# What is the Hilbert space in quantum field theory?

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:

1. Given a problem, how do we know what that Hilbert space is? Is it some sort of Fock space?
2. Does every $$\hat{\phi}(x,t)$$ act on the same Hilbert space?

Quantum fields (as defined by the Wightman axioms) are operator-valued distributions. We must smear them with a test function $$f$$ (usually a Schwartz space function) to obtain an (in general unbounded) operator on a Hilbert space $$\mathcal{H}$$: $$\phi(f) := \int_{\mathbb{R}^{d+1}} f(x,t) \phi(x,t) \ \mathrm{d} x \mathrm{d} t.$$ The Hilbert space $$\mathcal{H}$$ is part of the data that defines a QFT model. This Hilbert space does not need to be a Fock space.
To answer your second question, the Wightman axiom W1 in the linked Wikipedia entry demands that a dense subspace $$D \subset \mathcal{H}$$ exists such that, for each test function $$f$$, the smeared quantum field $$\phi(f)$$ is an operator with domain $$D$$. Thus, the smeared quantum fields act on the same Hilbert space with common dense domain $$D$$.
• What do you mean by: "$\mathcal{H}$ is usually a Fock space"? As far as I recall, a Fock space is the infinite direct sum of direct sums of Hilbert spaces. Apr 16 at 13:01
• A Fock space $\mathcal{F}$ is an infinite direct sum of tensor products of a (one-particle) Hilbert space $\mathfrak{h}$, i.e. $\mathcal{F} = \bigoplus_{n\in\mathbb{N}} \mathfrak{h}^{\otimes n}$, and therefore $\mathcal{F}$ is also a Hilbert space. Apr 16 at 20:27