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Consider two massless representations of the connected Poincare group $ISO_0(1,3)$ with helicities $s$ and $t$. What is the decomposition of their tensor product into irreducibles?

Massless representations with helicity s are defined in Wigner's classification of irreducible unitary representations of the connected Poincare group, the semidirect product $ISO_0(1,3)$ of the connected Lorentz group $SO_0(1,3)$ and the 4-dimensional translation group.

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    $\begingroup$ What is "massless"? what is "helicity"? representation means irreducible complex finite-dimensional representation? does "$ISO_0(3,1)$" mean the same as "$SO_0(3,1)$"? (SO stands for standard orthogonal, no idea about ISO). $\endgroup$
    – YCor
    Commented Oct 10, 2018 at 8:10
  • $\begingroup$ @YCor I means inhomogeneous, the semidirect product with the translation group. massless representations with helicity s are defined in Wigner's classification of irreducible unitary representations of the Poincare group. $\endgroup$
    – Arnold Neumaier
    Commented Oct 10, 2018 at 9:33
  • $\begingroup$ @YCor Infinite-dimensional unitary representations on a Hilbert space. $\endgroup$
    – Robert Furber
    Commented Oct 10, 2018 at 10:15

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I think the answer is given in the paper https://aip.scitation.org/doi/10.1063/1.1703659 (Decomposition of Direct Products of Representations of the Inhomogeneous Lorentz Group, by J. S. Lomont).

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    $\begingroup$ Thanks! There is a second paper by Lomont and Moses that gives a more explicit reduction: Reduction of Reducible Representations of the Infinitesimal Generators of the Proper, Orthochronous, Inhomogeneous Lorentz Group,aip.scitation.org/doi/pdf/10.1063/1.1705287 $\endgroup$
    – Arnold Neumaier
    Commented Oct 10, 2018 at 10:20
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    $\begingroup$ +1. Also Moussa–Stora (1965), Schaaf (1970), Barut–Rączka (1977, p. 553): $$U^{0,\,J_1}\otimes U^{0,\,J_2}\cong\int_0^\infty dM\sum_{J=|J_1-J_2|}^{\infty} \oplus U^{M,\,J}.$$ $\endgroup$
    – Francois Ziegler
    Commented Oct 10, 2018 at 10:37
  • $\begingroup$ @FrancoisZiegler: But this is the massive case, whereas I had asked for the massless case. $\endgroup$
    – Arnold Neumaier
    Commented Oct 10, 2018 at 10:50
  • $\begingroup$ @Arnold No, massive would be $U^{M_1,\,J_1}\otimes U^{M_2,\,J_2}$. $\endgroup$
    – Francois Ziegler
    Commented Oct 10, 2018 at 10:52
  • $\begingroup$ @FrancoisZiegler: But massless cases have positive and negative helicity for each J, and even ignoring that, the formula you gave is not consistent with that in Lomont's paper - there is an extra contribution of massless representations with inifinite multiplicity. $\endgroup$
    – Arnold Neumaier
    Commented Oct 10, 2018 at 11:20

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