# The Fock Space vs the Hilbert space in the context of Quantum Field Theory

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, $F_1 = \oplus_{k=0}^{\infty} Sym^k(H_n)$ 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 two fold,

• 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, $\otimes_{i \in Fields} F^i_{p_i}$ where $p_i=$1 if the $i^{th}$ 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?