Feynman Path Integral Formulation

Chapter 3
Gravity in 2+ε Dimensions
3.1 Dimensional Expansion
In the previous sections it was shown that pure Einstein gravity is not perturbatively
renormalizable in the traditional sense in four dimensions. To one-loop order higher
derivative terms are generated, which, when included in the bare action, lead to potential
unitarity problems, whose proper treatment most likely lies outside the perturbative
regime. The natural question then arises: Are there any other field theories
where the standard perurbative treatment fails, yet for which one can find alternative
methods and from them develop consistent predictions The answer seems unequivocally
yes (Parisi, 1975; 1985). Outside of gravity, there are two notable examples
of 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 and
in 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 statistical
field 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 around
the non-trivial ultraviolet fixed point can be compared to high accuracy satellite
experiments on three-dimensional systems, and the results agree in some cases to
several decimals.
The next three sections will therefore discuss these models from the perspective
of those results which will have some relevance later for the gravity case. Of particular
interest are predictions for universal corrections to free field behavior, for
the scale dependence of couplings, and the role of the non-perturbative correlation
length which arises in the strong coupling regime.
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