## 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.

This is a preview of subscription content, access via your institution.

## Notes

- 1.
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.

- 2.
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]. - 3.
- 4.
- 5.
Of course, one could alternatively formulate one’s methodology solely in terms of these new structures but we will not do so here.

- 6.
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. - 7.
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]. - 8.
This list is not intend to be exhaustive, but rather cover all cases of physical interest.

- 9.
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.

- 10.
A related argument leading to the same conclusion is given in [42].

- 11.
- 12.
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.

- 13.
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.

- 14.
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]. - 15.
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.

- 16.
This is, for example, why ghost fields have no external legs.

- 17.
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.

- 18.
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.

- 19.
- 20.
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 \(\mathbb R^+\).

- 21.
- 22.
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 \(P\).

- 23.
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. - 24.
Note: this can change the uniqueness properties of these solutions [83] but not the existence of solutions in the spatially closed case.

## References

- 1.
Bergman, P.G.: Letter to P. A. M. Dirac, (Oct 9th 1959)

- 2.
DeWitt, B.: Quantum theory of gravity. I. The canonical theory. Phys. Rev.

**160**, 1113–1148 (1967) - 3.
Isham, C.: Canonical quantum gravity and the problem of time. Arxiv preprint gr-qc (1992). http://arxiv.org/abs/grqc/9210011

- 4.
Anderson, E.: The problem of time in quantum gravity. Ann. Phys.

**524**, 757–786 (2012) - 5.
Dirac, P.A.M.: The theory of gravitation in hamiltonian form. Proc. R Soc. Lond. Ser. A Math. Phys. Sci.

**246**, 333–343 (1958) - 6.
Arnowitt, R., Deser, S., Misner, C.W.: Canonical variables for general relativity. Phys. Rev.

**117**, 1595–1602 (1960) - 7.
Moncrief, V.: How solvable is (2 + 1)-dimensional Einstein gravity? J. Math. Phys.

**31**, 2978 (1990) - 8.
Kiefer, C., Zeh, H.: Arrow of time in a recollapsing quantum universe. Phys. Rev. D

**51**(8), 4145 (1995) - 9.
Kiefer, C.: Quantum Gravity. Oxford University Press, Oxford (2007)

- 10.
Kiefer, C.: Can the arrow of time be understood from quantum cosmology? In: The Arrows of Time, pp. 191–203. Springer, Heidelberg (2012)

- 11.
Bombelli, L., Lee, J., Meyer, D., Sorkin, R.D.: Space-time as a causal set. Phys. Rev. Lett.

**59**(5), 521–524 (1987) - 12.
Dowker, F.: Causal sets and the deep structure of spacetime. In: Ashtekar, A. (ed.) 100 Years of Relativity, Space-Time Structure: Einstein and Beyond, pp. 445–464. World Press Scientific, Singapore (2005)

- 13.
Henson, J.: The causal set approach to quantum gravity. arXiv, preprint gr-qc/0601121 (2006)

- 14.
Isham, C., Butterfield, J.: Some possible roles for topos theory in quantum theory and quantum gravity. Found. Phys.

**30**(10), 1707–1735 (2000) - 15.
Isham, C.: Some reflections on the status of conventional quantum theory when applied to quantum gravity. In: The Future of the Theoretical Physics and Cosmology (Cambridge, 2002): Celebrating Stephen Hawking’s 60th Birthday, pp. 384–408 (2002)

- 16.
Hardy, L.: Towards quantum gravity: a framework for probabilistic theories with non-fixed causal structure. J. Phys. A Math. Theor.

**40**(12), 3081 (2007) - 17.
Rovelli, C.: Quantum mechanics without time: a model. Phys. Rev. D

**42**, 2638–2646 (1990) - 18.
Rovelli, C.: Time in quantum gravity: an hypothesis. Phys. Rev. D

**43**, 442 (1991) - 19.
Rovelli, C.: Partial observables. Phys. Rev. D

**65**, 124013 (2002) - 20.
Dittrich, B.: Partial and complete observables for canonical general relativity. Class. Quant. Gravity

**23**, 6155 (2006) - 21.
Dittrich, B.: Partial and complete observables for hamiltonian constrained systems. General Relat. Gravit.

**39**, 1891 (2007) - 22.
Dittrich, B., Thiemann, T.: Testing the master constraint programme for loop quantum gravity: I. General framework. Class. Quant. Gravity

**23**, 1025–1065 (2006) - 23.
Thiemann, T.: The phoenix project: master constraint programme for loop quantum gravity. Class. Quant. Gravity

**23**, 2211 (2006) - 24.
Thiemann, T.: Modern Canonical Quantum General Relativity. Cambridge University Press, Cambridge (2007)

- 25.
Husain, V., Pawlowski, T.: Time and a physical Hamiltonian for quantum gravity. Phys. Rev. Lett.

**108**, 141301 (2012). arXiv:1108.1145[gr-qc] - 26.
Giesel, K., Thiemann, T.: Scalar material reference systems and loop quantum gravity. arXiv:1206.3807[gr-qc]

- 27.
Brown, J.D., Kuchar, K.V.: Dust as a standard of space and time in canonical quantum gravity. Phys. Rev. D 51 (1995) 5600–5629. arXiv:gr-qc/9409001[gr-qc]

- 28.
Isham, C., Kuchar, K.: Representations of space-time diffeomorphisms 2. Canonical geometrodynamcis. Ann. Phys.

**164**, 316 (1985) - 29.
Loll, R.: Discrete lorentzian quantum gravity. Nucl. Phys. B Proc. Suppl.

**94**(1), 96–107 (2001) - 30.
Ambjørn, J., Jurkiewicz, J., Loll, R.: Dynamically triangulating lorentzian quantum gravity. Nucl. Phys. B

**610**(1), 347–382 (2001) - 31.
York, J.: Boundary terms in the action principles of general relativity. Found. Phys.

**16**(3), 249–257 (1986) - 32.
Gryb, S., Thébault, K.: The role of time in relational quantum theories. Found. Phys.

**42**, 1210–1238 (2012) - 33.
Anderson, E., Barbour, J., Foster, B.Z., Kelleher, B., O’Murchadha, N.: The physical gravitational degrees of freedom. Class. Quant. Gravity

**22**, 1795–1802 (2005). arXiv:gr-qc/0407104 - 34.
Gomes, H., Gryb, S., Koslowski, T.: Einstein gravity as a 3D conformally invariant theory. Class. Quant. Gravity

**28**, 045005 (2011). arXiv:1010.2481[gr-qc] - 35.
Brown, J.D., York, J.W.J.: Jacobi’s action and the recovery of time in general relativity. Phys. Rev. D

**40**, 3312–3318 (1989) - 36.
Henneaux, M., Teitelboim, C.: The cosmological constant and general covariance. Phys. Lett. B

**222**, 195–199 (1989) - 37.
Unruh, W.G.: A unimodular theory of canonical quantum gravity. Phys. Rev. D

**40**, 1048 (1989) - 38.
Unruh, W.G., Wald, R.M.: Time and the interpretation of canonical quantum gravity. Phys. Rev.

**D40**, 2598 (1989) - 39.
Smolin, L.: The quantization of unimodular gravity and the cosmological constant problem. arXiv:0904.4841[hep-th]

- 40.
Kuchar, K.V.: Does an unspecified cosmological constant solve the problem of time in quantum gravity? Phys. Rev. D

**43**, 3332–3344 (1991) - 41.
Barbour, J.B.: The Timelessness of quantum gravity. 1: The evidence from the classical theory. Class. Quant. Gravity

**11**, 2853–2873 (1994) - 42.
Barbour, J., Foster, B.Z.: Constraints and gauge transformations: Dirac’s theorem is not always valid. (Aug., 2008). arXiv:0808.1223[gr-qc]

- 43.
Pons, J., Salisbury, D., Sundermeyer, K.A.: Observables in classical canonical gravity: folklore demystified. J. Phys. A Math. General

**222**, 12018 (2010) - 44.
Pitts, J.B.: Change in hamiltonian general relativity from the lack of a time-like killing vector field. (Oct., 2013). http://philsci-archive.pitt.edu/10094/

- 45.
Kuchař, K.: The Problem of Time in Quantum Geometrodynamics, pp. 169–195. Oxford University Press, New York (1999)

- 46.
Kuchar̆, K.: The problem of time in canonical quantization of relativistic systems. In: Ashtekar, A., Stachel, J. (eds.) Conceptual Problems of Quantum Gravity, p. 141. Birkhauser, Boston (1991)

- 47.
Kuchař, K.V.: Time and interpretations of quantum gravity. In: Kunstatter G., Vincent D., Williams J. (eds.) Proceedings of the Fourth Canadian Conference on General Relativity and Relativistic Astrophysics, held 16–18 May, 1991 at University of Winnipeg, pp. 211–314. World Scientific, Singapore (1992)

- 48.
Fatibene, L., Francaviglia, M., Mercadante, S.: Noether symmetries and covariant conservation laws in classical, relativistic and quantum physics. arXiv:1001.2886[gr-qc]

- 49.
Gomes, H., Koslowski, T.: The link between general relativity and shape dynamics. Class. Quant. Gravity

**29**, 075009 (2012). arXiv:1101.5974[gr-qc] - 50.
Poincaré, H.: Science et Hypothèse. Ernest Flammarion, Paris (1902)

- 51.
Barbour, J.B., Bertotti, B.: Mach’s principle and the structure of dynamical theories. Proc. R. Soc. A

**382**(1783), 295–306 (1982) - 52.
Barbour, J.: Dynamics of pure shape, relativity and the problem of time. In: Decoherence and Entropy in Complex Systems (Proceedings of the Conference DICE, Piombino 2002, Elze H.-T. (ed.)). Springer Lecture Notes in Physics. Springer, New York (2003)

- 53.
Gryb, S.B.: A definition of background Independence. Class. Quant. Gravity.

**27**, 215018 (2010). arXiv:1003.1973[gr-qc] - 54.
Dirac, P.A.M.: Lectures on Quantum Mechanics. Dover Publications, Yeshivea University, New York (1964)

- 55.
Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems. University Press, Princeton (1992)

- 56.
Vytheeswaran, A.: Gauge unfixing in second class constrained systems. Ann. Phys.

**236**, 297–324 (1994) - 57.
York, J.J.W.: Gravitational degrees of freedom and the initial-value problem. Phys. Rev. Lett.

**26**, 1656–1658 (1971) - 58.
York, J.J.W.: Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial value problem of general relativity. J. Math. Phys.

**14**, 456–464 (1973) - 59.
York, J.J.W.: Role of conformal three geometry in the dynamics of gravitation. Phys. Rev. Lett.

**28**, 1082–1085 (1972) - 60.
Cook, G.B.: Initial data for numerical relativity. Living Rev. Relativ. 3(5), (2000). http://www.livingreviews.org/lrr-2000-5

- 61.
Mukhanov, V.F., Feldman, H., Brandenberger, R.H.: Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions. Phys. Rept.

**215**, 203–333 (1992) - 62.
Barbour, J.: Shape dynamics. An introduction. arXiv:1105.0183

- 63.
Barbour, J.: Scale-invariant gravity: particle dynamics. Class. Quant. Gravity

**20**, 1543–1570 (2003). arXiv:gr-qc/0211021 - 64.
Isham, C., Kakas, A.: A group theoretical approach to the canonical quantisation of gravity. I. Construction of the canonical group. Class. Quant. Gravity

**1**(6), 621 (1984) - 65.
Isham, C., Kakas, A.: A group theoretical approach to the canonical quantisation of gravity. II. Unitary representations of the canonical group. Class. Quant. Gravity

**1**(6), 633 (1984) - 66.
Giulini, D., Marolf, D.: On the generality of refined algebraic quantization. Class. Quant. Gravity

**16**, 2479–2488 (1999) - 67.
Giulini, D., Marolf, D.: A uniqueness theorem for constraint quantization. Class. Quant. Gravity

**16**, 2489–2505 (1999) - 68.
Kuchař, K.: General relativity: dynamics without symmetry. J. Math. Phys.

**22**, 2640 (1981) - 69.
Kuchar, K.V.: Canonical quantum gravity. arXiv:gr-qc/9304012[gr-qc]

- 70.
Torre, C.: Gravitational observables and local symmetries. Phys. Rev. D Part. Fields

**48**(6), R2373 (1993) - 71.
Hájícek, P.: Choice of gauge in quantum gravity. Nucl. Phys. B Proc. Suppl.

**80**, 1213 (2000) - 72.
Anderson, E.: Relational quadrilateralland. II. Analogues of isospin and hypercharge. arXiv:1202.4187[gr-qc]

- 73.
Gomes, H.deA.: Gauge theory in Riem: classical. Accepted to J. Math. Phys. arXiv:0807.4405[gr-qc]

- 74.
Arnowitt, R.L., Deser, S., Misner, C.W.: The dynamics of general relativity. In Witten, L. (ed.) Gravitation: An Introduction to Current Research, chap. 7, pp. 227–265. arXiv:gr-qc/0405109.

- 75.
Teitelboim, C.: How commutators of constraints reflect the space-time structure. Ann. Phys.

**79**, 542–557 (1973) - 76.
Thiemann, T.: Modern canonical quantum general relativity. Cambridge University Press, Cambridge (2007). arXiv:gr-qc/0110034[gr-qc]

- 77.
Isham, C.J.: Canonical quantum gravity and the problem of time. arXiv:gr-qc/9210011

- 78.
Dirac, P.A.M.: Fixation of coordinates in the Hamiltonian theory of gravitation. Phys. Rev.

**114**, 924–930 (1959) - 79.
Gomes, H.: The coupling of shape dynamics to matter. J. Phys. Conf. Ser.

**360**, 012058 (2012). arXiv:1112.0374[gr-qc] - 80.
Gomes, H.: Breaking the uniqueness of the shape dynamics Hamiltonian. arXiv:1201.3969[gr-qc]

- 81.
O’Murchadha, N., York, J.J.W.: Existence and uniqueness of solutions of the hamiltonian constraint of general relativity on compact manifolds. J. Math. Phys.

**4**, 1551–1557 (1973) - 82.
Barbour, J., Koslowski, T., Mercati, F.: The solution to the problem of time in shape dynamics. arXiv:1302.6264[gr-qc]

- 83.
Gomes, H., Koslowski, T.: Coupling shape dynamics to matter gives spacetime. General Relat. Gravity

**44**(2012) 1539–1553. arXiv:1110.3837[gr-qc] - 84.
Barbour, J., Lostaglio, M., Mercati, F.: Scale anomaly as the origin of time. arXiv:1301.6173[gr-qc]

- 85.
Strominger, A.: Inflation and the dS/CFT correspondence. JHEP

**0111**, 049 (2001). arXiv:hep-th/0110087[hep-th] - 86.
McFadden, P., Skenderis, K.: The holographic universe. J. Phys. Conf. Ser.

**222**, 012007 (2010). arXiv:1001.2007[hep-th]

## 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.

## Author information

### Affiliations

### Corresponding author

## Rights and permissions

## About this article

### Cite this article

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

Received:

Accepted:

Published:

Issue Date:

### Keywords

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