Fock space

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.

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. 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,

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 ${\displaystyle F_{-}(H)={\overline {{\bigwedge }^{*}H}}}$). For every basis for H there is a natural basis of the Fock space, the Fock states.

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