Theta representation

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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[edit]

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[edit]

Let f(z) be a holomorphic function, let a and b be real numbers, and let \tau be fixed, but arbitrary complex number in the upper half-plane; that is, so that the imaginary part of \tau is positive. Define the operators Sa and Tb 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( 2\pi i bz + b \tau \partial_z) ~ f (z) .

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 (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 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 \circ T_b f)(z) = 
\lambda \exp (i\pi b^2 \tau +2\pi ibz) f(z+a+b\tau)

where \lambda \in U(1). U(1) = Z(H) is the center of H, the 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[edit]

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 τ, define a norm on entire functions of the complex plane as

\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, \Im \tau is the imaginary part of \tau and the domain of integration is the entire complex plane. Let \mathcal{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 \mathcal{H}_\tau forms a Hilbert space. The action of U_\tau(\lambda,a,b) given above is unitary on \mathcal{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 \mathcal{H}_\tau is irreducible.

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

Isomorphism[edit]

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}_\tau and L2(R) are isomorphic as H-modules. Let

 \operatorname{M}(a,b,c) = \begin{bmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{bmatrix}

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 L2(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 Planck's constant. Each such representation is unitarily inequivalent. The corresponding theta representation is:

M(a,0,0) \to S_{ah}
M(0,b,0) \to T_{b/2\pi}
M(0,0,c) \to e^{ihc}

Discrete subgroup[edit]

Define the subgroup \Gamma_\tau\subset H_\tau as

\Gamma_\tau = \{ U_\tau(1,a,b) \in H_\tau : a,b \in \mathbb{Z} \}.

The Jacobi theta function is defined as

\vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp (\pi i n^2 \tau + 2 \pi i n z).

It is an entire function of z that is invariant under \Gamma_\tau. This follows from the properties of the theta function:

\vartheta(z+1; \tau) = \vartheta(z; \tau)

and

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

when a and b are integers. It can be shown that the Jacobi theta is the unique such function.

See also[edit]

References[edit]