# 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

$(T_b f)(z) = \exp (i\pi b^2 \tau +2\pi ibz) f(z+b\tau)= \exp( 2\pi i bz + b \tau \partial_z) ~ f (z) .$

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

$S_{a_1} (S_{a_2} f) = (S_{a_1} \circ S_{a_2}) f = S_{a_1+a_2} f$

and

$T_{b_1} (T_{b_2} f) = (T_{b_1} \circ T_{b_2}) f = T_{b_1+b_2} f.$

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 L2(R) are isomorphic as H-modules. Let

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

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 L2(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

$\vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp (\pi i n^2 \tau + 2 \pi i n z).$

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

$\vartheta(z+a+b\tau;\tau) = \exp(-\pi i b^2 \tau -2 \pi i b z)\vartheta(z;\tau)$

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