Heisenberg Algebra and Fock Models

The Heisenberg Algebra

Heisenberg (Oscillator) Algebra

The Heisenberg algebra or Oscillator Algebra $\mathfrak{H}$ is a complex Lie algebra with a basis \newcommand{\R}{\mathbb{R}}\newcommand{\Z}{\mathbb{Z}}\{\hbar 1_{\Hei}\}\cup\{a_{n}\}_{n\in\Z} and Lie bracket defined by $[\hslash 1_{\text{Hei}},a_n]=0,[a_m,a_n]=m{\delta }_{m,-n}\hslash 1_{\text{Hei}}.$
Here $\hslash$ is the reduced Planck constant, and is often set to $1\in \mathbb{C}$ in mathematical treatments.

The Heisenberg algebra exists by the following construction:

Proposition

A central extension of the abelian algebra of trigonometric polynomials on the circle, i.e., functions $S^1\to \text{C}$ with finite Fourier series, forms a Heisenberg algebra through the Lie bracket $[f,g]:=\left(\frac{1}{2\pi }{\int }_0^{2\pi }{f}^{\prime }(t)g(t)\mathrm{d}t\right)\cdot z,$
where $z$ is a central element.

Proof Let us denote the standard basis elements as $e_n=x\mapsto x^n$. Then $[e_m,e_n]=\frac{1}{2\pi }{\int }_0^{2\pi }{e}^{i(m+n)t}im\text{dd}t\cdot z=im{\delta }_{m,-n}z.$Setting $a_n\mapsto e_n$ and $1_{\text{Hei}}\mapsto z$ gives the desired isomorphism. $\square$

Fock Space Representation of $\text{Hei}$

Introduce the bosonic Fock space $\mathcal{F}=\text{C}[x_1,x_2,\dots ]$, the space of polynomials in infinitely many variables. Given $\lambda \in \text{C}$, define the representation $({\rho }_{\lambda },\text{F})$ of $\text{Hei}$ through ${\rho }_{\lambda }(a_n)=\epsilon (n)\frac{\partial }{\partial x_n},{\rho }_{\lambda }(a_{-n})=f\mapsto \epsilon (n{)}^{-1}n{x}_{n}f,{\rho }_{\lambda }(a_0)=\lambda I,{\rho }_{\lambda }(1_{\text{Hei}})=I,$for all $n>0$ and some $\epsilon :\mathbb{Z}\to \mathbb{R}$. Usually (and here) we set $\epsilon (n)=n$ for all $n\in \mathbb{Z}$.

Proposition

The representation $({\rho }_{\lambda },\text{F})$ is irreducible for all $\lambda \in \mathbb{R}$ and $\epsilon :\mathbb{Z}\to \mathbb{R}$ such that $\epsilon (n)\ne 0$ for all $n\in \mathbb{Z}$.

Proof Any polynomial in $\text{F}$ can be reduced to a constant by applying ${\rho }_{\lambda }(a_n)$ for sufficiently large $n$. Hence, every nonzero subrepresentation must contain the constants. By repeatedly applying ${\rho }_{\lambda }(a_{-n})$​, one generates all other polynomials, showing that the subrepresentation is the entire space $\text{F}$. $\square$

Oscillator Representation of $\text{vir}$

$\text{vir}$ Subalgebra in $U(\text{Hei})$

In $U(\text{Hei})$, we can define a subalgebra $\mathfrak{v}$ which is spanned by $d_n:=\frac{1}{2}{\sum }_{m\in \mathbb{Z}}:a_{n-m}{a}_m:$where $:a_m{a}_n:=a_m{a}_n$ if $m\le n$ and $:a_m{a}_n:=a_n{a}_m$ otherwise (this is called the normal ordering).
Then this subalgebra is isomorphic to the Virasoro algebra $\text{vir}$ through the map $\theta :\text{vir}\to \mathfrak{v}$, $L_n\mapsto d_n$ and $\kappa \mapsto 1_{\text{Hei}}$.

Oscillator Representation of $\text{vir}$

We now define a family of representations $({\phi }_{\lambda ,\mu },\text{F})$ of $\text{vir}$ for $\lambda ,\mu \in \text{C}$ by
${\phi }_{\lambda ,\mu }(L_n):=({\rho }_{\lambda }\circ \theta )(L_n)+i\mu n{\rho }_{\lambda }(a_n)+\frac{{\mu }^2}{2}{\delta }_{n,0}I,{\phi }_{\lambda ,\mu }(\kappa )=I,$for all $n\in \mathbb{Z}$. This is called the oscillator representation of $\text{vir}$.

Proposition

The operators $\{{\phi }_{\lambda ,\mu }(L_n){\}}_{n\in \mathbb{Z}}$ satisfy

1. ${\phi }_{\lambda ,\mu }(L_0)\cdot 1_{\text{F}}=\frac{1}{2}({\lambda }^2+{\mu }^2)$;
2. $[{\phi }_{\lambda ,\mu }(L_n),{\phi }_{\lambda ,\mu }(L_m)]=(n-m){\phi }_{\lambda ,\mu }(L_{n+m})+{\delta }_{m,-n}\frac{n^3-n}{12}(1+12{\mu }^2)I$;
3. If $f,g\in \text{F}$ and $x\in \text{vir}$, then $⟨{\phi }_{\lambda ,\mu }(x)f,g⟩=⟨f,{\phi }_{\bar{\lambda },\bar{\mu }}(x^{\ast })g⟩$.

Therefore $({\phi }_{\lambda ,\mu },\text{F})$ is a highest weight representation of $\text{vir}$ with central charge $1+12{\mu }^2$ and highest weight $\frac{1}{2}({\lambda }^2+{\mu }^2)$. Moreover, it is unitary if $\lambda ,\mu \in \mathbb{R}$.

Proof

1. $\begin{array}{rl}{\phi }_{\lambda ,\mu }(L_0)\cdot 1& ={\rho }_{\lambda }\left(\frac{1}{2}\sum _{j\in \mathbb{Z}}:a_{-j}{a}_j:\right)\cdot 1+\frac{1}{2}{\mu }^2\\ & =\sum _{j\in {\mathbb{Z}}_{{\ge }_0}}{\rho }_{\lambda }(a_{-j}{a}_j)\cdot 1+\frac{1}{2}{\rho }_{\lambda }(a_0{a}_0)\cdot 1+\frac{1}{2}{\mu }^2\\ & =\sum _{j\in {\mathbb{Z}}_{{\ge }_0}}{x}_{j}n\text{pddf}{x}_j\cdot 1+\frac{1}{2}{\lambda }^2+\frac{1}{2}{\mu }^2\\ & =\frac{1}{2}({\lambda }^2+{\mu }^2).\end{array}$
2. $\begin{array}{rl}[][{\phi }_{\lambda ,\mu }(L_n),{\phi }_{\lambda ,\mu }(L_m)]& =[{\rho }_{\lambda }(d_n),{\rho }_{\lambda }(d_m)]+im\mu [{\rho }_{\lambda }(d_n),{\rho }_{\lambda }(a_m)]\\ & +in\mu [{\rho }_{\lambda }(a_n),{\rho }_{\lambda }(d_m)]-mn{\mu }^2[{\rho }_{\lambda }(a_n),{\rho }_{\lambda }(a_m)]\\ & =(n-m){\rho }_{\lambda }(d_{n+m})+{\delta }_{m,-n}\frac{n^3-n}{12}I\\ & +i\mu (n^2-m^2){\rho }_{\lambda }(a_{m+n})-mn{\mu }^{2}n{\delta }_{m,-n}I\\ & =(n-m){\phi }_{\lambda ,\mu }(L_{n+m})+{\delta }_{m,-n}\frac{n^3-n}{12}(1+12{\mu }^2)I.\end{array}$

$\square$

Corollary

If $c\ge 1$ and $h\ge (c-1)/24$, then the irreducible highest weight representation $L(h,c)$ of $\text{vir}$ is unitarizable. In general, if $c\ge n$ and $h\ge (c-n)/24$ for some $n\in {\mathbb{Z}}_{>0}$, then $L(h,c)$ is unitarizable.

Proof Since $L(nh,nc)\cong L(h,c{)}^{\otimes n}$ is unitarizable for all $n\in {\mathbb{Z}}_{>0}$, given $nc=c^{\prime }\ge n$ and $nh=h^{\prime }\ge (c^{\prime }-n)/24$, let $\mu =\sqrt{(c^{\prime }-n)/12n}$ and $\lambda =\sqrt{2h^{\prime }/n-{\mu }^2}$. Then $\lambda ,\mu \in \mathbb{R}$ and $({\phi }_{\lambda ,\mu },\text{F})\cong L(h^{\prime },c^{\prime })$ is a unitary highest weight representation of $\text{vir}$ with central charge $nc$ and highest weight $nh$. $\square$

$L(h,c)$ is also unitarizable for $c\ge 1$ and $0\le h<(c-1)/24$, but we do not have an explicit oscillator construction yet. This is still an open problem in mathematics.

In fact, the irreducible highest representation