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.

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 fixed, but arbitrary 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

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