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 <math>H_3(\R)</math> 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 <math>\tau</math> be an arbitrary fixed complex number in the upper half-plane; that is, so that the imaginary part of <math>\tau</math> is positive. Define the operators ''S<sub>a</sub>'' and ''T<sub>b</sub>'' such that they act on holomorphic functions as :<math> (S_a f)(z) = f(z+a)= \exp(a\partial_z)f(z) </math> and :<math> (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 i bz + b\tau\partial_z) f(z). </math>
It can be seen that each operator generates a one-parameter subgroup: :<math> S_{a_1}\left(S_{a_2}f\right)=S_{a_1+a_2}f </math> and :<math> T_{b_1}\left(T_{b_2}f\right)=T_{b_1+b_2}f. </math> However, ''S'' and ''T'' do not commute: :<math> S_aT_b=\exp(2\pi iab)T_bS_a. </math>
Thus <math>S</math> and <math>T</math> together with a unitary phase form a nilpotent Lie group, the continuous real Heisenberg group, parametrizable as <math>H=U(1)\times\R\times\R</math>, where <math>U(1)</math> is the unitary group.
A general group element <math>U_\tau(\lambda,a,b)\in H</math> then acts on a holomorphic function ''f''(''z'') as :<math> U_\tau(\lambda,a,b) f(z) = \lambda(S_aT_bf)(z) = \lambda\exp(i\pi b^2\tau+2\pi ibz)f(z+a+b\tau), </math> where <math>\lambda\in U(1)</math>. The subgroup <math>U(1)</math> is the center <math>Z(H)</math> of <math>H</math>, and is also its commutator subgroup <math>[H,H]</math>. The parameter <math>\tau</math> on <math>U_\tau(\lambda,a,b)</math> serves only to remind that every different value of <math>\tau</math> gives rise to a different representation of the action of the group.
=== Hilbert space ===
The action of the group elements <math>U_\tau(\lambda,a,b)</math> is unitary and irreducible on a certain Hilbert space of functions. For a fixed value of <math>\tau</math>, define a norm on entire functions of the complex plane as :<math> \Vert f \Vert_\tau ^2 = \int_{\C} \exp\left(\frac{-2\pi y^2}{\Im\tau}\right) |f(x+iy)|^2\,dx\,dy. </math>
Here, <math>\Im\tau</math> is the imaginary part of <math>\tau</math> and the domain of integration is the entire complex plane. Let <math>\mathcal H_\tau</math> be the set of entire functions ''f'' with finite norm. The subscript <math>\tau</math> is used only to indicate that the space depends on the choice of parameter <math>\tau</math>. This <math>\mathcal H_\tau</math> forms a Hilbert space. The action of <math>U_\tau(\lambda,a,b)</math> given above is unitary on <math>\mathcal H_\tau</math>, that is, <math>U_\tau(\lambda,a,b)</math> preserves the norm on this space. Finally, the action of <math>U_\tau(\lambda,a,b)</math> on <math>\mathcal H_\tau</math> is irreducible.
This norm is closely related to that used to define the Segal–Bargmann space.{{Citation needed|date=July 2014}}
=== 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 <math>\mathcal H_\tau</math> and <math>L^2(\R)</math> are isomorphic as <math>H</math>-modules. Let :<math> M(a,b,c)= \begin{bmatrix} 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1 \end{bmatrix} </math> stand for a general group element of <math>H_3(\R)</math>. In the canonical Weyl representation, for every real number ''h'', there is a representation <math>\rho_h</math> acting on <math>L^2(\R)</math> as :<math> \rho_h(M(a,b,c))\psi(x)=\exp(ibx+ihc)\psi(x+ha) </math> for <math>x\in\R</math> and <math>\psi\in L^2(\R)</math>.
Here, ''h'' is the Planck constant. Each such representation is unitarily inequivalent. The corresponding theta representation is: :<math> M(a,0,0)\mapsto S_{ah}, </math> :<math> M(0,b,0)\mapsto T_{b/2\pi}, </math> :<math> M(0,0,c)\mapsto e^{ihc}. </math>
== Theta functions ==
The Heisenberg group can be used to give a unified account of theta functions in complex analysis and algebraic geometry. For <math>\tau</math> in the upper half-plane, the standard Jacobi theta function is :<math> \vartheta(z,\tau)= \sum_{n\in\mathbb Z} \exp(\pi i n^2\tau+2\pi inz). </math> It is an entire function of <math>z</math> satisfying the transformation laws :<math> \vartheta(z+1,\tau)=\vartheta(z,\tau), \qquad \vartheta(z+\tau,\tau) = e^{-\pi i\tau-2\pi iz}\vartheta(z,\tau). </math> More generally, for integers <math>a</math> and <math>b</math>, :<math> \vartheta(z+a+b\tau,\tau) = \exp(-\pi ib^2\tau-2\pi ibz)\vartheta(z,\tau). </math>
Define the subgroup <math>\Gamma_\tau\subset H</math> as :<math> \Gamma_\tau=\{U_\tau(1,a,b)\in H:a,b\in\mathbb Z\}. </math> The preceding transformation laws say exactly that <math>\vartheta(z,\tau)</math> is invariant under <math>\Gamma_\tau</math>. It can be shown that the Jacobi theta function is the unique such entire function, up to scalar multiple.<ref name="MumfordThetaI">{{cite book |last=Mumford |first=David |author-link=David Mumford |title=Tata Lectures on Theta I |series=Progress in Mathematics |volume=28 |publisher=Birkhäuser |year=1983 |isbn=978-0-8176-4577-9}}</ref>
Thus <math>\vartheta(z,\tau)</math> is not an ordinary function on the elliptic curve :<math> E_\tau=\mathbb C/(\mathbb Z+\tau\mathbb Z), </math> but rather a section of a line bundle on <math>E_\tau</math>. In one common convention, this line bundle is obtained from <math>\mathbb C\times\mathbb C</math> by the identifications :<math> (z,w)\sim(z+1,w), \qquad (z,w)\sim(z+\tau,e^{-\pi i\tau-2\pi iz}w). </math> 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.<ref name="MumfordThetaI" />
The Heisenberg group appears because translations of <math>z</math> 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 <math>\vartheta</math>. For <math>a,b\in\mathbb Q</math>, define :<math> \vartheta_{a,b}(z,\tau) = (S_bT_a\vartheta)(z,\tau) = e^{\pi ia^2\tau+2\pi ia(z+b)} \vartheta(z+a\tau+b,\tau). </math> Equivalently, :<math> \vartheta_{a,b}(z,\tau) = \sum_{n\in\mathbb Z} \exp\!\left(\pi i(n+a)^2\tau +2\pi i(n+a)(z+b)\right). </math> Changing <math>a</math> and <math>b</math> by integers changes these functions only by simple scalar factors, so the characteristics are naturally considered modulo <math>\mathbb Z</math>.
This gives a concrete finite-dimensional form of the Heisenberg representation. If <math>L</math> is the degree-one theta line bundle on <math>E_\tau</math>, then :<math> \dim H^0(E_\tau,L^{\otimes N})=N. </math> A basis of this space may be chosen from theta functions with characteristics. The action of translations by <math>N</math>-torsion points, together with the necessary scalar factors, gives a finite Heisenberg group acting on <math>H^0(E_\tau,L^{\otimes N})</math>. This is the finite-dimensional analogue of the Schrödinger representation.<ref name="MumfordThetaI" />
== Theta groups ==
More generally, one can associate to a line bundle <math>L</math> on an abelian variety <math>A</math> its theta group <math>G(L)</math>. Let :<math> K(L)=\{x\in A\mid t_x^*L\simeq L\}, </math> where <math>t_x:A\to A</math> denotes translation by <math>x</math>. The theta group consists of pairs <math>(x,\phi)</math>, where <math>x\in K(L)</math> and <math>\phi:L\to t_x^*L</math> is an isomorphism of line bundles. It fits into a central extension :<math> 1\longrightarrow \mathbb G_m \longrightarrow G(L) \longrightarrow K(L) \longrightarrow 1. </math> When <math>L</math> is ample, <math>K(L)</math> is finite, and <math>G(L)</math> is a finite algebraic analogue of the Heisenberg group.<ref name="Mumford1966">{{cite journal |last=Mumford |first=David |author-link=David Mumford |title=On the equations defining abelian varieties. I |journal=Inventiones Mathematicae |volume=1 |year=1966 |pages=287–354 |doi=10.1007/BF01389737}}</ref><ref name="MumfordThetaIII">{{cite book |last=Mumford |first=David |author-link=David Mumford |title=Tata Lectures on Theta III |series=Progress in Mathematics |volume=97 |publisher=Birkhäuser |year=1991 |isbn=978-0-8176-4579-3}}</ref>
The group <math>G(L)</math> acts naturally on the vector space of sections <math>H^0(A,L)</math>. This action is the algebro-geometric analogue of the Schrödinger representation of the real Heisenberg group. When <math>L</math> is ample, the commutator in <math>G(L)</math> induces a nondegenerate alternating pairing on <math>K(L)</math>, 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.<ref name="MumfordThetaIII" />
David Mumford used this Heisenberg-group formalism to give an algebraic theory of theta functions and to study equations defining abelian varieties.<ref name="Mumford1966" /><ref name="MumfordThetaIII" />
== See also == * Segal–Bargmann space * Hardy space
== References == {{reflist}} * David Mumford, ''Tata Lectures on Theta I'' (1983), Birkhäuser, Boston {{isbn|3-7643-3109-7}}
Category:Elliptic functions Category:Theta functions Category:Lie groups Category:Mathematical quantization