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The most important part of the new spin-foam loop quantum gravity formulationis the map $Y$: $H^{SU(2)} \rightarrow H^{SL(2,C)}$. It was only recently shownthat the Y-Map is convergent in spite of the fact that the classical Peter-Weyltheorem is not applicable to it, as Lorentz group is not compact. In this paperwe provide an alternative map $\tilde{Y}$. The $\tilde{Y}$ map has an advantageof 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 thesquare integrable functions of $SL(2,C)$ with the measure $L^2(g,e^{-|Y|^2/\hbar}\eta(g) du \,dY )$ according to the HolomorphicHuebschmann-Peter-Weyl theorem, which is applicable to the rationalrepresentations of the non-unitary groups, particularly non-unitary finiteLorenz representations. Since in LQG the unitary evolution is not mandatory asit does not follow from the Wheeler-DeWitt dynamics equation, the choice of thenon-unitary representation is valid. As it was stated in the original LQGformulation: "there is no sense in which conventional unitarity is necessary inthe theory".

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