using lower case indices for convenience.
For an elegant treatment of both Jacobi and parameterized particle models the reader is referred to .
In particular, in GR it is still the case that standard gauge theory methods lead to a classical theory without even minimal temporal structure. See §3 of .
For simplicity, we will assume that Σ is compact without boundary.
The solutions must admit at least one foliation where the spatial Cauchy hypersurfaces have constant mean (extrinsic) curvature—i.e. be ‘CMC foliable.’ One may argue that all physically reasonable solutions will satisfy this condition which, in any case, is not dramatically stronger than the global hyperbolicity assumption that is fundamental to ADM GR.
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We would like to thank Julian Barbour for helpful comments and criticisms and Lee Smolin for encouraging us to look more closely at the Hamilton–Jacobi formalism. We would also like to thank the participants of the 2nd PIAF conference in Brisbane (in particular Hans Westman) for their inspiring questions on the problem of time. Finally, we would like to extend a special thanks to Tim Koslowski for many useful discussions and for making valuable contributions to our overall understanding of this work. Research at the Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MEDT.
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Gryb, S., Thébault, K. The Role of Time in Relational Quantum Theories. Found Phys 42, 1210–1238 (2012). https://doi.org/10.1007/s10701-012-9665-5
- Problem of time
- Quantum gravity
- Constraint quantization
- Reparametrization invariance
- Internal time