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We show that proper time, when defined in the quantum theory of 2D gravity, becomes identical to the stochastic time associated with the stochastic quantization of space. This observation was first made by Kawai and collaborators in the context of 2D Euclidean quantum gravity, but the relation is even simpler and more transparent in the context of 2D gravity formulated in the framework of CDT (causal dynamical triangulations).
Since time plays such a prominent role in ordinary flat space quantum field theory defined by a Hamiltonian, it is of interest to study the role of time even in toy models of quantum gravity where the role of time is much more enigmatic. The model we will describe in this chapter is the so-called causal dynamical triangulation (CDT) model of quantum gravity. It starts by providing an ultraviolet regularization in the form of a lattice theory, the lattice link length being the (diffeomorphism-invariant) UV cut-off. In addition, the lattice respects causality. It is formulated in the spirit of asymptotic safety, where it is assumed that quantum gravity is described entirely by “conventional” quantum field theory, in this case by approaching a non-trivial fixed point [1, 2]. It is formulated in space-times with Lorentzian signature, but the regularized space-times which are used in the path integral defining the theory allow a rotation to Euclidean space-time. The action used is the Regge action for the piecewise linear geometry.
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