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.

ConstructionEdit

The theta representation is a representation of the continuous Heisenberg group   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 generatorsEdit

Let f(z) be a holomorphic function, let a and b be real numbers, and let   be fixed, but arbitrary complex number in the upper half-plane; that is, so that the imaginary part of   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   where U(1) is the unitary group.

A general group element   then acts on a holomorphic function f(z) as

 

where     is the center of H, the commutator subgroup  . The parameter   on   serves only to remind that every different value of   gives rise to a different representation of the action of the group.

Hilbert spaceEdit

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

 

Here,   is the imaginary part of   and the domain of integration is the entire complex plane. Let   be the set of entire functions f with finite norm. The subscript   is used only to indicate that the space depends on the choice of parameter  . This   forms a Hilbert space. The action of   given above is unitary on  , that is,   preserves the norm on this space. Finally, the action of   on   is irreducible.

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

IsomorphismEdit

The above theta representation of the Heisenberg group is isomorphic to the canonical Weyl representation of the Heisenberg group. In particular, this implies that   and   are isomorphic as H-modules. Let

 

stand for a general group element of   In the canonical Weyl representation, for every real number h, there is a representation   acting on   as

 

for   and  

Here, h is Planck's constant. Each such representation is unitarily inequivalent. The corresponding theta representation is:

 
 
 

Discrete subgroupEdit

Define the subgroup   as

 

The Jacobi theta function is defined as

 

It is an entire function of z that is invariant under   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.

See alsoEdit

ReferencesEdit

  • David Mumford, Tata Lectures on Theta I (1983), Birkhäuser, Boston ISBN 3-7643-3109-7