Lorentz Spin-Foam with Non Unitary Representations by use of Holomorphic Peter-Weyl Theorem

Abstract

In quantum gravity the unitary evolution does not follow from the Wheeler-DeWitt dynamics equation as it follows from the Schr\"odinger equation in non-relativistic quantum mechanics. Therefore we can define a spin-foam model based on SL(2,C) spinor finite non-unitary representations. The recently discovered holomorphic Peter-Weyl theorem \cite{Huebschmann} made it possible to decompose the delta function of a non-compact Lorentz group into the convergent sum of the matrix coefficients. We calculate the vertex amplitude with the help of that theorem and obtain a simple expression for our model. The $SL(2,C)$ Hilbert space is defined from $SU(2)$ Hilbert space by Huebschmann-Kirillov transform \cite{Huebschmann}. A new transform is simpler than the well known Hall transform as it does not contain a heat kernel convolution. We do not set Barbero-Immirzi constant $\gamma$ a priori, instead we obtain it as a solution of the diagonal and off-diagonal simplicity constraints being $\gamma = \frac{-in}{(|n| + 2p)}$ where $p$ is a non-negative half-integer. When $p=0$ the solution corresponds to the Ashtekar's self-dual connections. We point out that the Barbero-Immirzi becomes real when one chooses a unitary representation. It is complex when the representation is non-unitary principal series or non-unitary spinor representation.