Feynman Path Integral Formulation

250 7 Analytical Lattice Expansion Methodslimit, one needs still to develop some set of approximation methods. In principlethe reliability of the approximations can later be tested by numerical means, forexample by integrating directly over edges using the explicit lattice measure givenabove.One approach that appears natural in the gravity context follows along the lines ofwhat is normally done in gauge theories, namely an integration over compact groupvariables, using the invariant measure over the gauge group. It is of this method thatwe wish to take advantage here, as we believe that it is well suited for gravity aswell. In order to apply such a technique to gravity one needs (i) to formulate thelattice theory in such a way that group variables are separated and therefore appearexplicitly; (ii) integrate over the group variables using an invariant measure; and(iii) approximate the relevant correlation functions in such a way that the groupintegration can be performed exactly, using for example mean field methods for theparts that appear less tractable. In such a program one is aided by the fact that in thestrong coupling limit one is expanding about a well defined ground state, and thatthe measure and the interactions are local, coupling only lattice variable (edges orrotations) which are a few lattice spacings apart. The downside of such methods isthat one is no longer evaluating the functional integral for quantum gravity exactly,even in the strong coupling limit; the upside is that one obtains a clear analyticalestimate, which later can be in principle systematically tested by numerical methods(which are in principle exact).In the gravity case the analogs of the gauge variables of Yang-Mills theories aregiven by the connection, so it is natural therefore to look for a first order formulationof Regge gravity (Caselle D’Adda and Magnea, 1989), discussed in Sect. (4.2). Themain feature of this approach is that one treats the metric g μν and the affine connectionΓμν λ as independent variables. There one can safely consider functionallyintegrating separately over the affine connection and the metric, treated as independentvariables, with the correct relationship between metric and connection arisingthen as a consequence of the dynamics. In the lattice theory we will follow a similarspirit, separating out explicitly in the lattice action the degrees of freedom correspondingto local rotations (the analogs of the Γ ’s in the continuum), which we willfind to be most conveniently described by orthogonal matrices R.The next step is a use of the properties of local rotation matrices in the contextof the lattice theory, and how these relate to the lattice gravitational action. It wasshown in Sect. (6.4) that with each neighboring pair of simplices s,s + 1 one canassociate a Lorentz transformation R μ ν(s,s+1), which describes how a given vectorV μ transforms between the local coordinate systems in these two simplices, andthat the above transformation is directly related to the continuum path-ordered (P)exponential of the integral of the local affine connection Γμν(x) λ viaR μ ν =∫[Pepathbetween simplicesΓ λ dx λ ] μν , (7.93)with the connection having support only on the common interface between the twosimplices. Also, for a closed elementary path C h encircling a hinge h and passing