Feynman Path Integral Formulation

7.4 Strong Coupling Expansion 247δ A hδA h 'Fig. 7.8 Correlations between action contributions on hinge h and hinge h ′ arise to lowest orderin the strong coupling expansions from diagrams describing a narrow tube connecting the twohinges. Here vertices represent points in the dual lattice, with the tube-like closed surface tiledwith parallel transport polygons. For each link of the dual lattice, the SO(4) parallel transportmatrices R of Sect. 6.4 are represented by an arrow.with the correlation length ξ = 1/|logk| →0 to lowest order as k → 0 (here wehave used the usual definition of the correlation length ξ , namely that a generic correlationfunction is expected to decay as exp(−distance/ξ ) for large separations). 2This last result is quite general, and holds for example irrespective of the boundaryconditions (unless of course ξ ∼ L, where L is the linear size of the system, in whichcase a path can be found which wraps around the lattice).But further thought reveals that the above result is in fact not completely correct,due to the fact that in order to achieve a non-vanishing correlation one needs, at leastto lowest order, to connect the two hinges by a narrow tube (Hamber and Williams,2006). The previous result should then read correctly as< (δ A) h (δ A) h ′ > C ∼ (k n d) l , (7.86)where n d l represents the minimal number of dual lattice polygons needed to form aclosed surface connecting the hinges h and h ′ , with l the actual distance (in latticeunits) between the two hinges. Fig. 7.8 provides an illustration of the situation.With some additional effort many additional terms can be computed in the strongcoupling expansion. In practice the method is generally not really competitive withdirect numerical evaluation of the path integral via Monte Carlo methods. But itdoes provide a new way of looking at the functional integral, and provide the basisfor new approaches, such as the large d limit to be discussed in the second half ofthe next section.2 This statement, taken literally, oversimplifies the situation a bit, as depending on the spin (ortensor structure) of the operator appearing in the correlation function, the large distance decay ofthe corresponding correlator is determined by the lightest excitation in that specific channel. But inthe gravitational context one is mostly concerned with correlators involving spin two (transversetraceless)objects, evaluated at fixed geodesic distance.