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 00 0 0 0For fermions the corresponding spin rotation matrix is then obtained from[ ]S = exp i4γ αβ θ αβ , (6.23)with generators γ αβ = 1 2i [γα ,γ β ], and with the Dirac matrices γ α satisfying as usualγ α γ β +γ β γ α = 2η αβ . Taking appropriate traces, one can obtain a direct relationshipbetween the original rotation matrix R(s,s ′ ) and the corresponding spin rotationmatrix 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 byR(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 ofsimplex 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 pathC(s 1 ,...,s n ), for which the associated expectation value can be considered as thegravitational analog of the Wilson loop. Its combined rotation is given byR(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 pairwisecancellation of the Λ(s i ) matrices except at the endpoints, giving in the closedloop caseR(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 suchpolygonal 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 numberm of d-simplices, sequentially labeled by s 1 ...s m , and whose common facesf (s 1 ,s 2 )... f (s m−1 ,s m ) will also contain the hinge h. Thus in four dimensions severalfour-simplices will contain, and therefore encircle, a given triangle (hinge).