Feynman Path Integral Formulation

3.6 Phases of Gravity in 2+ε Dimensions 95μ ddμ m ≡ μ d [] A m μ |G(μ) − G c | ν = 0 , (3.122)dμprovided G runs in accordance with Eq. (3.117). To one-loop order one has fromEqs. (3.106) and (3.120) ν = 1/(d −2). When the bare (lattice) coupling G(Λ)=G cone has achieved criticality, m = 0. How far the bare theory is from the critical pointis determined by the choice of G(Λ), the distance from criticality being measuredby the deviation ΔG = G(Λ) − G c .Furthermore Eq. (3.121) shows how the (lattice) continuum limit is to be taken.In order to reach the continuum limit a ≡ 1/Λ → 0 for fixed physical correlationξ = 1/m, the bare coupling G(Λ) needs to be tuned so as to approach the ultravioletfixed point at G c ,Λ → ∞ , m fixed , G → G c . (3.123)The fixed point at G c thus plays a central role in the cutoff theory: together with theuniversal scaling exponent ν it determines the correct unique quantum continuumlimit in the presence of an ultraviolet cutoff Λ. Sometimes it can be convenient tomeasure all quantities in units of the cutoff and set Λ = 1/a = 1. In this case thequantity m measured in units of the cutoff (i.e. m/Λ) has to be tuned to zero in orderto construct the lattice continuum limit: for a fixed lattice cutoff, the continuumlimit is approached by tuning the bare lattice G(Λ) to G c . In other words, the latticecontinuum limit has to be taken in the vicinity of the non-trivial ultraviolet point.The discussion given above is not altered significantly, at least in its qualitativeaspects, by the inclusion of the two-loop correction of Eq. (3.110). From the expressionfor the two-loop β-functionμ ∂∂μ G = β(G)=ε G − 2 3 (25 − c)G2 − 20 3 (25 − c)G3 + ... (3.124)for c massless real scalar fields minimally coupled to gravity, one computes the rootsβ(G c )=0 to obtain the location of the ultraviolet fixed point, and from it on canthen determine the universal exponent ν = −1/β ′ (G c ). One findsG c =32(25 − c) ε − 452(25 − c) 2 ε2 + ...ν −1 = ε + 1525 − c ε2 + ... (3.125)which gives, for pure gravity without matter (c = 0) in four dimensions, to lowestorder ν −1 = 2, and ν −1 ≈ 4.4 at the next order. 2Also, in general higher order corrections to the results of the linearized renormalizationgroup equations of Eq. (3.119) are present, which affect the scaling awayfrom the fixed point. Let us assume that close to the ultraviolet fixed point at G c onecan write for the β-function the following expansion2 If one does not expand the solution in ε, one finds from the two-loop result ν −1 = 2d −(1/6)(19+√60d − 95) which gives a smaller value ≈ 2.8ind = 4, as well as rough estimate of the uncertainty.