Feynman Path Integral Formulation

6.5 Invariant Lattice Action 183
where the ω(n−1) α ’s (here referring to two (n − 1) simplices part of the same n-
simplex) are given in Eqs. (6.6) and (6.11). In the above expression the ω’s are
meant to be associated with vertex labels 0,...,i − 1,i + 1,...,n for ω n−1 , and
0,..., j − 1, j + 1,...,n for ω n−1 ′ respectively.
Then in the absence of a cosmological term one finds the remarkably simple
expression for the lattice field equations
1
2 l p ∑ δ h cotθ ph = 0 , (6.55)
h⊃l p
where the sum is over hinges (triangles) labeled by h meeting on the common edge
l p , and θ ph is the angle in the hinge h opposite to the edge l p . This is illustrated in
Fig. 6.7.
θ ph
p
Fig. 6.7 Angles appearing in
the Regge equations.
The discrete equations given above represent the lattice analogs of the Einstein
field equations in a vacuum, for which suitable solutions can be searched for by adjusting
the edge lengths. Since the equations are in general non-linear, the existence
of multiple solutions cannot in general be ruled out (Misner, Thorne and Wheeler,
1973). A number of papers have addressed the general issue of convergence to the
continuum in the framework of the classical formulation (Brewin and Gentle, 2001).
Several authors have discussed non-trivial applications of the Regge equations to
problems in classical general relativity such as the Schwarzschild and Reissner-
Nordstrom geometries (Wong, 1971), the Friedmann and Tolman universes (Collins
and Williams, 1973; 1974), and the problem of radial motion and circular (actually
polygonal) orbits (Williams and Ellis, 1981; 1984). Spherically symmetric, as
well as more generally inhomogeneous, vacuum spacetimes were studied using a
discrete 3 + 1 formulation with a variety of time-slicing prescriptions in (Porter,
1987a,b,c), and later extended (Dubal, 1989a,b) to a systematic investigation of
the axis-symmetric non-rotating vacuum solutions and to the problem of relativistic
spherical collapse for polytropic perfect fluids.
In classical gravity the general time evolution problem plays of course a central
role. The 3 + 1 time evolution problem in Regge gravity was discussed originally in
(Sorkin, 1975a,b) and later re-examined from a numerical, practical prespective in