Investigations on Lorentzian Spin-foams and Semiclassical Space-times

TL;DR

This thesis advances the understanding of Lorentzian spin-foam models in quantum gravity, proposing new amplitudes, boundary states, and semiclassical analyses that connect to Regge theory and generalize Minkowski's theorem.

Contribution

It introduces a complete Lorentzian spin-foam amplitude for space- and time-like triangles and analyzes its semiclassical limit, extending Minkowski's theorem to Lorentzian geometries.

Findings

Proposed a Lorentzian spin-foam amplitude asymptoting to Regge theory.

Generalized Minkowski's theorem to Lorentzian signature.

Developed a method for asymptotic evaluation of parameter-dependent integrals.

Abstract

This thesis is developed in the context of the spin-foam approach to quantum gravity; all results are concerned with the Lorentzian theory and with semiclassical methods. A correspondence is given between Majorana 2-spinors and time-like hypersurfaces in Minkowski 3-space based on complexified quaternions. It is shown that the former suggest a symplectic structure on the spinor phase space which, together with an area-matching constraint, yields a symplectomorphism to $T^*\mathrm{SU}(1,1)$. A complete 3-dimensional Lorentzian spin-foam amplitude for both space- and time-like triangles is proposed. It is shown to asymptote to Regge theory in the semiclassical regime. The asymptotic limit of the 4-dimensional Conrady-Hnybida model for general polytopes is scrutinized. Minkowski's theorem on convex polyhedra is generalized to Lorentzian signature, and new boundary states for time-like polygons are introduced. It is found that the semiclassical amplitude for such polygons is insufficiently constrained. A method for the asymptotic evaluation of integrals subject to external parameters is discussed. The method is developed in detail for the special problem of spin-foam gluing constraints away from their dominant critical points. A relation to the gluing constraints of effective spin-foams is suggested.