Feynman Path Integral Formulation

266 7 Analytical Lattice Expansion Methods(∫Z latt (k) = dμ(l 2 ) e k ∑ ∞ ∫h δ h A h1= ∑n=0n! kn dμ(l 2 )) n∑δ h A h . (7.151)hThen one can show that dominant diagrams contributing to Z latt correspond toclosed surfaces tiled with elementary transport loops. In the case of the hingehingeconnected correlation function the leading contribution at strong couplingcome from closed surfaces anchored on the two hinges, as in Eq. (7.86).It will be advantageous to focus on general properties of the parallel transportmatrices R, discussed previously in Sect. 6.4. For smooth enough geometries, withsmall curvatures, these rotation matrices can be chosen to be close to the identity.Small fluctuations in the geometry will then imply small deviations in the R’s fromthe identity matrix. But for strong coupling (k → 0) the measure ∫ dμ(l 2 ) does notsignificantly restrict fluctuations in the lattice metric field. As a result we will assumethat these fields can be regarded, in this regime, as basically unconstrainedrandom variables, only subject to the relatively mild constraints implicit in the measuredμ. The geometry is generally far from smooth since there is no coupling termto enforce long range order (the coefficient of the lattice Einstein term is zero), andone has as a consequence large local fluctuations in the geometry. The matrices Rwill therefore fluctuate with the local geometry, and average out to zero, or a valueclose to zero. In the sense that, for example, the SO(4) rotation⎛⎞cosθ −sinθ 0 0⎜ sinθ cosθ 0 0⎟R θ = ⎝⎠ , (7.152)0 0 1 00 0 0 1averages out to zero when integrated over θ. In general an element of SO(n) isdescribed by n(n − 1)/2 independent parameters, which in the case at hand can beconveniently chosen as the six SO(4) Euler angles. The uniform (Haar) measureover the group is thendμ H (R)= 1 ∫ 2π ∫ π ∫ π ∫ π ∫ π ∫ π32π 9 dθ 1 dθ 2 dθ 3 dθ 4 sinθ 4 dθ 5 sinθ 5 dθ 6 sin 2 θ 6 .0 0 0 000(7.153)This is just a special case of the general n result, which reads)n n−1dμ H (R)=(∏Γ (i/2)/2 n π n(n+1)/2 ∏i=1i=1i∏j=1sin j−1 θ i j dθ i j , (7.154)with 0 ≤ θ 1 k < 2π,0≤ θ j k < π.These averaging properties of rotations are quite similar of course to what happensin SU(N) Yang-Mills theories, or even more simply in (compact) QED, wherethe analogs of the SO(d) rotation matrices R are phase factors U μ (x) =e iaA μ (x) .There one has ∫ dA μ2π U μ(x)=0 and ∫ dA μ2π U μ(x)U μ(x)=1. † In addition, for two contiguousclosed paths C 1 and C 2 sharing a common side one has