The Fock space is an algebraic system (Hilbert space) used in quantum mechanics to describe quantum states with a variable or unknown number of particles.
Technically the Fock space is the Hilbert space made from the direct sum of tensor products of single-particle Hilbert spaces:
An example of a state of the Fock space is
A useful and convenient basis for this space is the occupancy number basis. If |ψi> is a basis of H, then we can agree to denote the state with n0 particles in state |ψ0>, n1 particles in state |ψ1>, ..., nk particles in state |ψk> by
Such a state is called a Fock state. Since |ψi> are understood as the steady states of the free field, i.e., a definite number of particles, a Fock state describes an assembly of non-interacting particles in definite numbers. The most general pure state is the linear superposition of Fock states.
Two operators of paramount importance are the annihilation and creation operators, which upon acting on a Fock state respectively remove and add a particle, in the ascribed quantum state. They are denoted and respectively, with φ referring to the quantum state |φ> in which the particle is removed or added. It is often convenient to work with states of the basis of H so that these operators remove and add exactly one particle in the given state. These operators also serve as a basis for more general operators acting on the Fock space (for instance the operator 'number of particle in state |φ> is ).
WARNING: Fock space only describes noninteracting quantum fields. See Haag's theorem