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Weyl Symmetry in Stochastic Quantum Gravity

We propose that the gauge principle of d-dimensional Euclidean quantum gravity is Weyl invariance in its stochastic (d+1)-dimensional bulk. Observables are defined as depending only on conformal classes of d-dimensional metrics. We work with the second order stochastic quantization of Einstein equations in a (d+1)-dimensional bulk. There, the evolution is governed by the stochastic time, which foliates the bulk into Euclidean d-dimensional leaves. The internal metric of each leaf can be parametrized by its unimodular part and conformal factor. Additional bulk metric components are the ADM stochastic lapse and a stochastic shift. The Langevin equation determines the acceleration of the leaf as the sum of a quantum noise, a drift force proportional to Einstein equations and a viscous first order force. Using Weyl covariant decomposition, this Langevin equation splits into irreducible stochastic equations, one for the unimodular part of the metric and one for its conformal factor. For the first order Langevin equation, the unphysical fields are the conformal factor, which is a classical spectator, and the stochastic lapse and shift. These fields can be gauge-fixed in a BRST invariant way in function of the initial data of the process. One gets observables that are covariant with respect to internal reparametrization in each leaf, and invariant under arbitrary reparametrization of the stochastic time. The interpretation of physical observable at finite stochastic time is encoded in a transitory (d+1)-dimensional phase where the Lorentz time cannot be defined. The latter emerges in the infinite stochastic time limit by an abrupt phase transition from quantum to classical gravity.