(3+1)-dimensional spin foam model of quantum gravity with spacelike and timelike components

We present a spin foam formulation of Lorentzian quantum general relativity. The theory is based on a simple generalization of a Euclidean model defined in terms of a field theory over a group. The model is an extension of a recently introduced Lorentzian model, in which both timelike and spacelike components are included. The spin foams in the model, corresponding to quantized 4-geometries, carry a natural nonperturbative local causal structure induced by the geometry of the algebra of the internal gauge $\left[\mathrm{sl}\right(2,C\left)\mathrm{}\right].$ Amplitudes can be expressed as integrals over the spacelike unit vectors hyperboloid in Minkowski space, or the imaginary Lobachevskian space.