The Einstein-Cartan-Dirac (ECD) theory

Abstract

There are various generalizations of Einstein's theory of gravity (GR); one of which is the Einstein- Cartan (EC) theory. It modifies the geometrical structure of manifold and relaxes the notion of affine connection being symmetric. The theory is also called $U_4$ theory of gravitation; where the underlying manifold is not Riemannian. The non-Riemannian part of the space-time is sourced by the spin density of matter. Here mass and spin both play the dynamical role. We consider the minimal coupling of Dirac field with EC theory; thereby calling the full theory as Einstein-Cartan-Dirac (ECD) theory. In the recent works by T.P Singh titled ``A new length scale in quantum gravity \cite{TP_1}", the idea of new unified mass dependent length scale $L_{cs}$ has been proposed. We discuss this idea and formulate ECD theory in both - standard as well as this new length scale. We found the non-relativistic limit of ECD theory using WKB-like expansion in $\sqrt{\hbar}/c$ of the ECD field equations with both the length scales. At leading order, ECD equations with standard length scales give Schr\"{o}dinger-Newton equation. With $L_{cs}$, in the low mass limit, it gives source-free Poisson equation, suggesting that small masses don't contribute to gravity at leading order. For higher mass limit, it reduces to Poisson equation with delta function source. Next, we formulate ECD theory with both the length scales (especially the Dirac equation which is also called hehl-Datta equation and Contorsion spin coefficients) in Newman-Penrose (NP) formalism. The idea of $L_{cs}$ suggests a symmetry between small and large masses. Formulating ECD theory with $L_{cs}$ in NP formalism is desirable because NP formalism happens to be the common vocabulary for the description of low masses (Dirac theory) and high masses (gravity theories). We propose a conjecture to establish this duality between small and large masses which is claimed to source the torsion and curvature of space-time respectively. We therefore call it ``Curvature-Torsion" duality conjecture. In the context of this conjecture, Solutions to HD equations on Minkowski space with torsion have been found and their implications for the conjecture are discussed. Three new works which we have done in this thesis [Non-relativistic limit of ECD theory, formulating ECD theory in NP formalism and attempts to find the solution to non-linear Dirac equation on $U_4$] are valid for standard theory and also the theory with $L_{cs}$. The conjecture to establish the Curvature-torsion duality is formulated in the context of idea of $L_{cs}$.