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.
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[edit] Construction
The theta representation is a representation of the continuous Heisenberg group <math>H_3(\mathbb{R})</math> 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 <math>\tau</math> be fixed, but arbitrary complex number in the upper half-plane; that is, so that the imaginary part of <math>\tau</math> is positive. Define the operators Sa and Tb such that they act on holomorphic functions as
- <math>(S_a f)(z) = f(z+a)</math>
and
- <math>(T_b f)(z) = \exp (i\pi b^2 \tau +2\pi ibz) f(z+b\tau).</math>
It can be seen that each operator generates a one-parameter subgroup:
- <math>S_{a_1} (S_{a_2} f) = (S_{a_1} \circ S_{a_2}) f = S_{a_1+a_2} f</math>
and
- <math>T_{b_1} (T_{b_2} f) = (T_{b_1} \circ T_{b_2}) f = T_{b_1+b_2} f,</math>
However, S and T do not commute:
- <math>S_a \circ T_b = \exp (2\pi iab) \; T_b \circ S_a.</math>
Thus we see that S and T together with a unitary phase form a nilpotent Lie group, the (continuous real) Heisenberg group, parametrizable as <math>H=U(1)\times\mathbb{R}\times\mathbb{R}</math> where U(1) is the unitary group. A general group element <math>U_\tau(\lambda,a,b)\in H</math> then acts on a holomorphic function f(z) as
- <math>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)</math>
where <math>\lambda \in U(1)</math>. U(1) = Z(H) is the center of H, the commutator subgroup [H, H]. The parameter <math>\tau</math> on <math>U_\tau(\lambda,a,b)</math> serves only to remind that every different value of <math>\tau</math> gives rise to a different representation of the action of the group.
[edit] Hilbert space
The action of the group elements <math>U_\tau(\lambda,a,b)</math> is unitary and irreducible on a certain Hilbert space space of functions. For a fixed value of τ, define a norm on entire functions of the complex plane as
- <math>\Vert f \Vert_\tau ^2 = \int_{\mathbb{C}}
\exp \left( \frac {-2\pi y^2} {\Im \tau} \right) |f(x+iy)|^2 \ dx \ dy. </math>
Here, <math>\Im \tau</math> is the imaginary part of <math>\tau</math> and the domain of integration is the entire complex plane. Let <math>\mathcal{H}_\tau</math> be the set of entire functions f with finite norm. The subscript <math>\tau</math> is used only to indicate that the space depends on the choice of parameter <math>\tau</math>. This <math>\mathcal{H}_\tau</math> forms a Hilbert space. The action of <math>U_\tau(\lambda,a,b)</math> given above is unitary on <math>\mathcal{H}_\tau</math>, that is, <math>U_\tau(\lambda,a,b)</math> preserves the norm on this space. Finally, the action of <math>U_\tau(\lambda,a,b)</math> on <math>\mathcal{H}_\tau</math> 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 <math>\mathcal{H}_\tau</math> and L2(R) are isomorphic as H-modules. Let
- <math> \operatorname{M}(a,b,c) = \begin{bmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{bmatrix} </math>
stand for a general group element of <math>H_3(\mathbb{R})</math>. In the canonical Weyl representation, for every real number h, there is a representation <math>\rho_h</math> acting on L2(R) as
- <math>\rho_h(M(a,b,c))\;\psi(x)= \exp (ibx+ihc) \psi(x+ha)</math>
for <math>x\in\mathbb{R}</math> and <math>\psi\in L^2(\mathbb{R})</math>.
Here, h is Planck's constant. Each such representation is unitarily inequivalent. The corresponding theta representation is:
- <math>M(a,0,0) \to S_{ah}</math>
- <math>M(0,b,0) \to T_{b/2\pi}</math>
- <math>M(0,0,c) \to e^{ihc}</math>
[edit] Discrete subgroup
Define the subgroup <math>\Gamma_\tau\subset H_\tau</math> as
- <math>\Gamma_\tau = \{ U_\tau(1,a,b) \in H_\tau : a,b \in \mathbb{Z} \}.</math>
The Jacobi theta function is defined as
- <math>\vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp (\pi i n^2 \tau + 2 \pi i n z).</math>
It is an entire function of z that is invariant under <math>\Gamma_\tau</math>. This follows from the properties of the theta function:
- <math>\vartheta(z+1; \tau) = \vartheta(z; \tau)</math>
and
- <math>\vartheta(z+a+b\tau;\tau) = \exp(-\pi i b^2 \tau -2 \pi i b z)\vartheta(z;\tau)</math>
when a and b are integers. It can be shown that the Jacobi theta is the unique such function.
[edit] References
- David Mumford, Tata Lectures on Theta I (1983), Birkhauser, Boston ISBN 3-7643-3109-7
