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.
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
be an arbitrary fixed complex number in the upper half-plane; that is, so that the imaginary part of
Define the operators Sa and Tb such that they act on holomorphic functions as
It can be seen that each operator generates a one-parameter subgroup:
Thus we see that S and T together with a unitary phase form a nilpotent Lie group, the (continuous real) Heisenberg group, parametrizable as
A general group element
f ) ( z ) = λ exp ( i π
gives rise to a different representation of the action of the group.
The action of the group elements
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
and the domain of integration is the entire complex plane.
be the set of entire functions f with finite norm.
is used only to indicate that the space depends on the choice of parameter
forms a Hilbert space.
preserves the norm on this space.
This norm is closely related to that used to define Segal–Bargmann space[citation needed].
The above theta representation of the Heisenberg group is isomorphic to the canonical Weyl representation of the Heisenberg group.
stand for a general group element of
The corresponding theta representation is:
The Jacobi theta function is defined as
It is an entire function of z that is invariant under
It can be shown that the Jacobi theta is the unique such function.