New Curved Spacetime Dirac Equations

arXiv (Cornell University), 2007

×ØÖ Øº I propose three new curved spacetime versions of the Dirac Equation. These equations have been developed mainly to try and account in a natural way for the observed anomalous gyromagnetic ratio of Fermions. The derived equations suggest that particles including the Electron which is thought to be a point particle do have a finite spatial size which is the reason for the observed anomalous gyromagnetic ratio. A serendipitous result of the theory, is that, two of the equation exhibits an asymmetry in their positive and negative energy solutions the first suggestion of which is clear that a solution to the problem as to why the Electron and Muon-despite their acute similarities-exhibit an asymmetry in their mass is possible. The Mourn is often thought as an Electron in a higher energy state. Another of the consequences of three equations emanating from the asymmetric serendipity of the energy solutions of two of these equations, is that, an explanation as to why Leptons exhibit a three stage mass hierarchy is possible.

arXiv (Cornell University), 2007

×ØÖ Øº I propose three new curved spacetime versions of the Dirac Equation. These equations have been developed mainly to try and account in a natural way for the observed anomalous gyromagnetic ratio of Fermions. The derived equations suggest that particles including the Electron which is thought to be a point particle do have a finite spatial size which is the reason for the observed anomalous gyromagnetic ratio. A serendipitous result of the theory, is that, two of the equation exhibits an asymmetry in their positive and negative energy solutions the first suggestion of which is clear that a solution to the problem as to why the Electron and Muon-despite their acute similarities-exhibit an asymmetry in their mass is possible. The Mourn is often thought as an Electron in a higher energy state. Another of the consequences of three equations emanating from the asymmetric serendipity of the energy solutions of two of these equations, is that, an explanation as to why Leptons exhibit a three stage mass hierarchy is possible.

2007

This reading is a continuation of the earlier reading Nyambuya (2008); where three new Curved Spacetime Dirac Equations have been derived mainly to try and account in a natural way for the observed anomalous gyromagnetic ratio of fermions and the suggestions is that particles including the Electron, which is thought to be a point particle, do have a finite spatial size and this is one of the reasons for the observed anomalous gyromagnetic ratio. Combining the idea in Nyambuya (2008) which lead to the derivation of the three new Curved Spacetime Dirac Equations, and the proposed Unified Field Theory (Nyambuya 2007), a total of 12 equations each with 16 sub-components are generated thus leading to a total of 192 equations for the Curved Spacetime Dirac Equation. Some symmetries of these equations are investigated, i.e., the Lorentz symmetry, charge conjugation symmetry (C), time reversal symmetry (T), Space reversal (P) and a combination of the C, P and T - symmetries. It is shown that these equations are Lorentz invariant, obey C-symmetry and that some violate T and P-symmetry while others do not and that they all obey PT-symmetry. These symmetries show (or modestly said -- seem to suggest) that anti-particles have positive mass and energy but a negative rest-mass and the opposite sign in electronic charge. Through the inspection of these symmetries, a suggestion is (here) made to the effect that the rest-mass of a particle must be related to the electronic charge of that particle thus leading us to a possible resolution of whether or not Neutrinos do have a none-zero rest-mass. Additionally, we demonstrate that these equations have the potency to explain naturally the observed lepton generation phenomena.

2009

×ØÖ Øº This reading is a continuation of the earlier reading Nyambuya (2008); where three new Curved Spacetime Dirac Equations have been derived mainly to try and account in a natural way for the observed anomalous gyromagnetic ratio of fermions and the suggestions is that particles including the Electron, which is thought to be a point particle, do have a finite spatial size and this is one of the reasons for the observed anomalous gyromagnetic ratio. Combining the idea in Nyambuya (2008) which lead to the derivation of the three new Curved Spacetime Dirac Equations, and the proposed Unified Field Theory (Nyambuya 2007), a total of 12 equations each with 16 sub-components are generated thus leading to a total of 192 equations for the Curved Spacetime Dirac Equation. Some symmetries of these equations are investigated, i.e., the Lorentz symmetry, charge conjugation symmetry (C), time reversal symmetry (T), Space reversal (P) and a combination of the C, P &T-symmetries. It is shown that these equations are Lorentz invariant, obey C-symmetry and that some violate T and P-symmetry while others do not and that they all obey PT-symmetry. These symmetries show (or modestly said-seem to suggest) that anti-particles have positive mass and energy but a negative rest-mass and the opposite sign in electronic charge. Through the inspection of these symmetries, a suggestion is (here) made to the effect that the rest-mass of a particle must be related to the electronic charge of that particle thus leading us to a possible resolution of whether or not Neutrinos do have a none-zero rest-mass. Additionally, we demonstrate that these equations have the potency to explain naturally the observed lepton generation phenomena.

arXiv (Cornell University), 2007

This paper is a continuation of my earlier paper (Nyambuya 2007a) in which the equivalent of the Dirac Equation in curved spacetime is derived. This equation has been developed mainly to account in a natural way for the observed anomalous gyromagnetic ratio of fermions and the suggestions is that particles including the Electron, which is thought to be a point particle, do have a finite spatial size which is the reason for the observed anomalous gyromagnetic ratio. In this reading, I investigate four symmetries of this equation, Lorentz invariance, charge conjugation (C), time (T) and space (P) reversal symmetries. I show that this equation is Lorentz invariant, obeys C invariance symmetry and violates T and P symmetry but is TP &, PT invariant. These symmetries show that anti-particles have positive mass and energy but a negative rest mass and the opposite sign in electronic charge. A suggestion is made that the rest mass of a particle must be related to the electronic charge of that particle. The equivalent Klein-Gordon and Schrodinger equation in curved spacetime are discussed. It is shown that these equations imply Bosons and atoms naturally must have a spin-orbit interaction when immersed in an ambient magnetic field. As currently understood, these equations can not account for spin-orbit interaction in a natural way.

arXiv (Cornell University), 2007

This paper is a continuation of my earlier paper (Nyambuya 2007a) in which the equivalent of the Dirac Equation in curved spacetime is derived. This equation has been developed mainly to account in a natural way for the observed anomalous gyromagnetic ratio of fermions and the suggestions is that particles including the Electron, which is thought to be a point particle, do have a finite spatial size which is the reason for the observed anomalous gyromagnetic ratio. In this reading, I investigate four symmetries of this equation, Lorentz invariance, charge conjugation (C), time (T) and space (P) reversal symmetries. I show that this equation is Lorentz invariant, obeys C invariance symmetry and violates T and P symmetry but is TP &, PT invariant. These symmetries show that anti-particles have positive mass and energy but a negative rest mass and the opposite sign in electronic charge. A suggestion is made that the rest mass of a particle must be related to the electronic charge of that particle. The equivalent Klein-Gordon and Schrodinger equation in curved spacetime are discussed. It is shown that these equations imply Bosons and atoms naturally must have a spin-orbit interaction when immersed in an ambient magnetic field. As currently understood, these equations can not account for spin-orbit interaction in a natural way.

2014

Quantum Electrodynamics (QED) is built on the original Dirac equation , an equation that exhibits perfect symmetry in that it is symmetric under charge conjugation (C), space (P) and time (T ) reversal and any combination of these discrete symmetries. We demonstrate herein that the proposed Lorentz invariant Curved Spacetime Dirac Equations (CSTD-Equations) [3], while they obey CPT and PT -Symmetries, these equations readily violate C, P, T , CP and CT -Symmetries. Realising this violation, namely the C-Violation, we take this golden opportunity to suggest that the Curved Spacetime Dirac Equations may help in solving the long standing riddle and mystery of the preponderance of matter over antimatter. We come to the tentative conclusion that if these CSTD-Equations are to explain the preponderance of matter over antimatter, then, photons are to be thought of as described by the flat version of this set of equations, while ordinary matter is to be explained by the positive and negatively curved spacetime versions of this same set of equations.

arXiv (Cornell University), 2023

General Relativity and Gravitation

We aim to give a mathematical and historical introduction to the 1932 paper "Dirac equation in the gravitational field I" by Erwin Schrödinger on the generalization of the Dirac equation to a curved spacetime and also to discuss the influence this paper had on subsequent work. The paper is of interest as the first place that the well-known formula g μν ∇ μ ∇ ν + m 2 + R/4 was obtained for the 'square' of the Dirac operator in curved spacetime. This formula is known by a number of names and we explain why we favour the name 'Schrödinger-Lichnerowicz formula'. We also aim to explain how the modern notion of 'spin connection' emerged from a debate in the physics journals in the period 1929-1933. We discuss the key contributions of Weyl, Fock and Cartan and explain how and why they were partly in conflict with the approaches of Schrödinger and several other authors. We reference and comment on some previous historical accounts of this topic.

2013

Abstract: We demonstrate how fermion rest masses may be understood on a strictly geometric footing, by showing how the Dirac equation is just a special case of the Einstein equation for gravitation in curved spacetime, in view of Weyl’s theory of gauge (phase) invariance.