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.
Construction.
The theta representation is a representation of the continuous Heisenberg group formula_1 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.
Let "f"("z") be a holomorphic function, let "a" and "b" be real numbers, and let formula_2 be fixed, but arbitrary complex number in the upper half-plane; that is, so that the imaginary part of formula_2 is positive. Define the operators "Sa" and "Tb" such that they act on holomorphic functions as
and
It can be seen that each operator generates a one-parameter subgroup:
and
However, "S" and "T" do not commute:
Thus we see that "S" and "T" together with a unitary phase form a nilpotent Lie group, the (continuous real) Heisenberg group, parametrizable as formula_9 where "U"(1) is the unitary group.
A general group element formula_10 then acts on a holomorphic function "f"("z") as
where formula_12 formula_13 is the center of "H", the commutator subgroup formula_14. The parameter formula_2 on formula_16 serves only to remind that every different value of formula_2 gives rise to a different representation of the action of the group.
Hilbert space.
The action of the group elements formula_16 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
Here, formula_20 is the imaginary part of formula_2 and the domain of integration is the entire complex plane. Let formula_22 be the set of entire functions "f" with finite norm. The subscript formula_2 is used only to indicate that the space depends on the choice of parameter formula_2. This formula_22 forms a Hilbert space. The action of formula_16 given above is unitary on formula_22, that is, formula_16 preserves the norm on this space. Finally, the action of formula_16 on formula_22 is irreducible.
This norm is closely related to that used to define Segal–Bargmann space.
Isomorphism.
The above "theta representation" of the Heisenberg group is isomorphic to the canonical Weyl representation of the Heisenberg group. In particular, this implies that formula_22 and formula_32 are isomorphic as "H"-modules. Let
stand for a general group element of formula_34 In the canonical Weyl representation, for every real number "h", there is a representation formula_35 acting on formula_32 as 
for formula_38 and formula_39
Here, "h" is Planck's constant. Each such representation is unitarily inequivalent. The corresponding theta representation is:
Discrete subgroup.
Define the subgroup formula_43 as
The Jacobi theta function is defined as
It is an entire function of "z" that is invariant under formula_46 This follows from the properties of the theta function:
and
when "a" and "b" are integers. It can be shown that the Jacobi theta is the unique such function.