Analog of the Peter-Weyl Expansion for Lorentz Group

Abstract

The expansion of a square integrable function on $SL(2,C)$ into the sum of the principal series matrix coefficients with the specially selected representation parameters was recently used in the Loop Quantum Gravity $\cite{RovelliBook2}$, $\cite{Rovelli2010}$. In this paper we prove that the sum $\sum _{j=1}^{\infty }\sum _{|m|\le j}\sum _{|n|\le j}\frac{D_{jm,jn}^{(j,\tau j)}(g)}{j^k}$, where $j,m,n\in Z,\tau \in C$ is convergent to a square integrable function on $SL(2,C)$. We also prove that for each fixed m: $\sum _{j=1}^{\infty }\frac{D_{jm,jm}^{(j,\tau j)}(g)}{j^k}$ is convergent and that the limit is a square integrable function on $SL(2,C)$. We then prove convergence of the sums $\sum _{j=|p|}^{\infty }\sum _{|m|\le j}\sum _{|n|\le j}{d}_{pm}^{\frac{j}{2}}{D}_{jm,jn}^{(j,\tau j)}(g)$, where $d_{|p|m}^{\frac{j}{2}}=(j+1{)}^{\frac{1}{2}}\underset{SU(2)}{\int }\varphi (u)\overline{D_{|p|m}^{\frac{j}{2}}(u)}du$ is $\varphi (u)$'s Fourier transform and $p,j,m,n\in Z,\tau \in C,u\in SU(2),g\in SL(2,C)$, thus establishing the map between the square integrable functions on $SU(2)$ and the space of the functions on $SL(2,C)$. Such maps were first used in $\cite{RovelliBook2}$.

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Cited by in corpus (2)

• Barbero-Immirzi parameter as a solution of the simplicity constraints
• Convergent $\tilde{Y}$-Map for a new covariant Loop Quantum Gravity formulation