Finitary Spacetime Sheaves of Quantum Causal Sets: Curving Quantum Causality

Abstract

A locally finite, causal and quantal substitute for a locally Minkowskian principal fiber bundle $\mathcal{P}$ of modules of Cartan differential forms $\omg$ over a bounded region $X$ of a curved $C^{\infty }$-smooth differential manifold spacetime $M$ with structure group $\mathbf{G}$ that of orthochronous Lorentz transformations $L^{+}:=SO(1,3{)}^{\uparrow }$, is presented. $\mathcal{P}$ is the structure on which classical Lorentzian gravity, regarded as a Yang-Mills type of gauge theory of a $sl(2,\com)$-valued connection 1-form $\mathcal{A}$, is usually formulated. The mathematical structure employed to model this replacement of $\mathcal{P}$ is a principal finitary spacetime sheaf ${\vec{\mathcal{P}}}_n$ of quantum causal sets $\amg_{n}$ with structure group ${\mathbf{G}}_n$, which is a finitary version of the group $\mathbf{G}$ of local symmetries of General Relativity, and a finitary Lie algebra ${\mathbf{g}}_n$-valued connection 1-form ${\mathcal{A}}_n$ on it, which is a section of its sub-sheaf $\amg^{1}_{n}$. ${\mathcal{A}}_n$ is physically interpreted as the dynamical field of a locally finite quantum causality, while its associated curvature ${\mathcal{F}}_n$, as some sort of `finitary Lorentzian quantum gravity.

This is a shortened version of the abstract. The paper is submitted to the International Journal of Theoretical Physics