My understanding is that in classical field theory, we study a classical field $\phi(x,t)$ where for each $x\in\mathbb{R}^3$, $t\in\mathbb{R}$, $\phi(x,t)$ is a scalar. In quantum field theory, we promote each $\phi(x,t)$ to an operator $\hat{\phi}(x,t)$ on an Hilbert space.
My questions are:
- Given a problem, how do we know what that Hilbert space is? Is it some sort of Fock space?
- Does every $\hat{\phi}(x,t)$ act on the same Hilbert space?