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").
[1][2] 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,
is the operator that symmetrizes or antisymmetrizes a tensor, depending on whether the Hilbert space describes particles obeying bosonic
statistics, and the overline represents the completion of the space.
fermionic) Fock space can alternatively be constructed as (the Hilbert space completion of) the symmetric tensors
The Fock space is the (Hilbert) direct sum of tensor products of copies of a single-particle Hilbert space
, the complex scalars, consists of the states corresponding to no particles,
to be the Hilbert space completion of the algebraic direct sum.
such that the norm, defined by the inner product is finite
is any state from the single particle Hilbert space
, we must bear in mind that in quantum mechanics identical particles are indistinguishable.
(To describe many species of particles, we take the tensor product of as many different Fock spaces as there are species of particles under consideration).
It is one of the most powerful features of this formalism that states are implicitly properly symmetrized.
This is a mathematical formulation of the Pauli exclusion principle that no two (or more) fermions can be in the same quantum state.
In fact, whenever the terms in a formal product are linearly dependent; the product will be zero for antisymmetric tensors.
Also, the product of orthonormal states is properly orthonormal by construction (although possibly 0 in the Fermi case when two states are equal).
Note that trailing zeroes may be dropped without changing the state.
are understood as the steady states of a free field, the Fock states describe an assembly of non-interacting particles in definite numbers.
Two operators of great importance are the creation and annihilation operators, which upon acting on a Fock state add or respectively remove a particle in the ascribed quantum state.
To create ("add") a particle, the quantum state
; and respectively to annihilate ("remove") a particle, an (even or odd) interior product is taken with
so that these operators remove and add exactly one particle in the given basis state.
These operators also serve as generators for more general operators acting on the Fock space, for instance the number operator giving the number of particles in a specific state
(strictly speaking, the equivalence classes of square integrable functions where functions are equivalent if they differ on a set of measure zero).
The Fock spaces then have a natural interpretation as symmetric or anti-symmetric square integrable functions as follows.
Consider the space of tuples of points which is the disjoint union
The identification follows directly from the isometric mapping
[3] of complex holomorphic functions square-integrable with respect to a Gaussian measure: