Feynman Path Integral Formulation

184 6 Lattice Regularized Quantum Gravity
(Barrett et al, 1997) using a discrete time step formulation, whereas in (Piran and
Williams, 1986) a continuous time fomalism was proposed. The choice of lapse and
shift functions in Regge gravity were discussed further in (Tuckey, 1989; Galassi,
1993) and in (Gentle and Miller, 1998), and applied to the Kasner cosmology in the
last reference. An alternative so-called null-strut approach was proposed in (Miller
and Wheeler, 1985) which builds up a spacelike-foliated spacetime with a maximal
number of null edges, but seems difficult to implement in practice. Finally in
(Khatsymovsky, 1991) and (Immirzi, 1996) a continuous time Regge gravity formalism
in the tetrad-connection variables was developed, in part targeted towards
quantum gravity calculations. A recent comprehensive review of classical applications
of Regge gravity can be found for example in (Gentle, 2002), as well as a more
complete set of references.
6.6 Lattice Diffeomorphism Invariance
Consider the two-dimensional flat skeleton shown in Fig. 6.8. It is clear that one
can move around a point on the surface, keeping all the neighbors fixed, without
violating the triangle inequalities and leave all curvature invariants unchanged.
Fig. 6.8 On a random simplicial
lattice there are in general
no preferred directions.
In d dimensions this transformation has d parameters and is an exact invariance
of the action. When space is slightly curved, the invariance is in general only an approximate
one, even though for piecewise linear spaces piecewise diffeomorphisms
can still be defined as the set of local motions of points that leave the local contribution
to the action, the measure and the lattice analogues of the continuum curvature
invariants unchanged (Hamber and Williams, 1998). Note that in general the gauge
deformations of the edges are still constrained by the triangle inequalities. The general
situation is illustrated in Figs. 6.8, 6.9 and 6.10. In the limit when the number
of edges becomes very large, the full continuum diffeomorphism group should be
recovered.