Feynman Path Integral Formulation

188 6 Lattice Regularized Quantum Gravity
“plaquette variables” associated with an elementary square (thereby involving the
ordered product of four group elements), provided the Bianchi identity constraint is
included as well in the lattice path integral.
6.8 Gravitational Wilson Loop
We have seen that with each neighboring pair of simplices s,s + 1 one can associate
a Lorentz transformation R μ ν(s,s + 1), which describes how a given vector
V μ transforms between the local coordinate systems in these two simplices, and
that the above transformation is directly related to the continuum path-ordered (P)
exponential of the integral of the local affine connection Γ λ
μν(x) via
R μ ν =
∫
[
Pe
path
between simplices
Γ λ dx λ ] μ
ν , (6.60)
with the connection having support only on the common interface between the two
simplices. Also, for a closed elementary path C h encircling a hinge h and passing
through each of the simplices that meet at that hinge one has for the total rotation
matrix R ≡ ∏ s R s,s+1 associated with the given hinge
[
∏
s
R s,s+1
] μ
ν = [
e δ(h)U(h)] μν . (6.61)
Equivalently, this last expression can be re-written in terms of a surface integral
of the Riemann tensor, projected along the surface area element bivector A αβ (C h )
associated with the loop,
∫
[ ] 1
μ
∏ R s,s+1
[e ≈ 2
R··αβ Aαβ (C h )] μ
S . (6.62)
s
ν ν
More generally one might want to consider a near-planar, but non-infinitesimal,
closed loop C, as shown in Fig. 7.9. Along this closed loop the overall rotation
matrix will still be given by
R μ ν(C) =
[ ] μ
∏ R s,s+1 . (6.63)
s⊂C
ν
In analogy with the infinitesimal loop case, one would like to state that for the overall
rotation matrix one has
R μ ν(C) ≈
[e δ(C)U(C))] μ
, (6.64)
ν