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Analog of the Peter-Weyl Expansion for L...
Perlov, Leonid...

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Analog of the Peter-Weyl Expansion for Lorentz Group by Perlov, Leonid ( Author )
Lemonjar Open Access
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04-09-2015
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 ∑j=1∞∑|m|≤j∑|n|≤jD(j,τj)jm,jn(g)jk, where j,m,n∈Z,τ∈C is convergent to a square integrable function on SL(2,C). We also prove that for each fixed m: ∑j=1∞D(j,τj)jm,jm(g)jk is convergent and that the limit is a square integrable function on SL(2,C). We then prove convergence of the sums ∑j=|p|∞∑|m|≤j∑|n|≤jdj2pmD(j,τj)jm,jn(g), where dj2|p|m=(j+1)12∫SU(2)ϕ(u)Dj2|p|m(u)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯du is ϕ(u)'s Fourier transform and p,j,m,n∈Z,τ∈C,u∈SU(2),g∈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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https://arxiv.org/abs/1509.01312
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