Feynman Path Integral Formulation

186 6 Lattice Regularized Quantum Gravity
6.7 Lattice Bianchi Identities
Consider therefore a closed path C h encircling a hinge h and passing through each
of the simplices that meet at that hinge. In particular one may take C h to be the
boundary of the polyhedral dual (or Voronoi) area surrounding the hinge. We recall
that the Voronoi polyhedron dual to a vertex P is the set of all points on the lattice
which are closer to P than any other vertex; the corresponding new vertices then
represent the sites on the dual lattice. A unique closed parallel transport path can
then be assigned to each hinge, by suitably connecting sites in the dual lattice.
With each neighboring pair of simplices s,s+1 one associates a Lorentz transformation
R α β (s,s+1), which describes how a given vector V μ transforms between the
local coordinate systems in these two simplices As discussed previously, the above
transformation is directly related to the continuum path-ordered (P) exponential of
the integral of the local affine connection Γμν λ via
R μ ν =
∫
[
P e
path
between simplices
Γ λ dx λ
] μ
ν . (6.56)
The connection here has support only on the common interface between the two
simplices.
4
δ
04
Fig. 6.11 Illustration of the
exact lattice Bianchi identity
in the case of three dimensions,
where several hinges
(edges) meet on a vertex. The
combined rotation for a path
that sequentially encircles
several hinges and which can
be shrunk to a point is given
by the identity matrix.
1
δ
01
2
0
δ
02
δ
03
3
Just as in the continuum, where the affine connection and therefore the infinitesimal
rotation matrix is determined by the metric and its first derivatives, on the lattice
one expects that the elementary rotation matrix between simplices R s,s+1 is fixed by
the difference between the g ij ’s of Eq. (6.3) within neighboring simplices.
For a vector V transported once around a Voronoi loop, i.e. a loop formed by
Voronoi edges surrounding a chosen hinge, the change in the vector V is given by