Discrete Wheeler-DeWitt equation

Abstract

We present a discrete form of the Wheeler-DeWitt equation for quantum gravitation, based on the lattice formulation due to Regge. In this setup, the infinite-dimensional manifold of 3-geometries is replaced by a space of three-dimensional piecewise linear spaces, with the solutions to the lattice equations providing a suitable approximation to the continuum wave functional. The equations incorporate a set of constraints on the quantum wave functional, arising from the triangle inequalities and their higher-dimensional analogs. The character of the solutions is discussed in the strong-coupling (large-G) limit, where it is shown that the wave functional only depends on geometric quantities, such as areas and volumes. An explicit form, determined from the discrete wave equation supplemented by suitable regularity conditions, shows peaks corresponding to integer multiples of a fundamental unit of volume. An application of the variational method using correlated product wave functions suggests a relationship between quantum gravity in n þ 1 dimensions, and averages computed in the Euclidean path integral formulation in n dimensions. The proposed discrete equations could provide a useful, and complementary, computational alternative to the Euclidean lattice path integral approach to quantum gravity.

Quantum gravity on foliated spacetimes: Asymptotically safe and sound

“…As a first application we studied the scale-dependence of Newton's coupling and the cosmological constant in the presence of a foliation structure. The resulting fixed point structure and universal critical exponents match the leading correction obtained from the perturbative computations in 2 + spacetime dimensions and show a good agree-ment with lattice quantum gravity in D = 2 + 1 dimensions [82] and the discretized Wheeler-de Witt equation [103,104]. In particular the real critical exponents of the non-Gaussian fixed points in D = 2 + 1 may provide a natural explanation for the apparent mismatch between the real critical exponents seen in Monte Carlo approaches and the complex critical exponents typically obtained from the functional renormalization group.…”

“…For D > 3.40 the real stability coefficients become complex (blue line) which reflects the typical characteristics of the UV-NGFP seen in the functional RG approach to Asymptotic Safety. This universality [102], the circles indicate the scaling of Newton's coupling found within lattice Quantum Gravity [82] and the square marks the scaling found from the exact solution of the discretized WheelerdeWitt equation [103,104]. In D = 4 the down-triangle indicates the critical exponents obtained from foliated spacetimes using the Matsubara formalism [88] while the diamond corresponds to the dynamical fixed point seen in the "geometrical" flow equation [24].…”

Wheeler-DeWitt equation in3+1dimensions

“…Here we will generally follow the procedure outlined in [1] and discretize the continuum Wheeler-DeWitt equation directly, a procedure that makes sense in the lattice formulation, as these equations are still formulated in terms of geometric objects, for which the Regge theory is very well suited. It is known that on a simplicial lattice [16,17,18,19,20,21,22] (see for example [23] for a more detailed presentation of the Regge-Wheeler lattice formulation)…”

“…Recently a Hamiltonian lattice formulation was written down based on the Wheeler-DeWitt equation, where the gravity Hamiltonian is expressed in the metric-space representation. Specifically, in [1,2] a general discrete Wheeler-DeWitt equation was given for pure gravity, based on the simplicial lattice transcription of gravity formulated by Regge and Wheeler. Here we extend the work initiated in [1,2] to the physical case of 3 + 1 dimensions, and show how nonperturbative vacuum solutions to the lattice Wheeler-DeWitt equations can be obtained for arbitrary values of Newton's constant G. The procedure we follow is similar to what was done earlier in 2 + 1 dimensions.…”

“…Specifically, in [1,2] a general discrete Wheeler-DeWitt equation was given for pure gravity, based on the simplicial lattice transcription of gravity formulated by Regge and Wheeler. Here we extend the work initiated in [1,2] to the physical case of 3 + 1 dimensions, and show how nonperturbative vacuum solutions to the lattice Wheeler-DeWitt equations can be obtained for arbitrary values of Newton's constant G. The procedure we follow is similar to what was done earlier in 2 + 1 dimensions. We solve the lattice equations exactly for several finite and regular triangulations of the three-sphere, and then extend the result to an arbitrarily large number of tetrahedra.…”