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 . It is named after V. A. Fock who first introduced it in his paper Konfigurationsraum und zweite Quantelung. 

Informally, a Fock space consists of a set of Hilbert spaces representing a zero particle state, a one particle state, a two particle state, and so on. If the identical particles are bosons, the -particle state is a symmetrized tensor product of  single particle Hilbert spaces. If the identical particles are fermions, the -particle state is an antisymmetrized tensor product of  single particle Hilbert spaces . A state in Fock space is a linear combination of states, where each state has a definite number of particles.

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  is the operator which symmetrizes or antisymmetrizes a tensor, depending on whether the Hilbert space describes particles obeying bosonic  or fermionic  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  (resp. alternating tensors ). For every basis for  there is a natural basis of the Fock space, the Fock states.