Local degrees of freedom and correlator

[spaghetti3451]

In quantum field theory, the degrees of freedom $\phi({\bf{x}},t)$ are local. This means that the the dynamics of the field in a given region of spacetime is not governed by events outside its lightcone.Is the local/non-local nature of degrees of freedom in a quantum field theory independent of the second quantisation of the classical fields in the theory to quantum fields?

Is the local/non-local nature of degrees of freedom in a quantum field theory independent of the transition from, for example, the time-ordered vacuum expectation value

$\langle \Omega | T\{q(t_{1})\cdots q(t_{n})\}|\Omega\rangle = \frac{\int\mathcal{D}q(t)\ e^{iS[q]}q(t_{1})\cdots q(t_{n})}{\int \mathcal{D}q(t)\ e^{iS[q]}}$

in quantum mechanics to

$\langle \Omega | T\{\phi(x_{1})\cdots q(x_{n})\}|\Omega\rangle = \frac{\int\mathcal{D}\phi(x)\ e^{iS[\phi]}\phi(x_{1})\cdots \phi(x_{n})}{\int \mathcal{D}\phi(x)\ e^{iS[x]}}$

in quantum field theory?

Does the local nature of a field theory manifest itself only in the action of the theory?

 

[eljose79]

I can say that the local/non-local nature of degrees of freedom in a quantum field theory is not necessarily dependent on the second quantization of classical fields or on the transition from time-ordered vacuum expectation values in quantum mechanics to those in quantum field theory. The local nature of a field theory is inherent in the very definition of a field, which is a quantity that varies continuously over space and time. This means that the dynamics of the field at a particular point in space and time is only influenced by events within its immediate vicinity, and not by events outside its lightcone.

However, the specific manifestation of this local nature in a field theory may depend on the specific formulation or mathematical framework used to describe the theory. For example, in the path integral formulation of quantum field theory, the local nature of degrees of freedom is reflected in the fact that the action is only dependent on the field values at a particular point in space and time, and not on the field values at all points in the entire spacetime.

In summary, the local nature of degrees of freedom in a quantum field theory is a fundamental aspect of the theory and is not necessarily dependent on the specific mathematical framework used to describe it.