Feynman Path Integral Formulation

296 8 Numerical Studies10861Ν420 0.2 0.4 0.6 0.8 1z ⩵ d 2d 1Fig. 8.9 Universal gravitational exponent 1/ν as a function of the dimension. The abscissa isz =(d − 2)/(d − 1), which maps d = 2toz = 0andd = ∞ to z = 1. The larger circles at d = 3andd = 4 are the lattice gravity results, interpolated (continuous curve) using the exact lattice results1/ν = 0ind = 2, and ν = 0atd = ∞ [from Eq. (7.159)]. The two curves close to the origin are the2 + ε expansion for 1/ν to one loop (lower curve) and two loops (upper curve). The lower almosthorizontal line gives the value for ν expected for a scalar field theory, for which it is known thatν = 1ind = 2andν = 1 2in d ≥ 4.the definition in Eq. (8.32), or even from the correlation of Wilson lines associatedwith the propagation of two heavy spinless particles. The outcome of such largescale numerical calculations is eventually a determination of the quantities ν, k c =1/8πG c and A ξ from first principles, to some degree of numerical accuracy.In either case one expects the scaling result of Eq. (8.76) close to the fixed point,which we choose to rewrite here in terms of the inverse correlation length m ≡ 1/ξm = Λ A m |k − k c | ν . (8.77)Note that in the above expression the correct dimension for m (inverse length) hasbeen restored by inserting explicitly on the r.h.s. the ultraviolet cutoff Λ. Herekand k c are of course still dimensionless quantities, and correspond to the bare microscopiccouplings at the cutoff scale, k ≡ k(Λ) ≡ 1/[8πG(Λ)]. A m is a calculablenumerical constant, related to A ξ in Eq. (8.50) by A m = A −1 . It is worth pointingξout that the above expression for m(k) is almost identical in structure to the one forthe non-linear σ-model in the 2 + ε expansion, Eq. (3.36) and in the large N limit,Eqs. (3.59), (3.60) and (3.64). It is of course also quite similar to 2 + ε result forcontinuum gravity, Eq. (3.121).The lattice continuum limit corresponds to the large cutoff limit taken at fixed m,Λ → ∞ , k → k c , m fixed , (8.78)