Analog of the Peter-Weyl Expansion for Lorentz Group

TL;DR

This paper proves the convergence of certain series expansions of functions on the Lorentz group $SL(2,C)$, establishing a mathematical foundation for their use in Loop Quantum Gravity and connecting functions on $SU(2)$ to those on $SL(2,C)$.

Contribution

It provides rigorous proofs of convergence for series expansions of functions on $SL(2,C)$, extending the Peter-Weyl theorem analog to the Lorentz group, with implications for quantum gravity.

Findings

Series expansions on $SL(2,C)$ converge to square integrable functions.

Convergence of sums involving Fourier transforms links $SU(2)$ functions to $SL(2,C)$ functions.

Established a mathematical map between functions on $SU(2)$ and $SL(2,C)$.

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 $\sum\limits_{j=1}^{\infty}\sum\limits_{|m| \le j}\sum\limits_{|n| \le j} \frac{D^{(j, \tau j)}_{jm, jn}(g)}{j^k}$, 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: $\sum\limits_{j=1}^{\infty}\frac{D^{(j, \tau j)}_{jm, jm}(g)}{j^k}$ is convergent and that the limit is a square integrable function on $SL(2,C)$. We then prove convergence of the sums $\sum\limits_{j=|p|}^{\infty}\sum\limits_{|m| \le j}\sum\limits_{|n| \le j} d^{\frac{j}{2}}_{pm} D^{(j, \tau j)}_{jm, jn}(g)$, where $d^{\frac{j}{2}}_{|p|m} = (j+1)^{\frac{1}{2}}\int\limits_{SU(2)}\phi(u)\overline{ D^{\frac{j}{2}}_{|p|m}(u)} \; du \; $ is $\phi(u)$'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 $\cite{RovelliBook2}$.