Feynman Path Integral Formulation

8.10 Curvature Scales 301for gravity, QED (massive via the Higgs mechanism) and a self-interacting scalarfield, respectively.A third argument suggesting the identification of the scale ξ with large scale curvatureand therefore with the observed scaled cosmological constant goes as follows.Observationally the curvature on large scale can be determined by parallel transportingvectors around very large loops, with typical size much larger than the latticecutoff l P . In gravity, curvature is detected by parallel transporting vectors aroundclosed loops. This requires the calculation of a path dependent product of Lorentzrotations R, in the Euclidean case elements of SO(4), as discussed in Sect. 6.4. Onthe lattice, the above rotation is directly related to the path-ordered (P) exponentialof the integral of the lattice affine connection Γμν λ viaR α β = [P e∫pathbetween simplicesΓ λ dx λ] αβ . (8.88)Now, in the strongly coupled gravity regime (G > G c ) large fluctuations in the gravitationalfield at short distances will be reflected in large fluctuations of the R matrices.Deep in the strong coupling regime it should be possible to describe thesefluctuations by a uniform (Haar) measure. Borrowing from the analogy with Yang-Mills theories, and in particular non-Abelian lattice gauge theories with compactgroups [see Eq. (3.145)], one would therefore expect an exponential decay of nearplanarWilson loops with area A of the type[ ∫W(Γ ) ∼ trexpS(C)R·· μν A μνC]∼ exp(−A/ξ 2 ) , (8.89)where A is the minimal physical area spanned by the near-planar loop. A derivationof this standard result for non-Abelian gauge theories can be found, for example, inthe textbook (Peskin and Schroeder, 1995).In summary, the Wilson loop in gravity provides potentially a measure for themagnitude of the large-scale, averaged curvature, operationally determined by theprocess of parallel-transporting test vectors around very large loops, and whichtherefore, from the above expression, is computed to be of the order R ∼ 1/ξ 2 .One would expect the power to be universal, but not the amplitude, leaving openthe possibility of having both de Sitter or anti-de Sitter space at large distances(as discussed previously in Sect. 8.8, the average curvature describing the paralleltransport of vectors around infinitesimal loops is described by a lattice version ofEuclidean anti-de Sitter space). A recent explicit lattice calculation indeed suggeststhat the de Sitter case is singled out, at least for sufficiently strong copuling (Hamberand Williams, 2007). Furthermore one would expect, based on general scalingarguments, that such a behavior would persists throughout the whole strong couplingphase G > G c , all the way up to the on-trivial fixed point. From it then followsthe identification of the correlation length ξ with a measure of large scale curvature,the most natural candidate being the scaled cosmological constant λ phys ,asinEq. (8.86). This relationship, taken at face value, implies a very large, cosmological