Causal spinfoam vertex for 4d Lorentzian quantum gravity

TL;DR

This work develops a causal spinfoam vertex for $4$-dimensional Lorentzian quantum gravity by encoding causal data with Toller $T$-matrices and a Feynman $i\varepsilon$ prescription. It clarifies how these causal structures relate to the EPRL vertex, showing that $\langle A_v^{\mathrm{EPRL}}|=\sum_{\kappa_{ab}=\pm1}\langle A_v^{(\kappa_{ab})}|$ but that a sum over causal orientations $\sigma_a$ does not reproduce the EPRL amplitude, and it recovers the Livine–Oriti Barrett–Crane limit in the appropriate regime. In the large-spin semiclassical limit for boundary data corresponding to a non-degenerate Lorentzian $4$-simplex, only geometries with $\sigma_a\sigma_b=+s_a s_b$ contribute with a single oscillatory phase $e^{+ i S_{\mathrm{Regge}}/\hbar}$, revealing a form of causal rigidity and a clean connection to Lorentzian Regge geometry. The results link the causal spinfoam framework to coherent-state representations, clarify relations to the EPRL and proper-vertex constructions, and outline future directions for multi-vertex amplitudes, finiteness questions, and phenomenological implications.

Abstract

We introduce a new causal spinfoam vertex for $4$d 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 $\mathrm{i}\varepsilon$ representation. We discuss how the Toller poles cancel in the EPRL 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 $\exp(+\mathrm{i}\, S_{\mathrm{Regge}}/\hbar)$ and a new form of causal rigidity.

Figures (2)

• Figure 1: Panel (a): The combinatorial causal data $\sigma_a$ are associated with each oriented edge (blue arrows) ingoing or outgoing from a vertex $v$ (purple dot). The figure depicts a $1\to 4$ transition from one spin-network node $n$ to four nodes (black dots). The wedge $(12)$ bounded by two edges (green plane) carries causal data $\sigma_1 \sigma_2=-1$. Panel (b): The Regge causal data $s_a$ are associated with the $4$-normals to the spacelike tetrahedra bounding a Lorentzian $4$-simplex. The figure depicts the Regge transitions $1\to 4$ and $2\to 3$. Of the two, only the $1\to 4$ transition---with one tetrahedron on the past boundary and four on the future boundary of the $4$-simplex---is compatible with the combinatorial data depicted in panel (a), as it satisfies the causal rigidity condition $\sigma_a \sigma_b=s_a s_b$, for all couples $(ab)$.
• Figure 2: Contours in the complex $\rho$-plane of the absolute value of the reduced Wigner and Toller matrices. For illustration, we display the case $k=j=l=1$, $m=0$, and $\beta=\pi/2$; their explicit analytic forms are provided in Appendix \ref{['app:BoosterFunctions']}. The markers $\times$ indicate the positions of the Toller poles at $\mathrm{i} \rho = -1, 0,+1$. The figure highlights the analytic structure, asymptotic fall-off, and matching conditions that characterize these functions.