The Fock Space

Full Fock Space

Full Fock Space

The full Fock space over a separable Hilbert space is the Hilbert space completion of the tensor algebra of : By definition is one-dimensional, and spanned by the vacuum vector, denoted or .

Fermionic Fock Space

Fermionic Fock Space

The fermionic Fock space over a Hilbert space is the Hilbert space defined as the direct sum of the exterior powers of : In other words, it is the full Fock space modulo the anticommutation relations for all .

Bosonic Fock Space

Bosonic Fock Space

The bosonic Fock space over a separable Hilbert space is the Hilbert space completion defined as the direct sum of the symmetric powers of : with inner product In other words, it is the full Fock space modulo the commutation relations for all . On the bosonic Fock space, we can define annihilator and creator as follows: for all and some . Usually (and here) we set for all . Even more formally, we can treat as a map that assigns each to an annihilation operator , and each to a creation operator , such that is linear in , is conjugate-linear in , and they satisfy the canonical commutation relations and are the annihilation and creation operators associated to some orthonormal basis of .

Canonical Quantization

Suppose is a bounded operator on , then the canonical quantization of is the operator defined as

Segal-Bargmann Space

Let be a separable complex Hilbert space. For each finite-dimensional subspace , let where
and is the Lebesgue measure on .
If , we identify with the closed subspace of consisting of functions depending only on the -variable.
Let Then the Segal-Bargmann space is the Hilbert completion of with respect to the norms above.

Proposition

The bosonic Fock space is isomorphic to the Segal-Bargmann space.