Foundations of Physics

, Volume 44, Issue 3, pp 305–348 | Cite as

Symmetry and Evolution in Quantum Gravity

  • Sean Gryb
  • Karim Thébaault


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.


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



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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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsUtrecht UniversityUtrechtThe Netherlands
  2. 2.Institute for Mathematics, Astrophysics and Particle PhysicsRadboud University NijmegenNijmegenThe Netherlands
  3. 3.Munich Center for Mathematical PhilosophyLudwig Maximilians Universität MunichMunichGermany

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