3+1 spinfoam model of quantum gravity with spacelike and timelike components
Alejandro Perez, Carlo Rovelli
TL;DR
The paper addresses constructing a Lorentzian, background-independent quantum gravity theory in the spinfoam framework that accommodates both spacelike and timelike surfaces. It introduces two SL(2,$\mathbb{C}$) based models, $S^{(+)}[\phi]$ and $S^{(-)}[\phi]$, implemented via distinct simplicity constraints (via $SU(2)$ and $SU(1,1)\times Z_2$) and yielding amplitudes expressed as integrals over Lobachevskian spaces with kernels $K^{\pm}$. The plus model reproduces a Barrett–Crane-like vertex on $H^{+}$, while the minus model includes discrete series through $H^{-}$, enabling a fully Lorentzian geometry with both spacelike and timelike components; harmonic analysis provides explicit forms for $K^{\pm}$ and the delta resolutions on the two spaces, underpinning finite edge and vertex amplitudes. The work offers a non-perturbative, covariant quantum gravity framework with a natural local causal structure and quantum geometrical spectra compatible with loop quantum gravity on spatial slices, while highlighting finite-regulator considerations and potential regularization via quantum groups.
Abstract
We present a spinfoam formulation of Lorentzian quantum General Relativity. The theory is based on a simple generalization of an 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 spinfoams in the model, corresponding to quantized 4-geometries, carry a natural non-perturbative local causal structure induced by the geometry of the algebra of the internal gauge (sl(2,C)). Amplitudes can be expressed as integrals over the spacelike unit-vectors hyperboloid in Minkowski space, or the imaginary Lobachevskian space.