# Theta representation

In mathematics, the theta representation is a particular representation of the Heisenberg group of quantum mechanics. It gains its name from the fact that the Jacobi theta function is invariant under the action of a discrete subgroup of the Heisenberg group. The representation was popularized by David Mumford.

## Construction

The theta representation is a representation of the continuous Heisenberg group $H_{3}(\mathbb {R} )$  over the field of the real numbers. In this representation, the group elements act on a particular Hilbert space. The construction below proceeds first by defining operators that correspond to the Heisenberg group generators. Next, the Hilbert space on which these act is defined, followed by a demonstration of the isomorphism to the usual representations.

### Group generators

Let f(z) be a holomorphic function, let a and b be real numbers, and let $\tau$  be fixed, but arbitrary complex number in the upper half-plane; that is, so that the imaginary part of $\tau$  is positive. Define the operators Sa and Tb such that they act on holomorphic functions as

$(S_{a}f)(z)=f(z+a)=\exp(a\partial _{z})f(z)$

and

$(T_{b}f)(z)=\exp(i\pi b^{2}\tau +2\pi ibz)f(z+b\tau )=\exp(i\pi b^{2}\tau +2\pi ibz+b\tau \partial _{z})f(z).$

It can be seen that each operator generates a one-parameter subgroup:

$S_{a_{1}}\left(S_{a_{2}}f\right)=\left(S_{a_{1}}\circ S_{a_{2}}\right)f=S_{a_{1}+a_{2}}f$

and

$T_{b_{1}}\left(T_{b_{2}}f\right)=\left(T_{b_{1}}\circ T_{b_{2}}\right)f=T_{b_{1}+b_{2}}f.$

However, S and T do not commute:

$S_{a}\circ T_{b}=\exp(2\pi iab)T_{b}\circ S_{a}.$

Thus we see that S and T together with a unitary phase form a nilpotent Lie group, the (continuous real) Heisenberg group, parametrizable as $H=U(1)\times \mathbb {R} \times \mathbb {R}$  where U(1) is the unitary group.

A general group element $U_{\tau }(\lambda ,a,b)\in H$  then acts on a holomorphic function f(z) as

$U_{\tau }(\lambda ,a,b)f(z)=\lambda (S_{a}\circ T_{b}f)(z)=\lambda \exp(i\pi b^{2}\tau +2\pi ibz)f(z+a+b\tau )$

where $\lambda \in U(1).$  $U(1)=Z(H)$  is the center of H, the commutator subgroup $[H,H]$ . The parameter $\tau$  on $U_{\tau }(\lambda ,a,b)$  serves only to remind that every different value of $\tau$  gives rise to a different representation of the action of the group.

### Hilbert space

The action of the group elements $U_{\tau }(\lambda ,a,b)$  is unitary and irreducible on a certain Hilbert space of functions. For a fixed value of τ, define a norm on entire functions of the complex plane as

$\Vert f\Vert _{\tau }^{2}=\int _{\mathbb {C} }\exp \left({\frac {-2\pi y^{2}}{\Im \tau }}\right)|f(x+iy)|^{2}\ dx\ dy.$

Here, $\Im \tau$  is the imaginary part of $\tau$  and the domain of integration is the entire complex plane. Let ${\mathcal {H}}_{\tau }$  be the set of entire functions f with finite norm. The subscript $\tau$  is used only to indicate that the space depends on the choice of parameter $\tau$ . This ${\mathcal {H}}_{\tau }$  forms a Hilbert space. The action of $U_{\tau }(\lambda ,a,b)$  given above is unitary on ${\mathcal {H}}_{\tau }$ , that is, $U_{\tau }(\lambda ,a,b)$  preserves the norm on this space. Finally, the action of $U_{\tau }(\lambda ,a,b)$  on ${\mathcal {H}}_{\tau }$  is irreducible.

This norm is closely related to that used to define Segal–Bargmann space[citation needed].

## Isomorphism

The above theta representation of the Heisenberg group is isomorphic to the canonical Weyl representation of the Heisenberg group. In particular, this implies that ${\mathcal {H}}_{\tau }$  and $L^{2}(\mathbb {R} )$  are isomorphic as H-modules. Let

$M(a,b,c)={\begin{bmatrix}1&a&c\\0&1&b\\0&0&1\end{bmatrix}}$

stand for a general group element of $H_{3}(\mathbb {R} ).$  In the canonical Weyl representation, for every real number h, there is a representation $\rho _{h}$  acting on $L^{2}(\mathbb {R} )$  as

$\rho _{h}(M(a,b,c))\psi (x)=\exp(ibx+ihc)\psi (x+ha)$

for $x\in \mathbb {R}$  and $\psi \in L^{2}(\mathbb {R} ).$

Here, h is Planck's constant. Each such representation is unitarily inequivalent. The corresponding theta representation is:

$M(a,0,0)\to S_{ah}$
$M(0,b,0)\to T_{b/2\pi }$
$M(0,0,c)\to e^{ihc}$

## Discrete subgroup

Define the subgroup $\Gamma _{\tau }\subset H_{\tau }$  as

$\Gamma _{\tau }=\{U_{\tau }(1,a,b)\in H_{\tau }:a,b\in \mathbb {Z} \}.$

The Jacobi theta function is defined as

$\vartheta (z;\tau )=\sum _{n=-\infty }^{\infty }\exp(\pi in^{2}\tau +2\pi inz).$

It is an entire function of z that is invariant under $\Gamma _{\tau }.$  This follows from the properties of the theta function:

$\vartheta (z+1;\tau )=\vartheta (z;\tau )$

and

$\vartheta (z+a+b\tau ;\tau )=\exp(-\pi ib^{2}\tau -2\pi ibz)\vartheta (z;\tau )$

when a and b are integers. It can be shown that the Jacobi theta is the unique such function.