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

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 operatorss 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.

Operators and group law

Let f(z) be a holomorphic function, let a and b be real numbers be an arbitrary fixed complex number in the upper half-plane 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:

together with a unitary phase form a nilpotent Lie group, the continuous real Heisenberg group is the unitary group.

then acts on a holomorphic function f(z) as

is the center, and is also its commutator subgroup gives rise to a different representation of the action of the group.

Hilbert space

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

be the set of entire functions f forms a Hilbert space is irreducible.

This norm is closely related to that used to define the Segal–Bargmann space.

Relation with the Weyl representation

The above theta representation of the Heisenberg group is isomorphic to the canonical Weyl representation and L^2(\R) are isomorphic as H-modules. Let

. In the canonical Weyl representation, for every real number h as

.

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

Theta functions

The Heisenberg group can be used to give a unified account of theta functions in complex analysis and algebraic geometry in the upper half-plane, the standard Jacobi theta function is

It is an entire function satisfying the transformation laws

,

as

is invariant. It can be shown that the Jacobi theta function is the unique such entire function, up to scalar multiple.

is not an ordinary function on the elliptic curve

but rather a section of a line bundle by the identifications

The exponential factors in the theta transformation laws are the corresponding factors of automorphy. In this sense, a theta function is a function on the universal cover whose transformation law allows it to descend as a section of a line bundle on the quotient torus.

must be accompanied by scalar factors in order to preserve these transformation laws. Thus translations in the two period directions define a projective representation, and the corresponding central extension is a Heisenberg group.

Theta functions with characteristics

Theta functions with rational characteristics, define

Equivalently,

.

, then

. This is the finite-dimensional analogue of the Schrödinger representation.

Theta groups

on an abelian variety. Let

is an isomorphism of line bundles. It fits into a central extension

is a finite algebraic analogue of the Heisenberg group.

acts naturally on the vector space, analogous to the symplectic form used in the construction of the ordinary Heisenberg group. A finite version of the Stone–von Neumann theorem describes the resulting irreducible representation with prescribed central character.

David Mumford used this Heisenberg-group formalism to give an algebraic theory of theta functions and to study equations defining abelian varieties.

See also

• Segal–Bargmann space

• Hardy space

References

• David Mumford, Tata Lectures on Theta I (1983), Birkhäuser, Boston