Feynman Path Integral Formulation

262 7 Analytical Lattice Expansion Methods1/d. The complete volume term λ 0 ∑V d appearing in the action can then be easilywritten down using the above expressions.In d dimensions the dihedral angle in a d-dimensional simplex of volume V d ,between faces of volume V d−1 and V ′ d−1, is obtained from Eq. (6.12)sinθ d =dd − 1V d V d−2V d−1 V ′ d−1. (7.136)In the equilateral case one has for the dihedral angle√dθ d = arcsin2 − 1 π∼d d→∞ 2 − 1 d − 1 + ... (7.137)6d3 which will require four simplices to meet on a hinge, to give a deficit angle of2π − 4 ×2 π ≈ 0 in large dimensions. One notes that in large dimensions the simpliceslook locally (i.e. at a vertex) more like hypercubes. Several d-dimensionalsimplices will meet on a (d − 2)-dimensional hinge, sharing a common face of dimensiond −1 between adjacent simplices. Each simplex has (d −2)(d −1)/2 edges“on” the hinge, some more edges are then situated on the two “interfaces” betweenneighboring simplices meeting at the hinge, and finally one edge lies “opposite” tothe hinge in question.In the large d limit one then obtains, to leading order for the dihedral angle at thehinge with vertices labelled by 1...d − 1√dθ d ∼ d→∞ arcsin2 − 1+ ε d,d+1 + ε 1,d ε 1,d+1 + ...d+d1 (−ε1,d + ... − 1 2 ε2 1,d + ... − 1 2 ε2 d,d+1−ε 12 ε 1,d+1 − ε 1,d ε 3,d+1 − ε 1,d ε d,d+1 + ... ) .+O(d −2 ) . (7.138)From the expressions in Eq. (7.135) for the volume and Eq. (7.138) for the dihedralangle one can then evaluate the d-dimensional Euclidean lattice action, involvingcosmological constant and scalar curvature terms as in Eq. (6.43)I(l 2 )=λ 0 ∑ V d − k ∑ δ d V d−2 , (7.139)where δ d is the d-dimensional deficit angle, δ d = 2π − ∑ simplices θ d . The latticefunctional integral is then∫Z(λ 0 , k) = [dl 2 ] exp ( −I(l 2 ) ) . (7.140)To evaluate the curvature term −k ∑δ d V d−2 appearing in the gravitational latticeaction one needs the hinge volume V d−2 , which is easily obtained from Eq. (7.135),by reducing d → d − 2.