Causally Confined Euclidean Saddles in Spin-Foam Quantum Gravity

We investigate the semiclassical structure of spin-foam transition amplitudes for boundary data that do not admit a real Lorentzian Regge geometry. Considering a fixed triangulation with a single dominant vertex, we demonstrate that when boundary tetrahedra carry mutually incompatible causal orientations, the closure equations have no real solution and the path integral is dominated by a complex Euclidean saddle of the Regge action. In this regime the vertex amplitude acquires a non-oscillatory factor of the form exp(−SE/ℏ), where SE is the Euclidean action evaluated at the complex saddle. We introduce a causal-obstruction criterion based on a convexity argument for the future timelike cone in R 3,1 , and establish a formal classification of boundary data into three types according to the existence and nature of the saddle-point solutions. We show that SE scales linearly with the spin parameter j in the semiclassical limit, SE = ℏ j C(α)/(8πG), where C(α) is a finite dimensionless geometric constant, providing explicit control over the suppression. Non-degeneracy of the Hessian at the complex saddle is verified after gauge fixing, confirming the validity of the saddle-point approximation. The results constitute a proof-of-concept demonstration that exponentially suppressed, causally confined quantum-geometric transitions emerge as a structural feature of the covariant formulation of loop quantum gravity, without additional postulates.