Fock space

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. It is named for V. A. Fock.

Technically the Fock space is the Hilbert space made from the direct sum of tensor products of single-particle Hilbert spaces:

<math>F_\nu(H)=\bigoplus_{n=0}^{\infty}S_\nu H^{\otimes n}<math>

where S_ν is the operator which symmetrize or antisymmetrize the space, whereby providing the Fock space describing 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.)

An example of a state of the Fock space is

<math>|\Psi\rangle_\nu=|\phi_1,\phi_2,\cdots,\phi_n\rangle_\nu<math>

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 feature 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.

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

<math>|n_0,n_1,\cdots,n_k\rangle_\nu<math>

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.

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 <math>a(\phi)<math> and <math>a^{\dagger}(\phi)<math> 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 <math>a^{\dagger}(\phi)a(\phi)<math>).

WARNING: Fock space only describes noninteracting quantum fields. See Haag's theorem. However, if the model has a mass gap, the asymptotic past and asymptotic future states can be described by a Fock space. Fock space does not describe finite temperature physics as well.

While Fock space is appropriate for free massive particles with finite energy (i.e. zero temperature) because for a collection of these particles, finite total energy and finite particle number mean the same thing, it is no longer appropriate for massless particles because an infinite number of them can still have finite energy. See soft photonsl:Fokov prostor