We introduce a new causal spinfoam vertex for 4D Lorentzian quantum gravity. The causal data are encoded in Toller $T$-matrices, which add to Wigner $D$-matrices $T^{(+)}+T^{(-)}=D$, and for which we provide a Feynman $\mathrm{i}ϵ$ representation. We discuss how the Toller poles cancel in the Engle-Pereira-Rovelli-Livine vertex, how the Livine-Oriti model is obtained in the Barrett-Crane limit, and how spinfoam causal data are distinct from Regge causal data. In the large-spin limit, we show that only Lorentzian Regge geometries with causal data compatible with the spinfoam data are selected, resulting in a single exponential $\exp (+\mathrm{i}{S}_{\text{Regge}}/\hslash )$ and a new form of causal rigidity.

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• Loop quantum gravity
• Quantum gravity