Feynman Path Integral Formulation

6.12 Scalar Matter Fields 205A ij is the dual (Voronoi) area associated with the edge ij, and the symbol < ij>denotes a sum over nearest neighbor lattice vertices. It is immediate to generalizethe action of Eq. (6.133) to higher dimensions, with the two-dimensional Voronoivolumes replaced by their higher dimensional analogs, leading to(I(l 2 ,φ) = 1 2 ∑ V (d) φi − φ) j 2ij. (6.134)l ijHere V (d)ijis the dual (Voronoi) volume associated with the edge ij, and the sum isover all links on the lattice.φ 3l 23h 1l 54φ 4φ 11l 1l 42l 3φ 2Fig. 6.15 Dual area associated with the edge l 1 (shaded area), and the corresponding dual link h 1 .In two dimensions, in terms of the edge length l ij and the dual edge lengthh ij , connecting neighboring vertices in the dual lattice, one has A ij = 1 2 h ijl ij (seeFig. 6.15). Other choices for the lattice subdivision will lead to a similar formulafor the lattice action, with the Voronoi dual volumes replaced by their appropriatecounterparts for the new lattice. Explicitly, for an edge of length l 1 the dihedral dualvolume contribution is given byA l1 = l2 1 (l2 2 + l2 3 − l2 1 )16A 123+ l2 1 (l2 4 + l2 5 − l2 1 )16A 234= 1 2 l 1h 1 , (6.135)with h 1 is the length of the edge dual to l 1 .On the other hand the barycentric dihedral area for the same edge would besimply (see Fig 6.16)A l1 =(A 123 + A 234 )/3 . (6.136)It is well known that one of the disadvantages of the Voronoi construction is the lackof positivity of the dual volumes, as pointed out in (Hamber and Williams, 1984).Thus some of the weights appearing in Eq. (6.133) can be negative for such anaction. For the barycentric subdivision this problem does not arise, as the areas A ij