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Wikipedia / 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 1932 paper "Konfigurationsraum und zweite Quantelung" ("Configuration space and second quantization").

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 -particle states are vectors in a symmetrized tensor product of single-particle Hilbert spaces . If the identical particles are fermions, the -particle states are vectors in an antisymmetrized tensor product of single-particle Hilbert spaces (see symmetric algebra and exterior algebra respectively). A general state in Fock space is a linear combination of -particle states, one for each .

Technically, the Fock space is (the Hilbert space completion of) the direct sum of the symmetric or antisymmetric tensors in the tensor powerss of a single-particle Hilbert space

is the operator that 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 (or fermionic) Fock space can alternatively be constructed as (the Hilbert space completion of) the symmetric tensors (or alternating tensors). For every basis of , there is a natural basis of the Fock space, the Fock states.

Definition

The Fock space is the (Hilbert) direct sum of tensor products

, the complex scalars the states of two identical particles etc.

is given by

where

  • is a complex coefficient,

  • is a complex coefficient,

  • is a complex coefficient, etc.

to be the Hilbert space completion of the algebraic direct sum. It consists of all infinite tuples such that the norm i.e., the restriction of the norm on the tensor product

the inner product.

Product states, indistinguishable particles, and a useful basis for Fock space

A product state of the Fock space is a state of the form

is any) is symmetric (resp. antisymmetric) multiplication in the symmetric (antisymmetric) tensor algebra. The general state in a Fock space is a linear combination of product states. A state that cannot be written as a convex sum of product states is called an entangled state.

When we speak of , we must bear in mind that in quantum mechanics identical particles are indistinguishable are equal because the antisymmetric (exterior). 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).

A useful and convenient basis for a Fock space is the occupancy number basis, and no particles in the remaining states, by defining

takes the value 0 or 1 for fermionic particles and 0, 1, 2, ... for bosonic particles. Note that trailing zeroes may be dropped without changing the state. Such a state is called a Fock state are understood as the steady states of a free field, the Fock states describe an assembly of non-interacting particles in definite numbers. The most general Fock state is a linear superposition of pure states.

Two operators of great importance are the creation and annihilation operators; and respectively to annihilate ("remove") a particle, an (even or odd) interior product 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.

Wave function interpretation

, the space of square-integrable functions with measure (strictly speaking, the equivalence classes of square integrable functions where functions are equivalent if they differ on a set of measure zero). The typical example is the free particle the space of square integrable functions on three-dimensional space. The Fock spaces then have a natural interpretation as symmetric or anti-symmetric square integrable functions as follows.

, etc. Consider the space of tuples of points which is the disjoint union

can be identified with the space of anti-symmetric functions. The identification follows directly from the isometric.

, the Slater determinant

are orthonormal. There is a similar "Slater permanent" with the determinant replaced with the permanent-sector of the even Fock space.

Representation theory

In representation theory, Fock spaces can be described as cyclic modules generated by a distinguished vector, the vacuum. Annihilation operators kill the vacuum, while repeated application of creation operators generates the remaining basis vectors. Thus a Fock space may be viewed as a representation space for algebras generated by creation and annihilation operators. For bosons, the symmetric Fock space gives the standard Fock representation of the canonical commutation relations, or equivalently of the Weyl algebra. For fermions, the antisymmetric Fock space gives the corresponding representation of the canonical anticommutation relations, equivalently of a Clifford algebra. The vacuum vector is cyclic in both cases.

In the bosonic case, for a finite-dimensional one-particle space, the canonical commutation relations define a Heisenberg algebra generated by the creation and annihilation operators together with a central element, which acts as the identity in the usual Fock representation. The corresponding Weyl algebra is the quotient of the universal enveloping algebra. Quadratic expressions in the creation and annihilation operators close under commutator and give a representation of the symplectic Lie algebra. At the group level this is the oscillator, or metaplectic representation.

, these two components have highest weights root system on a Cartan subalgebra. Equivalently, in the holomorphic Fock model, the two components may be viewed, depending on the choice of positive roots, as modules generated by the even and odd vacuum sectors.

, with symmetric bilinear form.

Quadratic expressions in the fermionic creation and annihilation operators generate an action of an orthogonal Lie algebra gives the spin representation.

Fock space constructions also occur for infinite-dimensional algebras. Fermionic Fock space can be realized as a semi-infinite exterior algebra and used to construct representations of infinite-rank matrix algebras and affine Lie algebras. Bosonic Fock spaces similarly appear as modules for Heisenberg algebras and in the construction of vertex algebras. The boson–fermion correspondence identifies certain bosonic and fermionic Fock representations and is a central example of the role of Fock space in representation theory.

Relation to the Segal–Bargmann space

Define the Segal–Bargmann space of complex holomorphic functions square-integrable with respect to a Gaussian measure:

, Segal and Bargmann showed

See also

References