Inmathematics, thetheta representation is a particular representation of theHeisenberg group ofquantum mechanics. It gains its name from the fact that theJacobi theta function is invariant under the action of a discrete subgroup of the Heisenberg group. The representation was popularized byDavid Mumford.

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 particularHilbert space. The construction below proceeds first by definingoperators that correspond to the Heisenberg group generators. Next, the Hilbert space on which these act is defined, followed by a demonstration of theisomorphism to the usual representations.

Letf(z) be aholomorphic function, leta andb bereal numbers, and let${\displaystyle \tau }$ be an arbitrary fixed complex number in theupper half-plane; that is, so that the imaginary part of${\displaystyle \tau }$ is positive. Define the operatorsS_a andT_b such that they act on holomorphic functions as$${\displaystyle (S_{a}f)(z)=f(z+a)=\exp (a{\mathrm{\partial }}_z)f(z)}$$and$${\displaystyle (T_{b}f)(z)=\exp (i\pi b^2\tau +2\pi ibz)f(z+b\tau )=\exp (i\pi b^2\tau +2\pi ibz+b\tau {\mathrm{\partial }}_z)f(z).}$$

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 andT do not commute:$${\displaystyle S_a\circ T_b=\exp (2\pi iab)T_b\circ S_a.}$$

Thus we see thatS andT together with aunitary phase form anilpotentLie group, the (continuous real)Heisenberg group, parametrizable as${\displaystyle H=U(1)\times \mathbb{R}\times \mathbb{R}}$ whereU(1) is theunitary group.

A general group element${\displaystyle U_{\tau }(\lambda ,a,b)\in H}$ then acts on a holomorphic functionf(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 thecenter ofH, thecommutator 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.

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 onentire functions of thecomplex plane as

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

Here,${\displaystyle \mathrm{\Im }\tau }$ is the imaginary part of${\displaystyle \tau }$ and the domain of integration is the entire complex plane. Let${\displaystyle {\mathcal{H}}_{\tau }}$ be the set of entire functionsf 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 aHilbert 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${\displaystyle {\mathcal{H}}_{\tau }}$ isirreducible.

This norm is closely related to that used to defineSegal–Bargmann space^{[citation needed]}.

The abovetheta representation of the Heisenberg group is isomorphic to the canonicalWeyl representation of the Heisenberg group. In particular, this implies that${\displaystyle {\mathcal{H}}_{\tau }}$ and${\displaystyle L^2(\mathbb{R})}$ areisomorphic asH-modules. Letstand for a general group element of${\displaystyle H_3(\mathbb{R}).}$ In the canonical Weyl representation, for every real numberh, 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 thePlanck constant. Each such representation isunitarily inequivalent. The corresponding theta representation is:$${\displaystyle M(a,0,0)\to S_{ah}}$$$${\displaystyle M(0,b,0)\to T_{b/2\pi }}$$$${\displaystyle M(0,0,c)\to e^{ihc}}$$

Define the subgroup${\displaystyle {\mathrm{\Gamma }}_{\tau }\subset H_{\tau }}$ as$${\displaystyle {\mathrm{\Gamma }}_{\tau }=\{U_{\tau }(1,a,b)\in H_{\tau }:a,b\in \mathbb{Z}\}.}$$

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

It is anentire function ofz that isinvariant under${\displaystyle {\mathrm{\Gamma }}_{\tau }.}$ This follows from the properties of the theta function:$${\displaystyle \vartheta (z+1;\tau )=\vartheta (z;\tau )}$$and$${\displaystyle \vartheta (z+a+b\tau ;\tau )=\exp (-\pi i{b}^2\tau -2\pi ibz)\vartheta (z;\tau )}$$whena andb are integers. It can be shown that the Jacobi theta is the unique such function.

• Segal–Bargmann space
• Hardy space

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

• Elliptic functions
• Theta functions
• Lie groups
• Mathematical quantization

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