Toller matrices and the Feynman $i\varepsilon$ in spinfoams

TL;DR

This paper analyzes the properties of Toller matrices in Lorentzian quantum gravity spinfoams, connecting their analyticity, integral representations, and the Feynman $i extepsilon$ prescription.

Contribution

It demonstrates the equivalence of Ruhl's analyticity-based definition of Toller matrices with the Feynman $i extepsilon$ prescription and provides explicit hypergeometric function expressions.

Findings

Toller matrices satisfy a key relation with the Wigner matrix D.

They can be represented as integrals over boost eigenvalues, leading to residue sums.

Explicit hypergeometric function formulas are derived for relevant representations.

Abstract

We study the analytic properties and three equivalent representations of the Toller matrices $T^{(\pm)}$ which appear in the causal formulation of spinfoam transition amplitudes for 4d Lorentzian quantum gravity. These are polynomially bounded functions on the Lorentz group which satisfy the relation $T^{(+)}+T^{(-)}=D$, where the Wigner matrix $D$ provides a unitary irreducible representation of $SL(2,C)$. Ruhl's definition of $T^{(\pm)}$ in terms of analyticity and asymptotic properties is shown to be equivalent to the recently introduced Feynman $i\varepsilon$ 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 Dona, Gozzini and Nicotra. We provide explicit expressions in terms of hypergeometric functions and specialize them to the $\gamma$-simple representations relevant for spinfoams.