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 S_a and T_b 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 ibz+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.

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 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.

References

David Mumford, Tata Lectures on Theta I (1983), Birkhäuser, Boston ISBN 3-7643-3109-7

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