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.

Operators and group law

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:

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

and

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

However, S and T do not commute:

$S_a{T}_b=\exp (2\pi iab)T_b{S}_a.$

Thus $S$ 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

$U_{\tau }(\lambda ,a,b)f(z)=\lambda (S_a{T}_{b}f)(z)=\lambda \exp (i\pi b^2\tau +2\pi ibz)f(z+a+b\tau ),$

where $\lambda \in U(1)$. The subgroup $U(1)$ is the center $Z(H)$ of $H$, and is also its 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 $\tau$, define a norm on entire functions of the complex plane as

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

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.

Relation with the Weyl representation

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 $L^2(\mathbb{ℝ})$ 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 $H_3(\mathbb{ℝ})$. In the canonical Weyl representation, for every real number h, there is a representation ${\rho }_h$ acting on $L^2(\mathbb{ℝ})$ 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 the Planck constant. Each such representation is unitarily inequivalent. The corresponding theta representation is:

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

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

$M(0,0,c)\mapsto e^{ihc}.$

Theta functions

The Heisenberg group can be used to give a unified account of theta functions in complex analysis and algebraic geometry. For $\tau$ in the upper half-plane, the standard Jacobi theta function is

$\vartheta (z,\tau )=\sum _{n\in \mathbb{\mathbb{Z}}}\exp (\pi i{n}^2\tau +2\pi inz).$

It is an entire function of $z$ satisfying the transformation laws

$\vartheta (z+1,\tau )=\vartheta (z,\tau ),\vartheta (z+\tau ,\tau )=e^{-\pi i\tau -2\pi iz}\vartheta (z,\tau ).$

More generally, for integers $a$ and $b$,

$\vartheta (z+a+b\tau ,\tau )=\exp (-\pi i{b}^2\tau -2\pi ibz)\vartheta (z,\tau ).$

Define the subgroup ${\mathrm{\Gamma }}_{\tau }\subset H$ as

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

The preceding transformation laws say exactly that $\vartheta (z,\tau )$ is invariant under ${\mathrm{\Gamma }}_{\tau }$. It can be shown that the Jacobi theta function is the unique such entire function, up to scalar multiple.

Thus $\vartheta (z,\tau )$ is not an ordinary function on the elliptic curve

$E_{\tau }=\mathbb{ℂ}/(\mathbb{\mathbb{Z}}+\tau \mathbb{\mathbb{Z}}),$

but rather a section of a line bundle on $E_{\tau }$. In one common convention, this line bundle is obtained from $\mathbb{ℂ}\times \mathbb{ℂ}$ by the identifications

$(z,w)\sim (z+1,w),(z,w)\sim (z+\tau ,e^{-\pi i\tau -2\pi iz}w).$

The exponential factors in the theta transformation laws are the corresponding factors of automorphy. In this sense, a theta function is a function on the universal cover whose transformation law allows it to descend as a section of a line bundle on the quotient torus.

The Heisenberg group appears because translations of $z$ must be accompanied by scalar factors in order to preserve these transformation laws. Thus translations in the two period directions define a projective representation, and the corresponding central extension is a Heisenberg group.

Theta functions with characteristics

Theta functions with rational characteristics are obtained by applying Heisenberg operators to $\vartheta$. For $a,b\in \mathbb{\mathbb{Q}}$, define

${\vartheta }_{a,b}(z,\tau )=(S_b{T}_a\vartheta )(z,\tau )=e^{\pi i{a}^2\tau +2\pi ia(z+b)}\vartheta (z+a\tau +b,\tau ).$

Equivalently,

${\vartheta }_{a,b}(z,\tau )=\sum _{n\in \mathbb{\mathbb{Z}}}\exp (\pi i(n+a{)}^2\tau +2\pi i(n+a)(z+b)).$

Changing $a$ and $b$ by integers changes these functions only by simple scalar factors, so the characteristics are naturally considered modulo $\mathbb{\mathbb{Z}}$.

This gives a concrete finite-dimensional form of the Heisenberg representation. If $L$ is the degree-one theta line bundle on $E_{\tau }$, then

$\dim H^0(E_{\tau },L^{\otimes N})=N.$

A basis of this space may be chosen from theta functions with characteristics. The action of translations by $N$-torsion points, together with the necessary scalar factors, gives a finite Heisenberg group acting on $H^0(E_{\tau },L^{\otimes N})$. This is the finite-dimensional analogue of the Schrödinger representation.

Theta groups

More generally, one can associate to a line bundle $L$ on an abelian variety $A$ its theta group $G(L)$. Let

$K(L)=\{x\in A\mid t_x^{*}L\simeq L\},$

where $t_x:A\to A$ denotes translation by $x$. The theta group consists of pairs $(x,\varphi )$, where $x\in K(L)$ and $\varphi :L\to t_x^{*}L$ is an isomorphism of line bundles. It fits into a central extension

$1⟶{\mathbb{𝔾}}_m⟶G(L)⟶K(L)⟶1.$

When $L$ is ample, $K(L)$ is finite, and $G(L)$ is a finite algebraic analogue of the Heisenberg group.

The group $G(L)$ acts naturally on the vector space of sections $H^0(A,L)$. This action is the algebro-geometric analogue of the Schrödinger representation of the real Heisenberg group. When $L$ is ample, the commutator in $G(L)$ induces a nondegenerate alternating pairing on $K(L)$, analogous to the symplectic form used in the construction of the ordinary Heisenberg group. A finite version of the Stone–von Neumann theorem describes the resulting irreducible representation with prescribed central character.

David Mumford used this Heisenberg-group formalism to give an algebraic theory of theta functions and to study equations defining abelian varieties.

See also

• Segal–Bargmann space
• Hardy space

References

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