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 $au$ be fixed, but arbitrary complex number in the upper half-plane; that is, so that the imaginary part of $au$ is positive. Define the operators "S"_"a" and "T"_"b" such that they act on holomorphic functions as

:$(S\_a\; f)(z)\; =\; f(z+a)$

and

:$(T\_b\; f)(z)\; =\; exp\; (ipi\; b^2\; au\; +2pi\; ibz)\; f(z+b\; au).$

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:

:$S\_a\; circ\; T\_b\; =\; exp\; (2pi\; iab)\; ;\; T\_b\; circ\; S\_a.$

Thus we see that "S" and "T" together with a unitary phase form a nilpotent Lie group, the (continuous real) Heisenberg group, parametrizable as $H=U(1)\; imesmathbb\{R\}\; imesmathbb\{R\}$ where U(1) is the unitary group. A general group element $U\_\; au(lambda,a,b)in\; H$ then acts on a holomorphic function "f"("z") as

:$U\_\; au(lambda,a,b);f(z)=lambda\; (S\_a\; circ\; T\_b\; f)(z)\; =\; lambda\; exp\; (ipi\; b^2\; au\; +2pi\; ibz)\; f(z+a+b\; au)$

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

Hilbert space

The action of the group elements $U\_\; au(lambda,a,b)$ 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

:$Vert\; f\; Vert\_\; au\; ^2\; =\; int\_\{mathbb\{Cexp\; left(\; frac\; \{-2pi\; y^2\}\; \{Im\; au\}\; ight)\; |f(x+iy)|^2\; dx\; dy.$

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

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 $mathcal\{H\}\_\; au$ and L²(R) are isomorphic as "H"-modules. Let

:

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 $ho\_h$ acting on L²(R) as :$ho\_h(M(a,b,c));psi(x)=\; exp\; (ibx+ihc)\; psi(x+ha)$ for $xinmathbb\{R\}$ and $psiin\; L^2(mathbb\{R\})$.

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

:$M(a,0,0)\; o\; S\_\{ah\}$:$M(0,b,0)\; o\; T\_\{b/2pi\}$:$M(0,0,c)\; o\; e^\{ihc\}$

Discrete subgroup

Define the subgroup $Gamma\_\; ausubset\; H\_\; au$ as

:$Gamma\_\; au\; =\; \{\; U\_\; au(1,a,b)\; in\; H\_\; au\; :\; a,b\; in\; mathbb\{Z\}\; \}.$

The Jacobi theta function is defined as

:$vartheta(z;\; au)\; =\; sum\_\{n=-infty\}^infty\; exp\; (pi\; i\; n^2\; au\; +\; 2\; pi\; i\; n\; z).$

It is an entire function of "z" that is invariant under $Gamma\_\; au$. This follows from the properties of the theta function:

:$vartheta(z+1;\; au)\; =\; vartheta(z;\; au)$

and

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

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), Birkhauser, Boston ISBN 3-7643-3109-7