Feynman Path Integral Formulation

234 7 Analytical Lattice Expansion Methodswith the matrix V given by Eq. (7.21). Since the lattice cosmological term can alsobe expressed in terms of the matrix V ,√ g = 1 +12h μμ − 1 2 h αβV αβμν h μν + O(h 3 ) , (7.37)one finds, as in the continuum, for the combined Einstein and cosmological constanttermsλ 0 (1 + 1 2 h μμ)+ 1 2 · k2 h αβV αβμν (∂ 2 + 2λ 0k ) h μν + O(h 3 ) , (7.38)corresponding in this gauge to the exchange of a particle of “mass” μ 2 = −2λ 0 /k,in agreement with the continuum weak field result of Eq. (1.79). As for the Regge-Einstein term, there are higher order lattice corrections to the cosmological constantterm of O(k) (which are completely absent in the continuum, since no derivativesare present there). These should be irrelevant in the lattice continuum limit.7.3 Lattice Diffeomorphism InvarianceThe appearance of exact zero modes in the weak field expansion of the lattice gravitationalaction is not specific to an expansion about flat space. One can considerthe same procedure for variations about spaces which are classical solutions for thegravitational action with a cosmological constant term as in Eq. (6.43), such as theregular or irregular tessellations of the d-sphere. In principle it is possible to lookat a general d-dimensional case, but here, for illustrative purposes, only two dimensionswill be considered, in which case on is looking, in the simplest case, at theregular tessellations of the two-sphere. In the following the discussion will focustherefore at first on edge length fluctuations about the regular tetrahedron (with 6edges), octahedron (12 edges), and icosahedron (30 edges). One could consider irregulartessellations as well, but this will not be pursued here, although one believesthe results to have general validity for lattices sufficiently dense with points (Hartle,1985; Hamber and Williams, 1997).In two dimensions the lattice action for pure gravity isI(l 2 )=λ ∑hδhV h − 2k∑ δ h + 4a∑2 , (7.39)hhA hwith the two-dimensional volume element equated here to the dual area surroundingavertexV h = A h , and the local curvature given by R h = 2 δ h /A h . In the limit ofsmall fluctuations around a smooth background, the above lattice action describesthe continuum action∫I[g] = d 2 x √ g [ λ − kR+ aR 2] . (7.40)