Feynman Path Integral Formulation

8.7 Physical and Unphysical Phases 285χ R (k)∼k→k cA χR |k c − k| −(2−dν) . (8.54)At a critical point the fluctuation χ is in general expected to diverge, correspondingto the presence of a divergent correlation length. From such averages one cantherefore in principle extract the correlation length exponent ν of Eq. (8.50) withouthaving to compute a correlation function.An equivalent result, relating the quantum expectation value of the curvature tothe physical correlation length ξ , is obtained from Eqs. (8.50) and (8.53)R(ξ )∼k→k cξ 1/ν−4 , (8.55)again up to an additive constant. Matching of dimensionalities in this last equationis restored by inserting an appropriate power of the Planck length l P = √ G on ther.h.s..One can relate the critical exponent ν to the scaling behavior of correlations atlarge distances. The curvature fluctuation is related to the connected scalar curvaturecorrelator at zero momentum∫ ∫ √ √ dx dy < gR(x) gR(y) >cχ R (k) ∼< ∫ dx √ g >. (8.56)A divergence in the above fluctuation is then indicative of long range correlations,corresponding to the presence of a massless particle. Close to the critical point oneexpects for large separations l 0 ≪|x − y| ≪ξ a power law decay in the geodesicdistance, as in Eq. (8.31),< √ gR(x) √ gR(y) > ∼|x−y|→∞1|x − y| 2n . (8.57)Inserting the above expression in Eq. (8.56) and comparing with Eq. (8.54) determinesthe n as n = d − 1/ν. A priori one cannot exclude to possibility that somestates acquire a mass away from the critical point, in which case the correlationfunctions would have the behavior of Eq. (8.32) for |x − y|≫ξ .8.7 Physical and Unphysical PhasesAn important alternative to the analytic methods in the continuum is an attemptto solve quantum gravity directly via numerical simulations. The underlying ideais to evaluate the gravitational functional integral in the discretized theory Z bysumming over a suitable finite set of representative field configurations. In principlesuch a method given enough configurations and a fine enough lattice can provide anarbitrarily accurate solution to the original quantum gravity theory.In practice there are several important factors to consider, which effectively limitthe accuracy that can be achieved today in a practical calculation. Perhaps the most