Feynman Path Integral Formulation

Chapter 3Gravity in 2+ε Dimensions3.1 Dimensional ExpansionIn the previous sections it was shown that pure Einstein gravity is not perturbativelyrenormalizable in the traditional sense in four dimensions. To one-loop order higherderivative terms are generated, which, when included in the bare action, lead to potentialunitarity problems, whose proper treatment most likely lies outside the perturbativeregime. The natural question then arises: Are there any other field theorieswhere the standard perurbative treatment fails, yet for which one can find alternativemethods and from them develop consistent predictions? The answer seems unequivocallyyes (Parisi, 1975; 1985). Outside of gravity, there are two notable examplesof field theories, the non-linear sigma model and the self-coupled fermion model,which are not perturbatively renormalizable for d > 2, and yet lead to consistent andin some instances testable predictions above d = 2.The key ingredient to all of these results is, as originally recognized by Wilson,the existence of a non-trivial ultraviolet fixed point, a phase transition in the statisticalfield theory context, with non-trivial universal scaling dimensions (Wilson,1971a,b; Wilson and Fisher, 1972; Wilson, 1973; 1975; Gross, 1976). Furthermore,three quite different theoretical approaches are available for comparing predictions:the 2+ε expansion, the large-N limit, and the lattice approach. Within the lattice approach,several additional techniques are available: the strong coupling expansion,the weak coupling expansion and the numerically exact evaluation of the path integral.Finally, the results for the non-linear sigma model in the scaling regime aroundthe non-trivial ultraviolet fixed point can be compared to high accuracy satelliteexperiments on three-dimensional systems, and the results agree in some cases toseveral decimals.The next three sections will therefore discuss these models from the perspectiveof those results which will have some relevance later for the gravity case. Of particularinterest are predictions for universal corrections to free field behavior, forthe scale dependence of couplings, and the role of the non-perturbative correlationlength which arises in the strong coupling regime.67