Theta representation

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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.

1 Construction
- 1.1 Group generators
- 1.2 Hilbert space
2 Isomorphism
3 Discrete subgroup
4 References

[edit] Construction

The theta representation is a representation of the continuous Heisenberg group $H_3(\mathbb{R})$ 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.

[edit] Group generators

Let f(z) be a holomorphic function, let a and b be real numbers, and let $τ$ be fixed, but arbitrary complex number in the upper half-plane; that is, so that the imaginary part of $τ$ is positive. Define the operators S_a and T_b such that they act on holomorphic functions as

(S a f)(z) = f (z + a)

and

(T b f)(z) = exp(i π b 2 τ + 2π i b z) f (z + b τ).

It can be seen that each operator generates a one-parameter subgroup:

$S_{a_1} (S_{a_2} f) = (S_{a_1} \circ S_{a_2}) f = S_{a_1+a_2} f$

and

$T_{b_1} (T_{b_2} f) = (T_{b_1} \circ T_{b_2}) f = T_{b_1+b_2} f,$

However, S and T do not commute:

$S_a \circ T_b = \exp (2\pi iab) \; T_b \circ S_a.$

Thus we see that S and T together with a unitary phase form a nilpotent Lie group, the (continuous real) Heisenberg group, parametrizable as $H=U(1)\times\mathbb{R}\times\mathbb{R}$ where U(1) is the unitary group. A general group element $U_\tau(\lambda,a,b)\in H$ then acts on a holomorphic function f(z) as

$U_\tau(\lambda,a,b)\;f(z)=\lambda (S_a \circ T_b f)(z) = \lambda \exp (i\pi b^2 \tau +2\pi ibz) f(z+a+b\tau)$

where $\lambda \in U(1)$ . U(1) = Z(H) is the center of H, the commutator subgroup [H, H]. The parameter $τ$ on $U τ (λ, a, b)$ serves only to remind that every different value of $τ$ gives rise to a different representation of the action of the group.

[edit] Hilbert space

The action of the group elements $U τ (λ, a, b)$ 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

$\Vert f \Vert_\tau ^2 = \int_{\mathbb{C}} \exp \left( \frac {-2\pi y^2} {\Im \tau} \right) |f(x+iy)|^2 \ dx \ dy.$

Here, $\Im \tau$ is the imaginary part of $τ$ and the domain of integration is the entire complex plane. Let $\mathcal{H}_\tau$ be the set of entire functions f with finite norm. The subscript $τ$ is used only to indicate that the space depends on the choice of parameter $τ$ . This $\mathcal{H}_\tau$ forms a Hilbert space. The action of $U τ (λ, a, b)$ given above is unitary on $\mathcal{H}_\tau$ , that is, $U τ (λ, a, b)$ preserves the norm on this space. Finally, the action of $U τ (λ, a, b)$ on $\mathcal{H}_\tau$ is irreducible.

[edit] 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 $\mathcal{H}_\tau$ and L²(R) are isomorphic as H-modules. Let

$\operatorname{M}(a,b,c) = \begin{bmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{bmatrix}$

stand for a general group element of $H_3(\mathbb{R})$ . In the canonical Weyl representation, for every real number h, there is a representation $ρ h$ acting on L²(R) as

$\rho_h(M(a,b,c))\;\psi(x)= \exp (ibx+ihc) \psi(x+ha)$