A statistical field is often derived by averaging microscopic physics over mesoscopic lengthscales (in a particular way called, coarse graining). This results in a free enegy, $F$, which (when exponentiated) gives the weight factor over which we integrate to get the partition function, $Z$. As the averaging gives a (macroscopic) field (as an approximation to a lattice average), the integral for $Z$ is a Functional Integral, expressable as a Path Integral.

Origin of Universality (Why do many field theories look like each other?)

This free energy can be written as a power series in the field. It turns out that only a few terms (the renormalizable ones, and maybe a few non-renormalizable ones) contribute for a given precission of interest (this is understood via the Renormalization Group).

Thus, the only thing that fundamentally differentiates one theory/model from another are the symmetries of the field, which determine which terms can appear in the free energy. Dimensionality and transformation properties of the field (whether it is a scalar, a vector, a spinor, ...) also play a role.

The microphysics only enters through the parameters of the theory. But as these are often few, they can be and most often are determined experimentally. For this reason statistical field theories are often referred to as phenomenological.

Similar considerations apply in Quantum field theory

The Landau-Ginzbyurg Hamiltonian

Assumptions:

• Locality and uniformity
• Isotropy
• Stability