Black Holes

Fock model

The action of SU(1,1) on holomorphic Fock space was described by Bargmann (1970) and Itzykson (1967).

The metaplectic double cover of SU(1,1) can be constructed explicitly as pairs (g, γ) with

$\displaystyle{g=\begin{pmatrix} \alpha & \beta \\ \overline{\beta} & \overline{\alpha} \end{pmatrix}}$

and

$\displaystyle{\gamma^2=\alpha.}$

If g = g₁g₂, then

$\displaystyle{\gamma = \gamma_1\gamma_2\left(1 +{\beta_1\overline{\beta_2}\over \alpha_1\alpha_2}\right)^{1/2},}$

using the power series expansion of (1 + z)^1/2 for |z| < 1.

The metaplectic representation is a unitary representation π(g, γ) of this group satisfying the covariance relations

$\displaystyle{\pi(g,\gamma) W_{\mathcal F}(z) \pi(g,\gamma)^*= W_{\mathcal F}(g\cdot z),}$

where

$\displaystyle{g\cdot z=\alpha z + \beta \overline{z}.}$

Since $\mathcal F$ is a reproducing kernel Hilbert space, any bounded operator T on it corresponds to a kernel given by a power series of its two arguments. In fact if

$\displaystyle{K_T(a,b)=(TE_{\overline{b}},E_a),}$

and F in $\mathcal F$ , then

$TF(a)=(TF,E_a)=(F,T^*E_a)=\frac{1}{\pi} \iint_{\mathbf C} F(z)\overline{(T^*E_a,E_z)} e^{-|z|^2}\, dx dy=\frac{1}{\pi} \iint_{\mathbf C} K_T(a,\overline{z}) F(z)e^{-|z|^2}\, dxdy.$

The covariance relations and analyticity of the kernel imply that for S = π(g, γ),

$\displaystyle{K_S(a,z)=C \cdot \exp\,{1\over 2\alpha}(\overline{\beta} z^2 + 2az - \beta a^2)}$

for some constant C. Direct calculation shows that

$\displaystyle{C=\gamma^{-1}}$

leads to an ordinary representation of the double cover.

Coherent states can again be defined as the orbit of E₀ under the metaplectic group.

For w complex, set

$\displaystyle{ F_w(z)=e^{wz^2/2}.}$

Then F_w lies in $\mathcal F$ if and only if |w| < 1.

In particular F₀ = 1 = E₀.

Moreover

$\pi(g,\gamma)F_w= (\overline{\alpha} +\overline{\beta}w)^{-\frac{1}{2}} F_{gw}=\frac{1}{\overline{\gamma}} \left(1+{\overline{\beta}\over \overline{\alpha}}w\right)^{-1/2}F_{gw},$

where

$\displaystyle{gw={\alpha w + \beta\over \overline{\beta}w + \overline{\alpha}} .}$

Similarly the functions zF_w lie in $\mathcal F$ and form an orbit of the metaplectic group:

$\displaystyle{\pi(g,\gamma)[zF_w](z)= (\overline{\alpha} +\overline{\beta}w)^{-3/2} zF_{gw}(z).}$

Since (F_w, E₀) = 1, the matrix coefficient of the function E₀ = 1 is given by^[15]

$\displaystyle{(\pi(g,\gamma)1,1)=\gamma^{-1}.}$

Black Holes

Friday, 21 June 2013

Fock model

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