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Toller matrices and the Feynman iϵ in spinfoams

Eugenio Bianchi1,2,*, Chaosong Chen1,2,†, and Mauricio Gamonal1,2,‡

  • *Contact author: ebianchi@psu.edu
  • †Contact author: cchen@psu.edu
  • ‡Contact author: mgamonal@psu.edu
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Phys. Rev. D 114, 046014 – Published 13 August, 2026

DOI: https://doi.org/10.1103/v3kc-4n3n

Abstract

We study the analytic properties and three equivalent representations of the Toller matrices T(±), which appear in the causal formulation of spinfoam transition amplitudes for 4D Lorentzian quantum gravity. These are polynomially bounded functions on the Lorentz group that satisfy the relation T(+)+T(−)=D, where the Wigner matrix D provides a unitary irreducible representation of SL(2,C). Rühl’s definition of T(±) in terms of analyticity and asymptotic properties is shown to be equivalent to the recently introduced Feynman iϵ prescription in spinfoams. We show that, equivalently, they can be represented as an integral over eigenvalues of the boost operator, which results in a sum over residues. The latter reproduces the Wick rotation relating Euclidean Spin(4) to Lorentzian SL(2,C) spinfoams studied by Donà, Gozzini, and Nicotra. We provide explicit expressions in terms of hypergeometric functions and specialize them to the γ-simple representations relevant for spinfoams.

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Phys. Rev. D 114, 046014– Published 13 August, 2026
  • Received 1 May 2026
  • Accepted 29 June 2026
Article part of CHORUS
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