The Fock Space

Full Fock Space

Full Fock Space

The full Fock space over a separable Hilbert space \newcommand{\H}{\mathcal{H}}\H is the Hilbert space completion of the tensor algebra of \H: \newcommand{\F}{\mathcal{F}}\F(\H):=\overline{\bigoplus_{n=0}^{\infty}\otimes^{n} \H}.
By definition \otimes^{0}\H is one-dimensional, and spanned by the vacuum vector, denoted $|0⟩$ or $\Omega$.

Fermionic Fock Space

Fermionic Fock Space

The fermionic Fock space over a Hilbert space \H is the Hilbert space defined as the direct sum of the exterior powers of \H: ${\mathcal{F}}_{-}(\stackrel{˝}{)}:={⨁}_{i=0}^{\infty }{\wedge }^i\stackrel{˝}{.}$
In other words, it is the full Fock space $\text{F}(\stackrel{˝}{)}$ modulo the anticommutation relations $x\otimes y=-y\otimes x$ for all x,y\in \H.

Bosonic Fock Space

Bosonic Fock Space

The bosonic Fock space over a separable Hilbert space \H is the Hilbert space completion defined as the direct sum of the symmetric powers of \H: \newcommand{\F}{\mathcal{F}}\F_{+}(\H):=\overline{\bigoplus_{i=0}^{\infty}S^{i} \H},
with inner product $⟨x^{\otimes n},y^{\otimes m}⟩={\delta }_{m,n}n!⟨x,y{⟩}^n.$
In other words, it is the full Fock space modulo the commutation relations $x\otimes y=y\otimes x$ for all x,y\in \H.
On the bosonic Fock space, we can define annihilator $\hat{a}$ and creator ${\hat{a}}^{\ast }$ as follows: ${\hat{a}}_n=\epsilon (n)\frac{\partial }{\partial x_n},{\hat{a}}_n^{\ast }=f\mapsto \epsilon (n{)}^{-1}n{x}_{n}f,$ for all $n>0$ and some $\epsilon :\mathbb{Z}\to \mathbb{R}$. Usually (and here) we set $\epsilon (n)=\sqrt{n}$ for all $n\in \mathbb{Z}$.
Even more formally, we can treat $\hat{a}$ as a map that assigns each x\in\H to an annihilation operator $\hat{a}(x)$, and each x\in \H to a creation operator ${\hat{a}}^{\ast }(x)$, such that $\hat{a}(x)$ is linear in $x$, ${\hat{a}}^{\ast }(x)$ is conjugate-linear in $x$, and they satisfy the canonical commutation relations $[\hat{a}(x),{\hat{a}}^{\ast }(y)]=⟨x,y⟩,[\hat{a}(x),\hat{a}(y)]=0,[{\hat{a}}^{\ast }(x),{\hat{a}}^{\ast }(y)]=0.$ ${\hat{a}}_n$ and ${\hat{a}}_n^{\ast }$ are the annihilation and creation operators associated to some orthonormal basis $\{e_n{\}}_{n\in \mathbb{N}}$ of \H.

Canonical Quantization

Suppose $A\in B(\stackrel{˝}{)}$ is a bounded operator on \H, then the canonical quantization of $A$ is the operator $\Gamma (A)\in B({\text{F}}_{+}(\stackrel{˝}{)})$ defined as $\Gamma (A)(x_1\odot \cdots \odot x_n):=A{x}_1\odot \cdots \odot A{x}_n.$

Segal-Bargmann Space

Let \H be a separable complex Hilbert space. For each finite-dimensional subspace E\subset \H, let $\mathcal{B}(E):=\left\{f:E\to \text{C}\text{ holomorphic }:\Vert f{\Vert }_E^2:={\int }_E|f(z){|}^2\mathrm{d}{\mu }_E(z)<\infty \right\},$
where
$\text{d}{\mu }_E(z):={\pi }^{-\dim E}{e}^{-\Vert z{\Vert }^2}\text{dd}{\lambda }_E(z)$
and $\text{d}{\lambda }_E$ is the Lebesgue measure on $E$.
If $E\subset F$, we identify $\mathcal{B}(E)$ with the closed subspace of $\mathcal{B}(F)$ consisting of functions depending only on the $E$-variable.
Let \mathcal{P}_{\mathrm{cyl}}(\H):=\bigcup_{E\subset \H_+,\ \dim E<\infty}\{\text{holomorphic polynomials on }E\}.
Then the Segal-Bargmann space $\mathcal{B}(\stackrel{˝}{)}$ is the Hilbert completion of \mathcal P_{\mathrm{cyl}}(\H_+) with respect to the norms above.

Proposition

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