Feynman Path Integral Formulation

6.9 Lattice Regularized Path Integral 193and with again λ ≠ −2/d. This procedure defines a metric on the tangent space ofpositive real symmetric matrices g ij (s). After computing the determinant of G, theresulting functional measure is∫∫dμ[l 2 ]=∏ [ detG[g(s)] ] 1 2 ∏ dg ij (s) , (6.80)si≥ jwith the determinant of the super-metric G ijkl [g(s)] given by the local expressiondetG[g(s)] ∝ (1 + 1 2 dλ) [g(s)](d−4)(d+1)/4 . (6.81)Using Eq. (6.69), and up to irrelevant constants, one obtains again the standardlattice measure of Eq. (6.76). Of course the same procedure can be followed for theMisner-like measure, leading to a similar result for the lattice measure, but with adifferent power σ.One might be tempted to try to find alternative lattice measures by looking directlyat the discrete form for the supermetric, written as a quadratic form in thesquared edge lengths (instead of the metric components), and then evaluating theresulting determinant. The main idea, inspired by work described in a paper (Lundand Regge, 1974) on the 3 + 1 formulation of simplicial gravity, is as follows. Firstone considers a lattice analog of the DeWitt supermetric by writing‖δl 2 ‖ 2 = ∑ijG ij (l 2 ) δl 2 i δl 2 j , (6.82)with G ij (l 2 ) now defined on the space of squared edge lengths (Hartle, Miller andWilliams, 1997). The next step is to find an appropriate form for G ij (l 2 ) expressed interms of known geometric objects. One simple way of constructing the explicit formfor G ij (l 2 ), in any dimension, is to first focus on one simplex, and write the squaredvolume of a given simplex in terms of the induced metric components within thesame simplex s,V 2 (s) = ( )1 2 [d!detgij l 2 (s) ] . (6.83)One computes to linear order1V (l 2 ) ∑ i∂V 2 (l 2 )∂l 2 iδl 2 i = 1 d!√det(g ij ) g ij δg ij , (6.84)and to quadratic order1V (l 2 ) ∑ ij∂ 2 V 2 (l 2 )∂l 2 i ∂l2 jδl 2 i δl 2 j = 1 d!√ [ ]det(g ij ) g ij g kl δg ij δg kl − g ij g kl δg jk δg li.(6.85)