Convergent $\tilde{Y}$-Map for a new covariant Loop Quantum Gravity formulation

Abstract

The most important part of the new spin-foam loop quantum gravity formulation is the map $Y$: $H^{SU(2)} \rightarrow H^{SL(2,C)}$. It was only recently shown that the Y-Map is convergent in spite of the fact that the classical Peter-Weyl theorem is not applicable to it, as Lorentz group is not compact. In this paper we provide an alternative map $\tilde{Y}$. The $\tilde{Y}$ map has an advantage of preserving the Lorentz covariance, which gets broken in the case of Y-Map. The image of a new map $\tilde{Y}$ contains the weighted infinite sum of $SL(2,C)$ matrix coefficients. The sum is convergent and its limit is the square integrable functions of $SL(2,C)$ with the measure $L^2(g, e^{-|Y|^2/\hbar}\eta(g) du \,dY )$ according to the Holomorphic Huebschmann-Peter-Weyl theorem, which is applicable to the rational representations of the non-unitary groups, particularly non-unitary finite Lorenz representations. Since in LQG the unitary evolution is not mandatory as it does not follow from the Wheeler-DeWitt dynamics equation, the choice of the non-unitary representation is valid. As it was stated in the original LQG formulation: "there is no sense in which conventional unitarity is necessary in the theory".