General Spin Dirac Equation (II) (original) (raw)
On a General Spin Dirac Equation
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.
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). 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.
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.
Generalization of the Lorentz-Dirac equation to include spin
Physical Review A, 1989
For the classical point electron with Zitterbe~egung (hence spin) we derive, after regularization, the radiation reaction force and covariant equations for the dynamical variables (x", n. ", U", and S".), which reduce to the Lorentz-Dirac equation in the spinless limit.
General Relativity and Gravitation, 2014
It is well-known that in the Newman-Penrose formalism the Riemann tensor can be expressed as a set of eighteen complex first-order equations, in terms of the twelve spin coefficients, known as Ricci identities. The Ricci tensor herein is determined via the Einstein equations. It is also known that the Dirac equation in a curved spacetime can be written in the Newman-Penrose formalism as a set of four first-order coupled equations for the spinor components of the wave-function. In the present article we suggest that it might be possible to think of the Dirac equations in the N-P formalism as a special case of the Ricci identities, after an appropriate identification of the four Dirac spinor components with four of the spin coefficients, provided torsion is included in the connection, and after a suitable generalization of the energy-momentum tensor. We briefly comment on similarities with the Einstein-Cartan-Sciama-Kibble theory. The motivation for this study is to take some very preliminary steps towards developing a rigorous description of the hypothesis that dynamical collapse of the wave-function during a quantum measurement is caused by gravity.
2008
Spin of elementary particles is the only kinematic degree of freedom not having classical correspondence. It arises when seeking for the finite-dimensional representations of the Lorentz group, which is the only symmetry group of relativistic quantum field theory acting on multiple-component quantum fields non-unitarily. We study linear transformations, acting on the space of spatial and proper-time velocities rather than on coordinates. While ensuring the relativistic in-variance, they avoid these two exceptions: they describe the spin degree of freedom of a pointlike particle yet at a classical level and form a compact group hence with unitary finite-dimensional representations. Within this approach changes of the velocity modulus and direction can be accounted for by rotations of two independent unit vectors. Dirac spinors just provide the quantum description of these rotations.
On the derivation of the Dirac equation
We point out that the anticommutation properties of the Dirac matrices can be derived without squaring the Dirac hamiltonian, that is, without any explicit reference to the Klein-Gordon equation. We only require the Dirac equation to admit two linearly independent plane wave solutions with positive energy for all momenta. The necessity of negative energies as well as the trace and determinant properties of the Dirac matrices are also a direct consequence of this simple and minimal requirement.
Advances in Applied Clifford Algebras, 2017
Using Clifford and Spin-Clifford formalisms we prove that the classical relativistic Hamilton Jacobi equation for a charged massive (and spinning) particle interacting with an external electromagnetic field is equivalent to Dirac-Hestenes equation satisfied by a class of spinor fields that we call classical spinor fields. These spinor fields are characterized by having the Takabayashi angle function constant (equal to 0 or π). We also investigate a nonlinear Dirac-Hestenes like equation that comes from a class of generalized classical spinor fields. Finally, we show that a general Dirac-Hestenes equation (which is a representative in the Clifford bundle of the usual Dirac equation) gives a generalized Hamilton-Jacobi equation where the quantum potential satisfies a severe constraint and the "mass of the particle" becomes a variable. Our results can then eventually explain experimental discrepancies found between prediction for the de Broglie-Bohm theory and recent experiments. We briefly discuss de Broglie's double solution theory in view of our results showing that it can be realized, at least in the case of spinning free particles.The paper contains several Appendices where notation and proofs of some results of the text are presented.