Feynman Path Integral Formulation

226 7 Analytical Lattice Expansion Methods
I R ∝
∑ δ(l) A(l) , (7.1)
hinges
to quadratic order in the lattice weak fields one needs first and second variations
with respect to the edge lengths. In four dimensions the first variation of the lattice
Regge action is given by
δI R ∝
( )
∂A
∑ δ · ∑
hinges edges
∂l δl
, (7.2)
since Regge has shown that the term involving the variation of the deficit angle
δ vanishes (here the variation symbol should obviously not be confused with the
deficit angle). Furthermore in flat space all the deficit angles vanish, so that the
second variation is given simply by
δ 2 I R ∝
∑
hinges
(
∑
edges
) ( )
∂δ
∂l δl ∂A
· ∑
edges
∂l δl
. (7.3)
Next a specific lattice structure needs to be chosen as a background geometry. A
natural choice is to use a flat hypercubic lattice, made rigid by introducing face
diagonals, body diagonals and hyperbody diagonals, which results into a subdivision
of each hypercube into d! (here 4!=24) simplices. This subdivision is shown in
Fig. 7.1.
Fig. 7.1 A four-dimensional
hypercube divided up into
four-simplices.
By a simple translation, the whole lattice can then be constructed from this one
elemental hypercube. Consequently there will be 2 d −1 = 15 lattice fields per point,
corresponding to all the edge lengths emanating in the positive lattice directions
from any one vertex. Note that the number of degrees per lattice point is slightly
larger than what one would have in the continuum, where the metric g μν (x) has
d(d + 1)/2 = 10 degrees of freedom per spacetime point x in four dimensions
(perturbatively, the physical degrees of freedom in the continuum are much less:
1
2 d(d + 1) − 1 − d − (d − 1) = 1 2d(d − 3), for a traceless symmetric tensor, and after