The Dirac equation can be derived using the relativistic relation between mass and energy E = m (normalized units) and the symmetries of the universe, so it is necessary to have a firm grasp on the latter. The relativistic symmetry group is the orthochronous Poincare group SO(1, 3) n R, though we shall mainly use just the orthochronous Lorentz group SO(1, 3). Its Lie algebra is so(1, 3). We fir...

A Dirac particle is represented by a unitarily evolving state vector in a Hilbert space which factors as Hspin Hposition. Motivated by the similarity to simple models of decoherence consisting of a two state system coupled to an environment, we investigate the occurence of decoherence in the Dirac equation upon tracing over position. We conclude that the physics of this mathematically exact mod...

The magnetic field of an infinitely long solenoid is immensely important for various reasons. For example, it is used to study the Aharanov-Bohm [1] effect. Similarity with this field also helps one to understand the physics of cosmic strings [2]. It is therefore important to study the quantum mechanics of a Dirac fermion in the field of an infinitely long solenoid, which is what we do in the p...

The aim of this study is to combine four fundamental forces in a single equation. Dirac equation is written by putting the Yukawa potential as a representation of the strong and gravitational forces. The ordinary terms seen in the Dirac Equation are treated as the representations of the electromagnetic forces. The Lagrangian of the weak local interaction of the charged particles is converted to...

The Coulomb-gauge vector potential of a uniformly moving point charge is obtained by calculating the gauge function for the transformation between the Lorenz and Coulomb gauges. The expression obtained for the difference between the vector potentials in the two gauges is shown to satisfy a Poisson equation to which the inhomogeneous wave equation for this quantity can be reduced. The right-hand...

We consider possible dynamical models for a light fermion confined by a potential field. With the Dirac equation only Lorentz scalar confinement yields normalizable wavefunctions, while with the “no pair” variant of the Dirac equation only Lorentz vector confinement has normal Regge behaviour. A systematic investigation of Regge properties and phenomenological properties is carried out, includi...

Lecture 2. Index problem for manifolds with a boundary. Index of the Dirac operator and anomalies. Lecture 3. Spectral asymmetry and Riemannian geometry. Heat equation and asymptotic heat kernel. The η-and ζ-functions. Lecture 4. Two-component spinor calculus. Dirac and Weyl equations in two-component spinor form. Weyl equation with spectral or local boundary conditions. Potentials for massless...

This paper is a progress report on the foundations for the canonical proper-time approach to relativistic quantum theory. We first review the the standard square-root equation of relativistic quantum theory, followed by a review of the Dirac equation, providing new insights into the physical properties of both. We then introduce the canonical proper-time theory. For completeness, we give a brie...

Abstract: We prove that for a Dirac operator, with no resonance at thresholds nor eigenvalue at thresholds, the propagator satisfies propagation and dispersive estimates. When this linear operator has only two simple eigenvalues sufficiently close to each other, we study an associated class of nonlinear Dirac equations which have stationary solutions. As an application of our decay estimates, w...

The exact solution of the Dirac equation with the exact boundary condition is further investigated in the present paper. We introduce the unitary principle that can be used to disclose all logic paradoxes hidden in natural science and mathematics. By the unitary principle, we show that the existence of solution of differential equation requires an important modification to the constant of angul...