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 S ν {displaystyle S_{ u }} is the operator which symmetrizes or antisymmetrizes a tensor, depending on whether the Hilbert space describes particles obeying bosonic ( ν = + ) {displaystyle ( u =+)} or fermionic ( ν = − ) {displaystyle ( u =-)} 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 F + ( H ) = S ∗ H ¯ {displaystyle F_{+}(H)={overline {S^{*}H}}} (resp. alternating tensors F − ( H ) = ⋀ ∗ H ¯ {displaystyle F_{-}(H)={overline {{igwedge }^{*}H}}} ). For every basis for H there is a natural basis of the Fock space, the Fock states. Fock space is the (Hilbert) direct sum of tensor products of copies of a single-particle Hilbert space H {displaystyle H} Here C {displaystyle mathbb {C} } , the complex scalars, consists of the states corresponding to no particles, H {displaystyle H} the states of one particle, S ν ( H ⊗ H ) {displaystyle S_{ u }(Hotimes H)} the states of two identical particles etc. A typical state in F ν ( H ) {displaystyle F_{ u }(H)} is given by