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" ("Configuration space and second quantization"). 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, F ν ( H ) = ⨁ n = 0 ∞ S ν H ⊗ n ¯ . {\displaystyle F_{\nu }(H)={\overline {\bigoplus _{n=0}^{\infty }S_{\nu }H^{\otimes n}}}~.} Here S ν {\displaystyle S_{\nu }} is the operator that symmetrizes or antisymmetrizes a tensor, depending on whether the Hilbert space describes particles obeying bosonic ( ν = + ) {\displaystyle (\nu =+)} or fermionic ( ν = − ) {\displaystyle (\nu =-)} statistics, and the overline represents the completion of the space. The bosonic (or fermionic) Fock space can alternatively be constructed as (the Hilbert space completion of) the symmetric tensors F + ( H ) = S ∗ H ¯ {\displaystyle F_{+}(H)={\overline {S^{*}H}}} (or alternating tensors F − ( H ) =...