Time ordered product vs. commutator in path integral

[kharranger]

Suppose I want to bypass the entire Hamiltonian formulation of quantum field theory and define the theory using a path integral. Thus all I can calculate are Green's function which are time ordered products of local operators. Given only these (no expansions of the field in creation anihilation operators etc.), how can I show that the fields and local operators commute at space-like separation? [\phi(x),\phi(y)]=0, when x,y are spacelike separated?

KH

 

[Jim Kata]

Operator product expansion maybe?

 

[smallphi]

I think you will need extra assumtions since the fields have different order in the two parts of the commutator and the time ordered product will give only one of the parts.

So you will need an assumption connecting the expectations values of phi(x)phi(y) and phi(y)phi(x). Those must depend on x-y and y-x, due to homogeneity of spacetime (unless you have inhomogeneous external source in the theory).