Dynamically triangulating Lorentzian quantum gravity

Abstract

Fruitful ideas on how to quantize gravity are few and far between. In this paper, we give a complete description of a recently introduced non-perturbative gravitational path integral whose continuum limit has already been investigated extensively in d < 4, with promising results. It is based on a simplicial regularization of Lorentzian space-times and, most importantly, possesses a well-defined, non-perturbative Wick rotation. We present a detailed analysis of the geometric and mathematical properties of the discretized model in d = 3, 4. This includes a derivation of Lorentzian simplicial manifold constraints, the gravitational actions and their Wick rotation. We define a transfer matrix for the system and show that it leads to a well-defined self-adjoint Hamiltonian. In view of numerical simulations, we also suggest sets of Lorentzian Monte Carlo moves. We demonstrate that certain pathological phases found previously in Euclidean models of dynamical triangulations cannot be realized in the Lorentzian case.

The transfer matrix in four-dimensional CDT

“…In ordinary (Euclidean) lattice field theory reflection positiveness of the transfer matrix ensures a unitary time evolution. The CDT transfer matrix has a similar property [3][4][5]. 1 In this article we will analyze an "effective" transfer matrix of CDT to be defined below.…”

“…An interesting feature of the CDT model, rotated to Euclidean signature, is that it possesses a transfer matrix [3][4][5]. At (discrete) time t n we have a spatial hypersurface with a spatial geometry characterized by a three-dimensional triangulation T 3 (t n ).…”

“…In CDT we sum over all four-dimensional triangulations of the "slab" between t n and t n+1 compatible with the topology S 3 × [0, 1] and such that each four-simplex which "fills" the slab has subsimplices which are (sub)simplices of T 3 (t n ) as well as T 3 (t n+1 ). This leads to 4 types of four-simplices in the slab: type (4, 1), (2,3), (3,2) and (1,4), where the numbers 2 It has recently been suggested that there might be a continuum limit of DT which belongs to the same universality class as the CDT theory described below [58].…”

“…The number N (4,1) (t n ) of (4, 1) simplices in the slab is equal to the number N 3 (t n ) of three-simplices in T 3 (t n ), and similarly the number of (1,4) simplices in the slab is equal to the number N 3 (t n+1 ) of three-simplices in T 3 (t n+1 ). The transfer matrix M, i.e.…”

Quantum Gravity via Causal Dynamical Triangulations

“…For a more detailed description of the CDT construction we refer the interested reader to [22,23,24].…”