  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.
