[Math] The Fock space vs the Hilbert space in the context of quantum field theory

Question

Mathematically the definitions are as follows : if $H_n$ is a $n$-dimensional complex Hilbert space then its two different corresponding “Fock space"(s) are often denoted as $F_{1}$ and $F_{-1}$ defined as, $F_1 = \oplus_{k=0}^{\infty} \mathrm{Sym}^k(H_n)$ and $F_{-1}= \oplus_{k=0}^{\infty} \Lambda^k(H_n)$.

Physically for a quantum field theory one "defines" its so-called Hilbert space as the dual of an implicit vector space over $\mathbb{C}$ whose basis is in bijective correspondence to the set of all possible values for all the classical fields that occur in the underlying Lagrangian.

Now my question is twofold,

• Does this physical notion of a "Hilbert space of a QFT" correspond to the $H_n$ or some “total Fock space" that can be defined from the first mathematical definition as, $\otimes_{i \in \text{Fields}} F^i_{p_i}$ where $p_i=$1 if the $i^{\text{th}}$ field is bosonic or $-1$ if it is fermionic? (..I guess this tensoring is needed because the QFT can have both fermionic as well as bosonic fields..)

• If we agree as above that the states of a QFT live in such a "total Fock space" and not in the Hilbert space defined in the second paragraph then shouldn't the "quantum field" be mapping into a space of Hermitian operators on this total Fock space and not just the Hilbert space?

Answer 1

By saying "the Hilbert space of a QFT" physicists mean the "total Fock space" for all fields (not the $H_n$): $\cal{F} = \bigotimes_{k} \cal{F}_k$, where $\cal{F}_i$ are Fock spaces for separate quantum fields $\Psi_i$. Hence, the corresponding field operators $\Psi_i$ will effectively act only in their corresponding Fock sub-spaces $\cal{F}_i$ (and trivially on the other parts): $\Psi_i = \mathbb{1} \otimes \dotsb \otimes \mathbb{1} \otimes \psi_i \otimes \mathbb{1} \otimes \dotsb \otimes \mathbb{1}$.

This article treats on that question: Back and forth from Fock space to Hilbert space: a guide for commuters, arXiv:1805.04552