Feynman Path Integral Formulation

6.5 Invariant Lattice Action 179
6.5 Invariant Lattice Action
The first step in writing down an invariant lattice action, analogous to the continuum
Einstein-Hilbert action, is to find the lattice analogue of the Ricci scalar. From the
expression for the Riemann tensor at a hinge given in Eq. (6.36) one obtains by
contraction
R(h) =2 δ(h)
A C (h) . (6.37)
The continuum expression √ gR is then obtained by multiplication with the volume
element V (h) associated with a hinge. The latter is defined by first joining the vertices
of the polyhedron C, whose vertices lie in the dual lattice, with the vertices of
the hinge h, and then computing its volume.
By defining the polygonal area A C as A C (h)=dV(h)/V (d−2) (h), where V (d−2) (h)
is the volume of the hinge (an area in four dimensions), one finally obtains for the
Euclidean lattice action for pure gravity
I R (l 2 )=− k
∑
hinges h
δ(h)V (d−2) (h) , (6.38)
with the constant k = 1/(8πG). One would have obtained the same result for the
single-hinge contribution to the lattice action if one had contracted the infinitesimal
form of the rotation matrix R(h) in Eq. (6.32) with the hinge bivector ω αβ of
Eq. (6.8) (or equivalently with the bivector U αβ of Eq. (6.31) which differs from
ω αβ by a constant). The fact that the lattice action only involves the content of the
hinge V (d−2) (h) (the area of a triangle in four dimensions) is quite natural in view
of the fact that the rotation matrix at a hinge in Eq. (6.32) only involves the deficit
angle, and not the polygonal area A C (h).
An alternative form for the lattice action (Fröhlich, 1981) can be obtained instead
by contracting the elementary rotation matrix R(C) of Eq. (6.32), and not just its
infinitesimal form, with the hinge bivector of Eq. (6.8),
I com (l 2 )=− k
∑
hinges h
1
2
ω αβ (h)R αβ (h) . (6.39)
The above construction can be regarded as analogous to Wilson’s lattice gauge
theory, for which the action also involves traces of products of SU(N) color rotation
matrices (Wilson, 1973). For small deficit angles one can of course use
ω αβ =(d − 2)!V (d−2) U αβ to show the equivalence of the two lattice actions.
But in general, away from a situation of small curvatures, the two lattice action
are not equivalent, as can be( seen already in ) two dimensions. Writing the rotation
cosδ sinδ
matrix at a hinge as R(h)=
, expressed for example in terms of
−sinδ cosδ
Pauli matrices, and taking the appropriate trace (ω αβ = ε αβ in two dimensions) one
finds
tr [ 1
2
(−iσ y )(cosδ p + iσ y sinδ p ) ] = sinδ p , (6.40)