The Fock space is an algebraic system (Hilbert space) used in quantum mechanics to describe quantum states with a variable or unknown number of identical particles. It is named for V. A. Fock. Algebra is a branch of mathematics which studies structure and quantity. ... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... Fig. ... A quantum state is any possible state in which a quantum mechanical system can be. ... In physics, a particle is an object, or body, with only a few degrees-of-freedom, including position, and perhaps orientation in space. ... Vladimir Aleksandrovich Fock (or Fok, Владимир Александрович Фок) (22 December 1898 - December 27, 1974) was a Soviet physicist, who did foundational work on quantum mechanics. ...

Technically the Fock space is the Hilbert space made from the direct sum of tensor products of single-particle Hilbert spaces: In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...

where S_ν is the operator which symmetrize or antisymmetrize the space, whereby providing the Fock space describing identical particles obeying bosonic (ν=+) or fermionic (ν=-) algebra respectively. H is the single particle Hilbert space. It describes the quantum states for a single particle, and to describes the quantum states of systems with n particles, or superpositions of such states, one must use a larger Hilbert space, the Fock space, which contains states for unlimited and variable number of particles. Fock states are the natural basis of this space. (See also the Slater determinant.) Bosons, named after Satyendra Nath Bose, are particles which form totally-symmetric composite quantum states. ... Fermions, named after Enrico Fermi, are particles which form totally-antisymmetric composite quantum states. ... A quantum state is any possible state in which a quantum mechanical system can be. ... A Fock state, in quantum mechanics, is any state of the Fock space with a well-defined number of particles in each state. ... A Slater determinant (named after the physicist John C. Slater) is an expression in quantum mechanics for the wavefunction of a many-fermion system, which by construction satisfies the Pauli principle. ...

Example

An example of a state of the Fock space is

describing n particles, one of which has wavefunction φ₁, another φ₂ and so on up to the n^th particle, where each φ_i is any wavefunction from the single particle Hilbert space H. When we speak of one particle in state φ_i it must be born in mind that in quantum mechanics identical particles are indistinguishable, and in a same Fock space all particles are identical (to describe many species of particles, made the tensor products of as many different Fock spaces). It is one of the most powerful features of this formalism that states are intrinsically properly symmetrized. So that for instance, if the above state |Ψ>_- is fermionic, it will be 0 if two (or more) of the φ_i are equal, because by the Pauli exclusion principle no two (or more) fermions can be in the same quantum state. Also, the states are properly normalized, by construction. In quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued function ψ defined over a portion of space and normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude squared of the wavefunction |ψ(x)|2 is the... Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. ... The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925, which states that no two identical fermions may occupy the same quantum state. ...

A useful and convenient basis for this space is the occupancy number basis. If |ψ_i> is a basis of H, then we can agree to denote the state with n₀ particles in state |ψ₀>, n₁ particles in state |ψ₁>, ..., n_k particles in state |ψ_k> by

with of course if ν=-, each n_i taking only the value 0 or 1 (otherwise the state is zero).

Such a state is called a Fock state. Since |ψ_i> 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. A Fock state, in quantum mechanics, is any state of the Fock space with a well-defined number of particles in each state. ...

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 a(φ) 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 ). Quantum field theory (QFT) is the application of quantum mechanics to fields. ...