Feynman Path Integral Formulation

72 3 Gravity in 2 + ε DimensionsΛ ∂ g∂Λ= β(g) =ε g −N − 22π g2 + O ( g 3 ,εg 2) . (3.14)The above β-function determines the scale dependence (at least in perturbation theory)of g for an arbitrary scale, which from now on will be denoted as μ. Then thedifferential equation μ ∂g∂μ= β[g(μ)] uniquely determines how g(μ) flows as a functionof momentum scale μ. The scale dependence of g(μ) is such that if the initialg is less than the ultraviolet fixed point value g c , withg c =2πε + ... (3.15)N − 2then the coupling will flow towards the Gaussian fixed point at g = 0. The new phasethat appears when ε > 0 and corresponds to a low temperature, spontaneously brokenphase with finite order parameter. On the other hand if g > g c then the couplingg(μ) flows towards increasingly strong coupling, and eventually out of reach of perturbationtheory. In two dimensions the β-function has no zero and only the strongcoupling phase is present.The simplest way of obtaining explicitly the renormalization group behavior ofthe coupling g is as follows. One parametrises, for N > 2, the original N-componentfield φ, which is constrained by φ 2 = 1, in terms of two fields θ and τ, defined by√φ 1 = 1 − τ 2 cosθ√φ 2 = 1 − τ 2 sinθφ j = τ j j = 3...N . (3.16)In this new set of variables, the original action for the non-linear sigma model becomes∫√S = 1 2d d x[(1 ] − τ 2 )(∂ μ θ) 2 +(∂ μ 1 − τ 2 ) 2 +(∂ μ τ) 2 . (3.17)The θ variable now enters the action in a very simple way, through a quadratic term.Since the action now contains as unconstrained variables θ and τ, one can nowthink, at least for sufficiently small coupling g (low temperatures), of integratingfirst over the τ variables, after rescaling τ → √ gτ. The result is an effective actionfor the θ variables, and to lowest order the overall effect is to replace in the firstterm τ 2 by its average 〈τ 2 〉. Since τ is to lowest order a free field, one computes〈 τ i (x)τ j (y) 〉 = gΛ 2−d δ ij∫d d k e ik·(x−y)(2π) d k 2 . (3.18)After evaluating the k integral, taking the limit x → y, and doing the trace over theindex i = 3...N one obtains for d > 2〈τ 2 g〉 =(N − 2)2πε + O(g2 ) . (3.19)