Feynman Path Integral Formulation

70 3 Gravity in 2 + ε Dimensions∫h∫d d xσ(x) =h d d x[1 − π 2 (x)] 1/2∫= d d x[h − 1 2 hπ2 (x)+...] . (3.9)A proof can be found (David, 1982) that all O(N) invariant Green’s are infraredfinite in the limit h → 0.One can write down the same field theory on a lattice, where it correspondsto the O(N)-symmetric classical Heisenberg model at a finite temperature T ∼ g.The simplest procedure is to introduce a hypercubic lattice of spacing a, with siteslabeled by integers n =(n 1 ...n d ), which introduces an ultraviolet cutoff Λ ∼ π/a.On the lattice field derivatives are replaced by finite differences∂ μ φ(x) → Δ μ φ(n)=and the discretized path integral then reads∫Z[J ]= dφ(n)δ[φ 2 (n) − 1]∏nφ(n + μ) − φ(n)a[× exp − a2−d2g ∑ (Δμ φ(n) ) ]2 + ∑J(x) · φ(x) .n,μn, (3.10)(3.11)The above expression is recognized as the partition function for a ferromagneticO(N)-symmetric spin system at finite temperature. Besides ferromagnets, it can beused to describe systems which are related to it by universality, such as superconductorsand superfluid helium transitions, whose critical behavior is described bya complex phase, and which are therefore directly connected to the plane rotatorN = 2, or U(1), model.In addition the lattice model of Eq. (3.11) provides an explicit regularization forthe continuum theory, which makes expressions like the one in Eq. (3.5) acquire awell defined meaning. It is in fact the only regularization which allows a discussionof the role of the measure in perturbation theory (Zinn-Justin, 2002). At the sametime it provides an ultraviolet regularization for perturbation theory, and allows forvarious non-perturbative calculations, such as power series expansions in three dimensionsand explicit numerical integrations of the path integral via Monte Carlomethods.In two dimensions one can compute the renormalization of the coupling g fromthe action of Eq. (3.6) and one finds after a short calculation (Polyakov, 1975) forsmall g1g(μ) = 1 g + N − 28πln μ2+ ... (3.12)Λ 2where μ is an arbitrary momentum scale. Physically one can view the origin ofthe factor of N − 2 in the fact that there are N − 2 directions in which the spin can