Theta representation

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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 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.

Let f(z) be a holomorphic function, let a and b be real numbers, and let ${\displaystyle \tau }$ be an arbitrary fixed complex number in the upper half-plane; that is, so that the imaginary part of ${\displaystyle \tau }$ is positive. Define the operators S_a and T_b such that they act on holomorphic functions as $${\displaystyle (S_{a}f)(z)=f(z+a)=\exp(a\partial _{z})f(z)}$$ and $${\displaystyle (T_{b}f)(z)=\exp(i\pi b^{2}\tau +2\pi ibz)f(z+b\tau )=\exp(i\pi b^{2}\tau +2\pi ibz+b\tau \partial _{z})f(z).}$$

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 and T do not commute: $${\displaystyle S_{a}\circ T_{b}=\exp(2\pi iab)T_{b}\circ S_{a}.}$$

Thus we see that S and T together with a unitary phase form a nilpotent Lie group, the (continuous real) Heisenberg group, parametrizable as ${\displaystyle H=U(1)\times \mathbb {R} \times \mathbb {R} }$ where U(1) is the unitary group.

A general group element ${\displaystyle U_{\tau }(\lambda ,a,b)\in H}$ then acts on a holomorphic function f(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 the center of H, the commutator 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 on entire functions of the complex plane as

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

Here, ${\displaystyle \Im \tau }$ 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 functions f 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 a Hilbert 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 ${\displaystyle {\mathcal {H}}_{\tau }}$ is irreducible.

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

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The above theta representation of the Heisenberg group is isomorphic to the canonical Weyl representation of the Heisenberg group. In particular, this implies that ${\displaystyle {\mathcal {H}}_{\tau }}$ and ${\displaystyle L^{2}(\mathbb {R} )}$ are isomorphic as H-modules. Let $${\displaystyle M(a,b,c)={\begin{bmatrix}1&a&c\\0&1&b\\0&0&1\end{bmatrix}}}$$ stand for a general group element of ${\displaystyle H_{3}(\mathbb {R} ).}$ In the canonical Weyl representation, for every real number h, 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 the Planck constant. Each such representation is unitarily inequivalent. The corresponding theta representation is: $${\displaystyle M(a,0,0)\to S_{ah}}$$ $${\displaystyle M(0,b,0)\to T_{b/2\pi }}$$ $${\displaystyle M(0,0,c)\to e^{ihc}}$$