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 [C. Rovelli and F. Vidotto, Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory (Cambridge University Press, Cambridge, 2014) and C. Rovelli, Classical Quantum Gravity 28(11), 114005 (2011)]. In this paper, we prove that the sum used originally in the Loop Quantum Gravity: ∑j=0∞∑m≤j∑n≤jDjm,jn(j,τj)(g), where j, m, n ∈ Z, τ ∈ C is convergent to a function on SL(2, C); however, the limit is not a square integrable function; therefore, such sums cannot be used for the Peter-Weyl like expansion. We propose the alternative expansion and prove that for each fixed m: ∑j=m∞Djm,jm(j,τj)(g) is convergent and that the limit is a square integrable function on SL(2, C). We then prove the analog of the Peter-Weyl expansion: any ψ(g) ∈ L2(SL(2, C)) can be decomposed into the sum: ψ(g)=∑j=m∞j2(1+τ2)cjmmDjm,jm(j,τj)(g), with the Fourier coefficients cjmm=∫SL(2,C)ψ(g)Djm,jmj,τj(g)¯dg, g ∈ SL(2, C), τ ∈ C, τ ≠ i, − i, j, m ∈ Z, m is fixed. We also prove convergence of the sums ∑j=p∞∑m≤j∑n≤jdpmj2Djm,jn(j,τj)(g), where dpmj2=(j+1)12∫SU(2)ϕ(u)Dpmj2(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 Rovelli [Class. Quant. Grav. 28, 11 (2011)].