---
tag: 1TJ8TNH331
kind: object
name: Fock space
msc: [46N50, 46C05]
instance_of: [HM2HHE3J51]
uses: [EY0X2NKA3S, WJ7PDMGQQW]
sources:
  - url: https://en.wikipedia.org/wiki/Fock_space
same_as:
  wikidata: Q899044
  wikipedia: Fock space
---

## Description

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]].
For $n \ge 1$, let $H^{\otimes n}$ be the [[tensor product|EY0X2NKA3S]] of $n$ copies of $H$, and let $H^{\otimes 0} = \C$.
The *full Fock space* over $H$ is the [[Hilbert sum|WJ7PDMGQQW]]
$$\mathcal{F}(H) = \widehat{\bigoplus}_{n \ge 0} H^{\otimes n}.$$
Let $S_+$ and $S_-$ be the orthogonal projections of $H^{\otimes n}$ onto its symmetric and antisymmetric tensors.
The *bosonic Fock space* is $\mathcal{F}_+(H) = \widehat{\bigoplus}_{n \ge 0} S_+ H^{\otimes n}$, and the *fermionic Fock space* is $\mathcal{F}_-(H) = \widehat{\bigoplus}_{n \ge 0} S_- H^{\otimes n}$.
Each of them is a Hilbert space.
Indeed, each summand is a closed subspace of $H^{\otimes n}$, and a Hilbert sum of Hilbert spaces is a Hilbert space.

## Remarks

The summand of degree $n$ is the space of states with exactly $n$ particles; the vector $1 \in H^{\otimes 0}$ is the vacuum.
For $H = \C$, every symmetric power $S_+ \C^{\otimes n}$ has dimension $1$, so $\mathcal{F}_+(\C)$ is isomorphic to $\ell^2(\N)$.
For the same $H$, the antisymmetric part vanishes for $n \ge 2$, so $\mathcal{F}_-(\C) = \C \oplus \C$; this reflects the Pauli exclusion principle.
Creation and annihilation operators act on Fock space by raising and lowering the degree $n$.

## Explanations

### L1 human

Fock space is the state space for a quantum system in which the number of particles can change.
It stacks the spaces for $0, 1, 2, \ldots$ particles side by side.
For identical bosons one keeps only symmetric states, and for fermions only antisymmetric ones.