A Lorentzian Signature Model for Quantum General Relativity
John W. Barrett, Louis Crane
TL;DR
The paper develops a Lorentzian, spin-network approach to quantum gravity by promoting bivectors to unitary representations of the Lorentz group and its quantum deformation. It builds a state-sum (or state integral) model based on principal-series representations $R(k,p)$ of ${ m SO}(3,1)$ realized on hyperboloids via the Gelfand–Graev transform, and introduces a Lorentzian 4-simplex amplitude expressed through a 10J-like vertex built from $R(0,p)$ channels. It defines and regularizes relativistic spin-network evaluations using kernels $K_p(x,y)$, provides finite regularized results for the 10J symbol in the Lorentzian setting, and relates the Lorentzian theory to its Euclidean counterpart with careful normalization. Moving to the Quantum Lorentz Algebra, the work argues for a finite measure on spacelike sectors and outlines a program to define the necessary $3J$/$6J$/$10J$ symbols, highlighting both the promise of a finite, Lorentzian quantum gravity model and the technical challenges ahead. Overall, it lays foundational steps toward a finite, Lorentzian spin-foam-like framework for quantum general relativity and connects noncompact representation theory with quantum-group techniques.
Abstract
We give a relativistic spin network model for quantum gravity based on the Lorentz group and its q-deformation, the Quantum Lorentz Algebra. We propose a combinatorial model for the path integral given by an integral over suitable representations of this algebra. This generalises the state sum models for the case of the four-dimensional rotation group previously studied in gr-qc/9709028. As a technical tool, formulae for the evaluation of relativistic spin networks for the Lorentz group are developed, with some simple examples which show that the evaluation is finite in interesting cases. We conjecture that the `10J' symbol needed in our model has a finite value.