Feynman Path Integral Formulation

258 7 Analytical Lattice Expansion Methods( )k n n4∏ 2n A i ε α [p + qε 2 ] β , (7.116)i=1where α + β = n. IfĀ is of the order of the geometric or arithmetic mean of theindividual loops, this can be approximated by( ) n k Āε α [p + qε 2 ] β . (7.117)16This shows again that one should take ε > 0, otherwise the answer vanishes to thisorder, and one needs to go to higher order in the expansion in k. This is quite legitimate,as the correct lattice action is recovered irrespective of the value of ε, asinEq. (7.101). Then using n = A C /Ā, one can write the area-dependent first factor asexp[(A C /Ā) log(k Ā/16)] = exp(−A C /ξ 2 ) , (7.118)where ξ ≡ [Ā/|log(k Ā/16)|] 1/2 . Recall that this is in the case of strong coupling,when k → 0. The above is the main result so far. The rapid decay of the quantumgravitational Wilson loop as a function of the area is seen here simply as a generaland direct consequence of the assumed disorder in the uncorrelated fluctuations ofthe parallel transport matrices R(s,s ′ ) at strong coupling.We note here the important point that the gravitational correlation length ξ isdefined independently of the expectation value of the Wilson loop. Indeed a keyquantity in gauge theories as well as gravity is the correlation between differentplaquettes, which in simplicial gravity is given by see Eq. (7.81)∫dμ(l 2 )(δ A) h (δ A) h ′ e k ∑ h δ h A h< (δ A) h (δ A) h ′ > = ∫. (7.119)dμ(l 2 )e k ∑ h δ h A hIn order to achieve a non-vanishing correlation one needs, at least to lowest order,to connect the two hinges by a narrow tube, so that< (δ A) h (δ A) h ′ > C ∼ (k n t) l ∼ e −d(h,h′ )/ξ , (7.120)where the “distance” n t l represents the minimal number of dual lattice polygonsneeded to form a closed surface connecting the hinges h and h ′ (as an example,for a narrow tube made out of cubes connecting two squares one has n t =4). In theabove expression d(h,h ′ ) represents the actual physical distance between the twohinges, and the correlation length is given in this limit (k → 0) by ξ ∼ l 0 /n t |logk|.where l 0 is the average lattice spacing. Here we have used the usual definition of thecorrelation length ξ , namely that a generic correlation function is expected to decayas exp(−distance/ξ ) for large separations. Fig. (7.8) provides an illustration of thesituation.