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

In quantum gravity the unitary evolution does not follow from the Wheeler-DeWitt dynamics equation as it follows from the Schrödinger 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 γ a priori, instead we obtain it as a solution of the diagonal and off-diagonal simplicity constraints being γ= 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.