## Abstract

We propose an operator constraint equation for the wavefunction of the Universe that admits genuine evolution. While the corresponding classical theory is equivalent to the canonical decomposition of General Relativity, the quantum theory contains an evolution equation distinct from standard Wheeler–DeWitt cosmology. Furthermore, the local symmetry principle—and corresponding observables—of the theory have a direct interpretation in terms of a conventional gauge theory, where the gauge symmetry group is that of spatial conformal diffeomorphisms (that preserve the spatial volume of the Universe). The global evolution is in terms of an arbitrary parameter that serves only as an unobservable label for successive states of the Universe. Our proposal follows unambiguously from a suggestion of York whereby the independently specifiable initial data in the action principle of General Relativity is given by a conformal geometry and the spatial average of the York time on the spacelike hypersurfaces that bound the variation. Remarkably, such a variational principle uniquely selects the form of the constraints of the theory so that we can establish a precise notion of both symmetry and evolution in quantum gravity.

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## Notes

The first recorded discussion of the problem of time was at the Stevens Relativity Meetings in 1958. There Dirac gave a talk on quantizing GR. The Bergmann letter follows on from that discussion. Thanks to Dean Rickles for this information.

The list of references given here is not intended to be comprehensive. Readers are encouraged to consult the classic review [3] and a modern update [4] for a more detailed and wide–ranging catalogue of approaches to the

*Problem of Time*. For our purposes, we feel that it is best to treat the ‘Problem of Time’ as an entirely separate issue to the ‘Problem of the Arrow of Time’. However, for an interesting exploration of a possible connections between the two problems see [8–10].Of course, one could alternatively formulate one’s methodology solely in terms of these new structures but we will not do so here.

Because we are speaking about the properties of the action evaluated along curves on \({\mathcal {A}}\), which are insensitive to the internal structure of individual points on \({\mathcal {A}}\), these symmetries correspond to traditional

*global*symmetries.Note that this is a

*stronger*requirement then requiring that the variation of that degree of freedom on the endpoints of the variation is free, as suggested in [33].This list is not intend to be exhaustive, but rather cover all cases of physical interest.

In Sect. 5, we illustrate how General Relativity can be cast into a gauge theory of this kind where the symmetry group is that of the volume preserving conformal diffeomorphisms.

A related argument leading to the same conclusion is given in [42].

If \({\mathcal {A}}_\text {e}\) is thought of as a fiber bundle, where the \(\theta ^\alpha \)-directions are fibers over \({\mathcal {A}}\), then \(D\) defines a section of this bundle. We will see that, upon phase space reduction with free variation, this reduces to a covariant derivative on \({\mathcal {A}}\), as in Yang–Mills gauge theory.

In other gauges, the explicit expression for the conserved charged can be obtained locally on phase space by isolating the canonical variable conjugate to the constraint \(C^\alpha = 0\) using local Darboux coordinates. The different expressions for the conserved charge in different gauges are related by the canonical transformation generated by (19). They represent non-standard representations of the original ontology of the theory.

This additional requirement refines the motivations given in [33] for what was referred to as a “free

*endpoint*variation”, despite being a more restrictive requirement than standard free endpoint variation. This explains our terminology “condition for free variation” since our “free variation” is indeed distinct from standard “free*endpoint*variation”. For more details surrounding these issues, see [4, 53, 63].Here we again note that, with our prescription, the Dirac quantization methodology can be substituted for a number of more rigorous modern approaches to the quantization of gauge theories provided our classification of degrees of freedom and constraints is preserved. See §3.2.2 for references.

This is, for example, why ghost fields have no external legs.

Here we note, once more, that there are well known formal issues with Dirac quantization that render such a Hilbert space—strictly speaking—not well defined. In a fully rigorous application of relational quantisation to a reparametrization invariant system more powerful techniques, such as group averaging [66, 67], would need to be used. Such modifications would not imply any difference in the basic structure of our arguments.

A full treatment of spatial diffeomorphisms would require large diffeomorphisms, adding complications that don’t affect our main argument. For simplicity, we will restrict our discussion to infinitesimal diffeomorphisms.

Note that, in the quantum theory, taking \(P\), which takes values in \(\mathbb R\), as the independently specifiable degree of freedom instead of \(V_g\), which takes values in \(\mathbb R^+\), avoids the issue of having to deal with an operator valued in \(\mathbb R^+\).

Note that, by adding a boundary term to the extended ADM action, we can express \(\bar{\pi }_0\) as the configuration variable with \(\bar{\phi }_0\) as its conjugate momentum. This justifies \(\bar{\phi }_0\approx 0\) as the free variation condition of \(P\).

In the case with no cosmological constant and Higgs mass, this theory can be shown to obey a form of

*dynamical similarity*, as shown in [82], which may address some of these problems.Note: this can change the uniqueness properties of these solutions [83] but not the existence of solutions in the spatially closed case.

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## Acknowledgments

We would like to thank Edward Anderson, Julian Barbour, Brian Pitts, Henrique Gomes, Tom Pashby, Hans Westman and Ken Wharton for comments on the draft, and Igor Khavkine for useful discussions. SG would like to acknowledge support from and NSERC PDF grant, for travel support from Renate Loll, and for the hospitality of Utrecht and Radboud Universities. KT would like to acknowledge support from the Alexander von Humboldt foundation and the Munich Center for Mathematical Philosophy.

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Gryb, S., Thébaault, K. Symmetry and Evolution in Quantum Gravity.
*Found Phys* **44**, 305–348 (2014). https://doi.org/10.1007/s10701-014-9789-x

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DOI: https://doi.org/10.1007/s10701-014-9789-x

### Keywords

- Gauge symmetry
- Quantum gravity
- Problem of time
- Canonical quantization
- Variational principles
- quantum cosmology