In both statistical field theory and quantum field theory one computes average values / time ordered expectation values of functionals of fields with the path integral. I have two related questions:
Is it the path integral alone that make quantum fields non-commutative or are other constraints also required?
Are statistical fields non-commutative?
1) It's exactly the opposite of what you said - in the Hamiltonian operator formalism, the fields are non-commutative; the Lagrangian path integral is an equivalent reformulation of the problem in which the fields are commutative. Field ordering doesn't matter in the path integral; the original "non-commuting-ness" effectively gets absorbed into the integration measure.
2) I guess it depends on exactly what you mean by "statistical fields," but nope, anything that goes into a path integral is commutative.
