General Spin Dirac Equation (II) (original) (raw)

2009

In its bare and natural form, the Dirac Equation describes only spin-1/2 particles. The main purpose of this reading is to make a valid and justified mathematical modification to the Dirac Equation so that it describes any spin particle. We show that this mathematical modification is consistent with the Special Theory of Relativity (STR). We believe that the fact that this modification is consistent with the STR gives the present effort some physical justification that warrants further investigations. From the vantage point of unity, simplicity and beauty, it is natural to wonder why should there exist different equations to describe particles of different spins? For example, the Klein-Gordon equation describes spin-0 particles, while the Dirac Equation describes spin-1/2, and the Rarita-Schwinger Equation describes spin-3/2. Does it mean we have to look for another equation to describe spin-2 particles, and then spin-5/2 particles etc? This does not look beautiful, simple, or at the very least suggest a Unification of the Natural Laws. Beauty of a theory is not a physical principle but, one thing is clear to the searching mind-i.e., a theory that possesses beauty, appeals to the mind, and is (posteriori) bound to have something to do with physical reality if it naturally submits itself to the test of experience. The effort of the present reading is to make the attempt to find this equation.

General Spin Dirac Equation

The Dirac Equation Revisited

The Dirac equation is a cornerstone of quantum mechanics that fully describes the behaviour of spin ½ particles. Recently, the energy momentum relationship has been reconsidered such that |E|^2 = |(m0c^ 2)| 2 + |(pc)| 2 has been modified to: |E| 2 = |(m0c^2)|^2-|(pc)|^2 where E is the kinetic energy, moc^2 is the rest mass energy and pc is the wave energy for the spin ½ particle. This has been termed the 'Hamiltonian approach' and with a new starting point, the original Dirac equation has been derived: and the modified covariant form found is where h/2π = c = 1. The behaviour of spin ½ particles is found to be the same as for the original Dirac equation. The Dirac equation will also be expanded by setting the rest energy as a complex number, |(m0c 2)| e^jωt

Dirac Equation in Curved Spacetime II - Properties of the Curved Spacetime Dirac Equation

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.

Revisiting the Schrödinger–Dirac Equation

Symmetry

In flat spacetime, the Dirac equation is the “square root” of the Klein–Gordon equation in the sense that, by applying the square of the Dirac operator to the Dirac spinor, one recovers the equation duplicated for each component of the spinor. In the presence of gravity, applying the square of the curved-spacetime Dirac operator to the Dirac spinor does not yield the curved-spacetime Klein–Gordon equation, but instead yields the Schrödinger–Dirac covariant equation. First, we show that the latter equation gives rise to a generalization to spinors of the covariant Gross–Pitaevskii equation. Then, we show that, while the Schrödinger–Dirac equation is not conformally invariant, there exists a generalization of the equation that is conformally invariant but which requires a different conformal transformation of the spinor than that required by the Dirac equation. The new conformal factor acquired by the spinor is found to be a matrix-valued factor obeying a differential equation that in...

Dirac Equation in Curved Spacetime II - Properties of the Equation

arXiv (Cornell University), 2007

On the structure of the energy-momentum and the spin currents in Dirac's electron theory

arXiv (Cornell University), 1997

We consider a classical Dirac field in flat Minkowski spacetime. We perform a Gordon decomposition of its canonical energy-momentum and spin currents, respectively. Thereby we find for each of these currents a convective and a polarization piece. The polarization pieces can be expressed as exterior covariant derivatives of the two-formsMα and M αβ = −M βα , respectively. In analogy to the magnetic moment in electrodynamics, we identify these two-forms as gravitational moments connected with the translation group and the Lorentz group, respectively. We point out the relation between the Gordon decomposition of the energy-momentum current and its Belinfante-Rosenfeld symmetrization. In the non-relativistic limit, the translational gravitational moment of the Dirac field is found to be proportional to the spin covector of the electron. File schueck-ing7.tex, 1997-06-03

Dirac Equation for General Spin Particles Including Bosons

viXra, 2009

We demonstrate (show) that the Dirac equation-which is universally assumed to represent only spin + 1 2 particles; can be manipulated using legal mathematical operationsstarting from the Dirac equation-so that it describes any general spin particle. If our approach is acceptable and is what Nature employs, then, as currently obtaining, one will not need a unique and separate equation to describe particles of different spins, but only one equation is what is needed-the General Spin Dirac Equation. This approach is more economic and very much in the spirit of unificationi.e., the tie-ing together into a single unified garment-a number of phenomenon (or facets of physical and natural reality) using a single principle, which, in the present case is the bunching together into one theory (equation), all spin particles into the General Spin Dirac Equation.

Editorial note to: Erwin Schrödinger, Dirac electron in the gravitational field I

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.

Different Aspects of Spin in Quantum Mechanics and General Relativity

Symmetry

In this paper, different aspects of the concept of spin are studied. The most well-established one is, of course, the quantum mechanical aspect: spin is a broken symmetry in the sense that the solutions of the Dirac equation tend to have directional properties that cannot be seen in the equation itself. It has been clear since the early days of quantum mechanics that this has something to do with the indefinite metric in Lorentz geometry, but the mechanism behind this connection is elusive. Although spin is not the same as rotation in the usual sense, there must certainly be a close relationship between these concepts. And, a possible way to investigate this connection is to instead start from the underlying geometry in general relativity. Is there a reason why rotating motion in Lorentz geometry should be more natural than non-rotating motion? In a certain sense, the answer turns out to be yes. But, it is by no means easy to see what this should correspond to in the usual quantum m...

General Spin Dirac Equation (II) (original) (raw)

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