Full Fock Space
Full Fock Space
The full Fock space over a Hilbert space
is the Hilbert space defined as the direct sum of tensor powers 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 .
Second Quantization
Suppose
is a bounded operator on , then the second quantization of is the operator defined as
Segal–Bargmann Space
The Segal–Bargmann space is the space of holomorphic functions
Proposition
The bosonic Fock space is isomorphic to the Segal-Bargmann space.