Feynman Path Integral Formulation

264 7 Analytical Lattice Expansion MethodsFig. 7.12 Cross polytopeβ n with n = 8and2n = 16vertices, whose surface canbe used to define a simplicialmanifold of dimension d =n − 1 = 7. For general d, thecross polytope β d+1 will have2(d + 1) vertices, connectedto each other by 2d(d + 1)edges.δ d = 0 + 4 d − ( ε d,d+1 + 3terms + ε 1,d ε 1,d+1 + ... ) + ... (7.144)and therefore close to flat in the large d limit (due to our choice of an equilateralstarting configuration). Indeed if the choice of triangulation is such that the deficitangle is not close to zero, then the discrete model leads to an average curvaturewhose magnitude is comparable to the lattice spacing or ultraviolet cutoff, whichfrom a physical point of view does not seem very attractive: one obtains a spacetimewith curvature radius comparable to the Planck length.When evaluated on such a manifold the lattice action becomes√d 2d/22 ( λ 0 − kd 3) [ 1 − 1 d!8 ∑ εij 2 + 1 ( 1d 4 ∑ ε ij + 1 ) ]8 ∑ ε ij ε ik + O(1/d 2 ) .(7.145)Dropping the 1/d correction the action is proportional to(λ0 − kd 3) ∑εij 2 . (7.146)− 1 2Since there are 2d(d + 1) edges in the cross polytope, one finds therefore that, atthe critical point kd 3 = λ 0 , the quadratic form in ε, defined by the above action,develops 2d(d + 1) ∼ 2d 2 zero eigenvalues.This result is quite close to the d 2 /2 zero eigenvalues expected in the continuumfor large d, with the factor of four discrepancy presumably attributed to an underlyingintrinsic ambiguity that arises when trying to identify lattice points with pointsin the continuum.It is worth noting here that the competing curvature (k) and cosmological constant(λ 0 ) terms will have comparable magnitude whenk c = λ 0 l 2 0d 3 . (7.147)