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 ${\displaystyle H_{3}(\mathbb {R} )}$ 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 ${\displaystyle \tau }$ be an arbitrary fixed complex number in the upper half-plane; that is, so that the imaginary part of ${\displaystyle \tau }$ is positive. Define the operators S_a and T_b such that they act on holomorphic functions as

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

and

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

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

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

and

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

However, S and T do not commute:

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

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

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

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

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

Hilbert space

The action of the group elements ${\displaystyle U_{\tau }(\lambda ,a,b)}$ 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

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

Here, ${\displaystyle \Im \tau }$ is the imaginary part of ${\displaystyle \tau }$ and the domain of integration is the entire complex plane.

Mumford sets the norm as ${\displaystyle \int _{\mathbb {C} }\exp \left({\frac {-2\pi y^{2}}{\Im \tau }}\right)|f(x+iy)|^{2}\ dx\ dy}$, but in this way ${\displaystyle T_{b}}$ is not unitary.

Let ${\displaystyle {\mathcal {H}}_{\tau }}$ be the set of entire functions f with finite norm. The subscript ${\displaystyle \tau }$ is used only to indicate that the space depends on the choice of parameter ${\displaystyle \tau }$. This ${\displaystyle {\mathcal {H}}_{\tau }}$ forms a Hilbert space. The action of ${\displaystyle U_{\tau }(\lambda ,a,b)}$ given above is unitary on ${\displaystyle {\mathcal {H}}_{\tau }}$, that is, ${\displaystyle U_{\tau }(\lambda ,a,b)}$ preserves the norm on this space. Finally, the action of ${\displaystyle U_{\tau }(\lambda ,a,b)}$ on ${\displaystyle {\mathcal {H}}_{\tau }}$ 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 ${\displaystyle {\mathcal {H}}_{\tau }}$ and ${\displaystyle L^{2}(\mathbb {R} )}$ are isomorphic as H-modules. Let

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

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

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

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

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

${\displaystyle M(a,0,0)\to S_{ah}}$

${\displaystyle M(0,b,0)\to T_{b/2\pi }}$

${\displaystyle M(0,0,c)\to e^{ihc}}$

Discrete subgroup

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

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

The Jacobi theta function is defined as

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

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

${\displaystyle \vartheta (z+1;\tau )=\vartheta (z;\tau )}$

and

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

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

See also

• Segal–Bargmann space
• Hardy space

References

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