Feynman Path Integral Formulation

68 3 Gravity in 2 + ε Dimensions
Later sections will then discuss the 2 + ε expansion for gravity, and what can
be learned from it by comparing it to the analogous expansion in the non-linear
sigma model. The similarity between the two models is such that they both exhibit
a non-trivial ultraviolet fixed point, a two-phase structure, non-trivial exponents and
scale-dependent couplings.
3.2 Perturbatively Non-renormalizable Theories:
The Sigma Model
The O(N)-symmetric non-linear σ-model provides an instructive and rich example
of a theory which, above two dimensions, is not perturbatively renormalizable
in the traditional sense, and yet can be studied in a controlled way in the context
of Wilson’s 2 + ε expansion. Such framework provides a consistent way to calculate
nontrivial scaling properties of the theory in those dimensions where it is not
perturbatively renormalizable (for example d = 3 and d = 4), which can then be
compared to non-perturbative results based on the lattice theory, as well as to experiments,
since in d = 3 the model describes either a ferromagnet or superfluid helium
in the vicinity of its critical point. In addition, the model can be solved exactly in the
large N limit for any d, without any reliance on the 2+ε expansion. Remarkably, in
all three approaches it exhibits a non-trivial ultraviolet fixed point at some coupling
g c (a phase transition in statistical mechanics language), separating a weak coupling
massless ordered phase from a massive strong coupling phase.
The non-linear σ-model is described by an N-component field φ a satisfying a
unit constraint φ 2 (x)=1, with functional integral given by
∫
Z[J] = [dφ ] ∏ δ [φ(x) · φ(x) − 1]
× exp
(− Λ d−2
g
x
∫
S(φ)+
)
d d xJ(x) · φ(x)
.
(3.1)
The action is taken to be O(N)-invariant
∫
S(φ) = 1 2
d d x ∂ μ φ(x) · ∂ μ φ(x) . (3.2)
Λ here is the ultraviolet cutoff and g the bare dimensionless coupling at the cutoff
scale Λ; in a statistical field theory context g plays the role of a temperature.
In perturbation theory one can eliminate one φ field by introducing a convenient
parametrization for the unit sphere, φ(x)={σ(x),π(x)} where π a is an N −1-
component field, and then solving locally for σ(x)
σ(x) =[1 − π 2 (x)] 1/2 . (3.3)