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 $H_3(ℝ)$ 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 an arbitrary fixed 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 $$(T_{b}f)(z)=\exp (i\pi b^2\tau +2\pi ibz)f(z+b\tau )=\exp (i\pi b^2\tau +2\pi ibz+b\tau {\partial }_z)f(z).$$

It can be seen that each operator generates a one-parameter subgroup: $$S_{a_1}\left(S_{a_2}f\right)=(S_{a_1}\circ S_{a_2})f=S_{a_1+a_2}f$$ and $$T_{b_1}\left(T_{b_2}f\right)=(T_{b_1}\circ T_{b_2})f=T_{b_1+b_2}f.$$

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 $H=U(1)\times ℝ\times ℝ$ 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

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

Here, $\mathrm{\Im }\tau$ is the imaginary part of $\tau$ and the domain of integration is the entire complex plane. Let ${\mathscr{H}}_{\tau }$ 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 ${\mathscr{H}}_{\tau }$ forms a Hilbert space. The action of $U_{\tau }(\lambda ,a,b)$ given above is unitary on ${\mathscr{H}}_{\tau }$, that is, $U_{\tau }(\lambda ,a,b)$ preserves the norm on this space. Finally, the action of $U_{\tau }(\lambda ,a,b)$ on ${\mathscr{H}}_{\tau }$ is irreducible.

This norm is closely related to that used to define Segal–Bargmann spaceTemplate: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 ${\mathscr{H}}_{\tau }$ and $L^2(ℝ)$ are isomorphic as H-modules. Let $$M(a,b,c)=\left[\begin{array}{ccc}1& a& c\\ 0& 1& b\\ 0& 0& 1\end{array}\right]$$ stand for a general group element of $H_3(ℝ).$ In the canonical Weyl representation, for every real number h, there is a representation ${\rho }_h$ acting on $L^2(ℝ)$ as $${\rho }_h(M(a,b,c))\psi (x)=\exp (ibx+ihc)\psi (x+ha)$$ for $x\in ℝ$ and $\psi \in L^2(ℝ).$

Here, h is the Planck 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 ${\mathrm{\Gamma }}_{\tau }\subset H_{\tau }$ as $${\mathrm{\Gamma }}_{\tau }=\{U_{\tau }(1,a,b)\in H_{\tau }:a,b\in \mathbb{Z}\}.$$

The Jacobi theta function is defined as $$\vartheta (z;\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\exp (\pi i{n}^2\tau +2\pi inz).$$

It is an entire function of z that is invariant under ${\mathrm{\Gamma }}_{\tau }.$ This follows from the properties of the theta function: $$\vartheta (z+1;\tau )=\vartheta (z;\tau )$$ and $$\vartheta (z+a+b\tau ;\tau )=\exp (-\pi i{b}^2\tau -2\pi ibz)\vartheta (z;\tau )$$ 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 Template:Isbn