Feynman Path Integral Formulation

176 6 Lattice Regularized Quantum Gravity
⎛
⎞
0 1 0 0
⎜−1 0 0 0⎟
σ 12 = ⎝
⎠ . (6.22)
0 0 0 0
0 0 0 0
For fermions the corresponding spin rotation matrix is then obtained from
[ ]
S = exp i4
γ αβ θ αβ , (6.23)
with generators γ αβ = 1 2i [γα ,γ β ], and with the Dirac matrices γ α satisfying as usual
γ α γ β +γ β γ α = 2η αβ . Taking appropriate traces, one can obtain a direct relationship
between the original rotation matrix R(s,s ′ ) and the corresponding spin rotation
matrix S(s,s ′ )
R αβ = tr ( S † γ α Sγ β
)
/tr1 , (6.24)
which determines the spin rotation matrix up to a sign.
One can consider a sequence of rotations along an arbitrary path P(s 1 ,...,s n+1 )
going through simplices s 1 ...s n+1 , whose combined rotation matrix is given by
R(P) =R(s n+1 ,s n )···R(s 2 ,s 1 ) , (6.25)
and which describes the parallel transport of an arbitrary vector from the interior of
simplex s 1 to the interior of simplex s n+1 ,
φ μ (s n+1 )=R μ ν(P)φ ν (s 1 ) . (6.26)
If the initial and final simplices s n+1 and s 1 coincide, one obtains a closed path
C(s 1 ,...,s n ), for which the associated expectation value can be considered as the
gravitational analog of the Wilson loop. Its combined rotation is given by
R(C) =R(s 1 ,s n )···R(s 2 ,s 1 ) . (6.27)
Under Lorentz transformations within each simplex s i along the path one has a pairwise
cancellation of the Λ(s i ) matrices except at the endpoints, giving in the closed
loop case
R(C) → Λ(s 1 )R(C)Λ T (s 1 ) . (6.28)
Clearly the deviation of the matrix R(C) from unity is a measure of curvature. Also,
the trace trR(C) is independent of the choice of Lorentz frames.
Of particular interest is the elementary loop associated with the smallest nontrivial,
segmented parallel transport path one can build on the lattice. One such
polygonal path in four dimensions is shown in Fig. 6.6. In general consider a
(d − 2)-dimensional simplex (hinge) h, which will be shared by a certain number
m of d-simplices, sequentially labeled by s 1 ...s m , and whose common faces
f (s 1 ,s 2 )... f (s m−1 ,s m ) will also contain the hinge h. Thus in four dimensions several
four-simplices will contain, and therefore encircle, a given triangle (hinge).