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
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
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
![{\displaystyle (S_{a}f)(z)=f(z+a)=\exp(a\partial _{z})f(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a9ffdb09fd6d6b9ce9cc0b99f99d6de315c304d)
and
![{\displaystyle (T_{b}f)(z)=\exp(i\pi b^{2}\tau +2\pi ibz)f(z+b\tau )=\exp(i\pi b^{2}\tau +2\pi ibz+b\tau \partial _{z})f(z).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9781967cf38413a24553e0e4d4068f18c7be0f40)
It can be seen that each operator generates a one-parameter subgroup:
![{\displaystyle S_{a_{1))\left(S_{a_{2))f\right)=\left(S_{a_{1))\circ S_{a_{2))\right)f=S_{a_{1}+a_{2))f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/caa1a8dce24cbd1c8d2cbadcaac489249f84c9b9)
and
![{\displaystyle T_{b_{1))\left(T_{b_{2))f\right)=\left(T_{b_{1))\circ T_{b_{2))\right)f=T_{b_{1}+b_{2))f.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08210c02b6f054d0e770f9bf5aa5d6a2322497e3)
However, S and T do not commute:
![{\displaystyle S_{a}\circ T_{b}=\exp(2\pi iab)T_{b}\circ S_{a}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc34f81f2d94e8f6f4b671de3e3403842016e65)
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
![{\displaystyle U_{\tau }(\lambda ,a,b)f(z)=\lambda (S_{a}\circ T_{b}f)(z)=\lambda \exp(i\pi b^{2}\tau +2\pi ibz)f(z+a+b\tau )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3422468d42815884ccb525d46a0a7ed02dbe9719)
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 space
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
![{\displaystyle \Vert f\Vert _{\tau }^{2}=\int _{\mathbb {C} }\exp \left({\frac {-\pi y^{2)){\Im \tau ))\right)|f(x+iy)|^{2}\ dx\ dy.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/879102a7daaacb2aa020927aadada28be7140956)
Here,
is the imaginary part of
and the domain of integration is the entire complex plane.
Mumford sets the norm as
, but in this way
is not unitary.
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].
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
and
are isomorphic as H-modules. Let
![{\displaystyle M(a,b,c)={\begin{bmatrix}1&a&c\\0&1&b\\0&0&1\end{bmatrix))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/926fd248dde074f6e1829d271593b89fd06bfd0b)
stand for a general group element of
In the canonical Weyl representation, for every real number h, there is a representation
acting on
as
![{\displaystyle \rho _{h}(M(a,b,c))\psi (x)=\exp(ibx+ihc)\psi (x+ha)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17594fd7d769b13bec779fd7ec5b9c6d2c425f40)
for
and
Here, h is Planck's constant. Each such representation is unitarily inequivalent. The corresponding theta representation is:
![{\displaystyle M(a,0,0)\to S_{ah))](https://wikimedia.org/api/rest_v1/media/math/render/svg/56506917c6d66cd89efb6698e4d6c104de74a62c)
![{\displaystyle M(0,b,0)\to T_{b/2\pi ))](https://wikimedia.org/api/rest_v1/media/math/render/svg/556ecef20cf7d9ee4e41a957420a767d613402fe)
![{\displaystyle M(0,0,c)\to e^{ihc))](https://wikimedia.org/api/rest_v1/media/math/render/svg/689424c636c2bd05459584d2ea1bcca916946240)
Discrete subgroup
Define the subgroup
as
![{\displaystyle \Gamma _{\tau }=\{U_{\tau }(1,a,b)\in H_{\tau }:a,b\in \mathbb {Z} \}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a622e56932263c9c0e4be2566ff434cde250cbe6)
The Jacobi theta function is defined as
![{\displaystyle \vartheta (z;\tau )=\sum _{n=-\infty }^{\infty }\exp(\pi in^{2}\tau +2\pi inz).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b56398f818edd5933b21c2f8cefdf521567d779a)
It is an entire function of z that is invariant under
This follows from the properties of the theta function:
![{\displaystyle \vartheta (z+1;\tau )=\vartheta (z;\tau )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71844ca49db38dd459333a0e16e9201b3d0ebf37)
and
![{\displaystyle \vartheta (z+a+b\tau ;\tau )=\exp(-\pi ib^{2}\tau -2\pi ibz)\vartheta (z;\tau )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc094c89033d0c81583138fdb2f4f5305856d86a)
when a and b are integers. It can be shown that the Jacobi theta is the unique such function.