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 that satisfy the relation $T^{(+)}+T^{(-)}=D$, where the Wigner matrix $D$ provides a unitary irreducible representation of $SL(2,\mathbb{C})$. Rühl’s definition of $T^{(\pm )}$ in terms of analyticity and asymptotic properties is shown to be equivalent to the recently introduced Feynman $\mathrm{i}ϵ$ 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 $SL(2,\mathbb{C})$ spinfoams studied by Donà, Gozzini, and Nicotra. We provide explicit expressions in terms of hypergeometric functions and specialize them to the $\gamma$-simple representations relevant for spinfoams.

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• Loop quantum gravity
• Quantum gravity
• Group theory