New spinor representation of Maxwell's equations. I. Generalities

A new representation of the electromagnetic field tensor has been found. In this representation it is shown that an intimate relationship exists between electromagnetism and spin; the duality rotation of the already unified theory is shown to coincide with the Touschek-Nishijima transformation of the theory of leptons. A nonlinear spinor equation equivalent to Maxwell's equations is deduced.     

New spinor representation of Maxwell's equations. II. Algebraic properties

By using the spinor representation of the electromagnetic field tensor shown in a previous paper (Campolattaro, 1979), several typical algebraic properties of the field are deduced.     

Classical field theory and analogy between Newton's and Maxwell's equations

A bivertical classical field theory includes the Newtonian mechanics and Maxwell's electromagnetic field theory as the special cases. This unification allows one to recognize the formal analogies among Newtonian mechanics and Maxwell's electrodynamics.On leave of absence from University of Wrocaw, Poland. Work partially supported by Polish Committee for Scientific Research K.BN, Grant #2 P302 023 07.     

Maxwell's equations from geometrical optics

Maxwell's electromagnetic equations are derived from Fermat's principle in geometrical optics by the process of “wavization” analogous to quantization of classical mechanics.     

Classical theory of the interaction between a spinor field and the gravitational field: First-order field equations

The classical theory of the interaction of a neutral spin-1/2 Dirac field and the gravitational field is studied. For the purely gravitational part of the Lagrangian, written in terms of a vierbein and the local connection coefficient _ab , (regarded as independent field variables), the usual first-order form is adopted. For the Dirac part, however, a different choice is made, in which the covariant derivative of is built with the aid of the vierbein instead of with _ab . This still yields a first-order formalism, but one in which _ab is related to the vierbein in the same way as it would be in the absence of. This ensures that the global connection remains symmetric in andv in the presence of. The way in which the vierbein field equation leads to a familiar Einstein equation with a symmetric and conserved stress tensor on its right side is also analyzed.     

Classical limit of relativistic quantum mechanical equations in the Foldy-Wouthuysen representation

It is shown that, under the Wentzel-Kramers-Brillouin approximation conditions, using the Foldy-Wouthuysen (FW) representation allows the problem of finding a classical limit of relativistic quantum mechanical equations to be reduced to the replacement of operators in the Hamiltonian and quantum mechanical equations of motion by the respective classical quantities.     

Shear free solutions of Maxwell's equations

A twisting shear free solution of Maxwell's equations is obtained by transforming to a complex coordinate system in which the corresponding solution is complex but twist free. The equations in this system are easily solved, and, by transforming back to the original coordinates, a twisting shear free solution of Maxwell's equations is obtained.     

Eight-dimensional spinor representation of the Poincaré group

Finite-dimensional matrix representations of the Poincaré group are discussed with particular emphasis on the eight-dimensional spinor representation. It is speculated that the complex eight-dimensional representation space might be interpreted as a more fundamental entity than Minkowski space, being in a sense a square root of the latter. One can model the usual position, momentum, and angular momentum variables of a particle of nonzero rest mass and arbitrary spin by real bilinear forms in the 8-spinor components, and obtain their correct equations of motion by subjecting the spinor to a Schrödinger-like evolution equation.     

On conformal covariance of spinor field equations

It is shown that the equation γ^μ_?μψ (χ) = 0 is covariant with respect to proper conformal transformations if the four-component spinor field ψ(χ) is a Cartan conformal semispinor of the first or second kind. If, instead, ψ is a Dirac spinor, then no less than two such equations are needed in a covariant field theory where the conformal group may be represented linearly. Some consequences are outlined.     

Nondiffracting accelerating wave packets of Maxwell's equations

We present the nondiffracting spatially accelerating solutions of the Maxwell equations. Such beams accelerate in a circular trajectory, thus generalizing the concept of Airy beams to the full domain of the wave equation. For both TE and TM polarizations, the beams exhibit shape-preserving bending which can have subwavelength features, and the Poynting vector of the main lobe displays a turn of more than 90°. We show that these accelerating beams are self-healing, analyze their properties, and find the new class of accelerating breathers: self-bending beams of periodically oscillating shapes. Finally, we emphasize that in their scalar form, these beams are the exact solutions for nondispersive accelerating wave packets of the most common wave equation describing time-harmonic waves. As such, this work has profound implications to many linear wave systems in nature, ranging from acoustic and elastic waves to surface waves in fluids and membranes.     

First order generalized symmetries of Maxwell's equations

The results of the calculation of first order generalized symmetries of Maxwell's equations in vacuum are summarized. The Lie algebra structure of the symmetries is also discussed.     

Einstein field equations in spinor formalism: a clifford-algebra approach

Einstein equations describing gravitational fields are expressed as a compact exterior system of spinor-valued forms. It is pointed out that Clifford algebra-valued differential forms provide a natural formalism for Einstein gravitational fields. Finally we try to express these equations in a real spacetime.     

Conformal invariance of Maxwell's equations with magnetic charges

It is shown that Maxwell's equations when there exist isolated magnetic charges are invariant under conformal transformations.     

Galilean symmetry of Maxwell's equations in classical electrodynamics

It is shown that the Galilean group, like the Lorentz group, is a group of exact symmetry of Maxwell's equation. The Galilean group differs in that, while the field transformations are linear and global in the relativistic case, they are nonlinear in the Galilean and, generally speaking, depend on the coordinates of the event through some weight functions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 78–83, August, 1985.The author thanks D. P. Grechukhin, V. I. Man'ko, and V. I. Fushchich for discussion of the work and valuable comments.     

Vacuum electrodynamics of accelerated systems: Nonlocal Maxwell's equations

The nonlocal electrodynamics of accelerated systems is discussed in connection with the development of Lorentz‐invariant nonlocal field equations. Nonlocal Maxwell's equations are presented explicitly for certain linearly accelerated systems. In general, the field equations remain nonlocal even after accelerated motion has ceased.     

Macroscopic Maxwell's equations of the general theory of relativity

Covariant Maxwell's equations of the general theory of relativity for a system of electromagnetically and gravitationally interacting particles of the form