Feynman Path Integral Formulation

234 7 Analytical Lattice Expansion Methods
with the matrix V given by Eq. (7.21). Since the lattice cosmological term can also
be expressed in terms of the matrix V ,
√ g = 1 +
1
2
h 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 constant
terms
λ 0 (1 + 1 2 h μμ)+ 1 2 · k
2 h αβV αβμν (∂ 2 + 2λ 0
k ) 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 constant
term of O(k) (which are completely absent in the continuum, since no derivatives
are present there). These should be irrelevant in the lattice continuum limit.
7.3 Lattice Diffeomorphism Invariance
The appearance of exact zero modes in the weak field expansion of the lattice gravitational
action is not specific to an expansion about flat space. One can consider
the same procedure for variations about spaces which are classical solutions for the
gravitational action with a cosmological constant term as in Eq. (6.43), such as the
regular or irregular tessellations of the d-sphere. In principle it is possible to look
at a general d-dimensional case, but here, for illustrative purposes, only two dimensions
will be considered, in which case on is looking, in the simplest case, at the
regular tessellations of the two-sphere. In the following the discussion will focus
therefore at first on edge length fluctuations about the regular tetrahedron (with 6
edges), octahedron (12 edges), and icosahedron (30 edges). One could consider irregular
tessellations as well, but this will not be pursued here, although one believes
the 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 is
I(l 2 )=λ ∑
h
δh
V h − 2k∑ δ h + 4a∑
2 , (7.39)
h
h
A h
with the two-dimensional volume element equated here to the dual area surrounding
avertexV h = A h , and the local curvature given by R h = 2 δ h /A h . In the limit of
small fluctuations around a smooth background, the above lattice action describes
the continuum action
∫
I[g] = d 2 x √ g [ λ − kR+ aR 2] . (7.40)