Discrete Wheeler-DeWitt equation

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 $n+1$ 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.

Figure 1

A triangle with labels.

Figure 2

Neighbors of a given triangle. The above picture is supposed to illustrate the fact that the Laplacian ${\Delta }_{l^2}$ appearing in the kinetic term of the lattice Wheeler-DeWitt equation (here in $2+1$ dimensions) contains edges $a$, $b$, $c$ that belong both to the triangle in question, as well as to several neighboring triangles (here three of them) with squared edges denoted sequentially by $s_1=l_1^2\dots s_6=l_6^2$

Figure 3

A tetrahedron with labels.

Figure 4

A small section of a suitable spatial lattice for quantum gravity in $2+1$ dimensions.

Figure 5

A small section of a suitable spatial lattice for quantum gravity in $3+1$ dimensions.

Figure 6

Energy surface $|E(\alpha ){|}^2$ in $2+1$ dimensions at strong coupling, $G\gg G_c$ in the ($\mathrm{Re}\alpha$, $\mathrm{Im}\alpha$) plane. Note the presence of two almost purely imaginary, complex conjugate roots. The specific values used here are $\chi =2$, $N_T=10$, ${\sigma }_0=3$ and $\lambda =1$.

Figure 7

Energy surface $|E(\alpha ){|}^2$ in $2+1$ dimensions for weak coupling, $G\ll G_c$. In this case both roots are along the real $\alpha$ axis.

Figure 8

A cube divided into simplices.