Feynman Path Integral Formulation

206 6 Lattice Regularized Quantum Gravityare always positive due to the enforcement of the triangle inequalities. Thus froma practical point of view the barycentric volume subdivision is the simplest to dealwith.φ 3l 23A 23l 5A 244φ 4A 13φ 11A 11A 12l 1A 22l 42l 3φ 2Fig. 6.16 More dual areas appearing in the scalar field action.The scalar action of Eq. (6.134) has a very natural form: it involves the squareddifference of fields at neighboring points divided by their invariant distances (φ i −φ j )/l ij , weighted by the appropriate space-time volume element V (d)ijassociatedwith the lattice link ij. This suggests that one could just as well define the scalarfields on the vertices of the dual lattice, and write(I(l 2 ,φ) = 1 2 ∑ V rs(d) φr − φ) s 2, (6.137)l rswith l rs the length of the edge connecting the dual lattice vertices r and s, and consequentlyV rs(d) the spacetime volume fraction associated with the dual lattice edgers. One would expect both forms to be equivalent in the continuum limit.Continuing on with the two-dimensional case, mass and curvature terms such asthe ones appearing in Eq. (6.124) can be added to the action, so that the total scalarlattice action contribution becomesI = 1 2∑A ij( φi − φ jl ij) 2+12 ∑A i (m 2 + ξ R i )φi 2 . (6.138)iThe term containing the discrete analog of the scalar curvature involves the quantityA i R i ≡ ∑ δ h ∼ √ gR . (6.139)h⊃iIn the above expression for the scalar action, A ij is the area associated with theedge l ij , while A i is associated with the site i. Again there is more than one way to