Feynman Path Integral Formulation

96 3 Gravity in 2 + ε Dimensionsβ(G) =− 1 ν (G − G c) − ω (G − G c ) 2 + O[(G − G c ) 3 ] . (3.126)After integrating μ ∂∂μG = β(G) as before, one finds for the structure of the correctionto m [see for comparison Eq. (3.121)]( m) 1/ν []= Am (G(Λ) − G c ) − ων(G(Λ) − G c ) 2 + ... . (3.127)ΛThe hope of course is that these corrections to scaling are small, (ω ≪ 1); inthevicinity of the fixed point the higher order term becomes unimportant when |G −G c |≪1/(ων). For the effective running coupling one then hasG(μ)G c= 1 + a 0( mμ) 1/ν ( m) 2/ν (( m) 3/ν )+ a0 ων + Oμμ, (3.128)which gives an estimate for the size of the modifications to Eq. (3.117).Finally, as a word of caution, one should mention that in general the convergenceproperties of the 2 + ε expansion are not well understood. The poor convergencefound in some better known cases is usually ascribed to the suspected existence ofinfrared renormalon-type singularities ∼ e −c/G close to two dimensions, and whichcould possibly arise in gravity as well. At the quantitative level, the results of the2 + ε expansion for gravity therefore remain somewhat limited, and obtaining thethree- or four-loop term still represents a daunting task. Nevertheless they provide,through Eqs. (3.117) and (3.121), an analytical insight into the scaling propertiesof quantum gravity close and above two dimensions, including the suggestion ofa non-trivial phase structure and an estimate for the non-trivial universal scalingexponents [Eq. (3.125)]. The key question raised by the perturbative calculationsis therefore: what remains of the above phase transition in four dimensions, howare the two phases of gravity characterized there non-perturbatively, and what is thevalue of the exponent ν determining the running of G in the vicinity of the fixedpoint in four dimensions.Finally we should mention that there are other continuum renormalization groupmethods which can be used to estimate the scaling exponents. An approach whichis closely related to the 2 + ε expansion for gravity is the derivation of approximateflow equations from the changes of the Legendre effective action with respect to asuitably introduced infrared cutoff μ. The method can be regarded as a variation onWilson’s original momentum slicing technique for obtaining approximate renormalizationgroup equations for lattice couplings. In the simplest case of a scalar fieldtheory (Morris, 1994a,b) one starts from the partition function∫exp(W[J]) = [dφ] exp { − 1 2 φ ·C−1 · φ − I Λ [φ]+J · φ } . (3.129)The C ≡ C(k, μ) term is taken to be an “additive infrared cutoff term”. For it to bean infrared cutoff it needs to be small for k < μ, ideally tending to zero as k → 0,and such that k 2 C(k, μ) is large when k > μ. Since the method is only ultimatelyapplied to the vicinity of the fixed point, for which all physical relevant scales are