The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space H. It is named after V. A. Fock who first introduced it in his 1932 paper "Konfigurationsraum und zweite Quantelung" ("Configuration space and second quantization").^[1][2]

Informally, a Fock space is the sum of a set of Hilbert spaces representing zero particle states, one particle states, two particle states, and so on. If the identical particles are bosons, the n-particle states are vectors in a symmetrized tensor product of n single-particle Hilbert spaces H. If the identical particles are fermions, the n-particle states are vectors in an antisymmetrized tensor product of n single-particle Hilbert spaces H (see symmetric algebra and exterior algebra respectively). A general state in Fock space is a linear combination of n-particle states, one for each n.

Technically, the Fock space is (the Hilbert space completion of) the direct sum of the symmetric or antisymmetric tensors in the tensor powers of a single-particle Hilbert space H,

$${\displaystyle F_{\nu }(H)={\overline {\bigoplus _{n=0}^{\infty }S_{\nu }H^{\otimes n}}}~.}$$

Here ${\displaystyle S_{\nu }}$ is the operator which symmetrizes or antisymmetrizes a tensor, depending on whether the Hilbert space describes particles obeying bosonic ${\displaystyle (\nu =+)}$ or fermionic ${\displaystyle (\nu =-)}$ statistics, and the overline represents the completion of the space. The bosonic (resp. fermionic) Fock space can alternatively be constructed as (the Hilbert space completion of) the symmetric tensors ${\displaystyle F_{+}(H)={\overline {S^{*}H}}}$ (resp. alternating tensors ${\textstyle F_{-}(H)={\overline {{\bigwedge }^{*}H}}}$). For every basis for H there is a natural basis of the Fock space, the Fock states.

Definition

The Fock space is the (Hilbert) direct sum of tensor products of copies of a single-particle Hilbert space ${\displaystyle H}$

$${\displaystyle F_{\nu }(H)=\bigoplus _{n=0}^{\infty }S_{\nu }H^{\otimes n}=\mathbb {C} \oplus H\oplus \left(S_{\nu }\left(H\otimes H\right)\right)\oplus \left(S_{\nu }\left(H\otimes H\otimes H\right)\right)\oplus \cdots }$$

Here ${\displaystyle \mathbb {C} }$, the complex scalars, consists of the states corresponding to no particles, ${\displaystyle H}$ the states of one particle, ${\displaystyle S_{\nu }(H\otimes H)}$ the states of two identical particles etc.

A general state in ${\displaystyle F_{\nu }(H)}$ is given by

$${\displaystyle |\Psi \rangle _{\nu }=|\Psi _{0}\rangle _{\nu }\oplus |\Psi _{1}\rangle _{\nu }\oplus |\Psi _{2}\rangle _{\nu }\oplus \cdots =a|0\rangle \oplus \sum _{i}a_{i}|\psi _{i}\rangle \oplus \sum _{ij}a_{ij}|\psi _{i},\psi _{j}\rangle _{\nu }\oplus \cdots }$$

• ${\displaystyle |0\rangle }$ is a vector of length 1 called the vacuum state and ${\displaystyle a\in \mathbb {C} }$ is a complex coefficient,
• ${\displaystyle |\psi _{i}\rangle \in H}$ is a state in the single particle Hilbert space and ${\displaystyle a_{i}\in \mathbb {C} }$ is a complex coefficient,
• ${\textstyle |\psi _{i},\psi _{j}\rangle _{\nu }=a_{ij}|\psi _{i}\rangle \otimes |\psi _{j}\rangle +a_{ji}|\psi _{j}\rangle \otimes |\psi _{i}\rangle \in S_{\nu }(H\otimes H)}$, and ${\displaystyle a_{ij}=\nu a_{ji}\in \mathbb {C} }$ is a complex coefficient, etc.

The convergence of this infinite sum is important if ${\displaystyle F_{\nu }(H)}$ is to be a Hilbert space. Technically we require ${\displaystyle F_{\nu }(H)}$ to be the Hilbert space completion of the algebraic direct sum. It consists of all infinite tuples ${\displaystyle |\Psi \rangle _{\nu }=(|\Psi _{0}\rangle _{\nu },|\Psi _{1}\rangle _{\nu },|\Psi _{2}\rangle _{\nu },\ldots )}$ such that the norm, defined by the inner product is finite