Model Dirac-Maxwell equations with pseudounitary symmetry

Abstract

We introduce a new system of equations called a model system of Dirac-Maxwell equations, reproducing the main properties of the standard system. At the same time, the model system of equations differs from the standard system in several ways; in particular, it is a tensor system and has a new symmetry with respect to the pseudounitary group. We also propose a version of the model system of Dirac-Maxwell equations with local (gauge) pseudounitary symmetry. We show that any spinor solution of the standard system of Dirac-Maxwell equations can be obtained from the corresponding tensor solution of the model system.In mathematical and theoretical physics, the Dirac equation is used to describe particles with spin 1/2 [1]- [3]. The Maxwell equations describe the electromagnetic field (photons). The system of DiracMaxwell equations models the interaction of electrons with the electromagnetic field.Sommerfeld discovered ([4], pp. 219-229 in the Russian translation) that two approaches can be used to prove the covariance of the Dirac equation for electrons in the case of the Lorentzian replacements of coordinates. In the first case, it is assumed that the Dirac matrices γ µ transform as components of a vector and the wave function ψ of the electron remains unchanged. In this case, all the quantities in the Dirac equation are tensor fields (vectors, covectors, and scalars). In the second case, it is assumed that the Dirac γ-matrices remain unchanged and the wave function ψ of the electron transforms according to the spinor representation of the Lorentz group, i.e., ψ is a Dirac spinor.Accordingly, we can speak about the presence of two representations of the Dirac equation for the electron: tensor and spinor. We can conclude from Sommerfeld's arguments that he had no basis to consider one representation right and the other wrong. We note that Usachev [5] studied the problem of spins of particles described by the Dirac equation in the tensor representation. He showed that this equation describes particles with spin 1/2 just like the Dirac equation in the spinor representation. The spinor representation of the Dirac equation is generally accepted in modern theoretical and mathematical physics.Papers have been devoted to developing a new approach to the Dirac equation and systems of DiracMaxwell and Dirac-Yang-Mills equations [6]- [22]; this approach is based on using the Clifford algebra and the tensor representation of the Dirac equation.Here, we propose and consider a model system of Dirac-Maxwell equations constructed based on the matrix technique and the tensor representation of the Dirac equation for electrons. We show that in contrast to the previously discussed equations, this system of equations has a new global symmetry with respect to the pseudounitary group SU (2, 2). We also study a version of the model system of Dirac-Maxwell equations with a local (gauge) pseudounitary symmetry.

Mass generation mechanism for spin 1/2 fermions in Dirac--Yang--Mills model equations with a symplectic gauge symmetry

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In the Standard Model of electroweak interactions the fundamental fermions acquire masses by the Yukawa interaction with the (spin 0) Higgs field. In our model spin 1/2 fermions acquire masses by an interaction with (spin 1) gauge field with symplectic symmetry.In [2,3,4,5,8] we develop a new approach to field theory, which based on so called model equations of field theory. In this paper we introduce Dirac-Yang-Mills model equations for spin 1/2 fermions interacting with two gauge fields simultaneously.

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“…In [2,3,4,5,8] we develop a new approach to field theory, which based on so called model equations of field theory. In this paper we introduce Dirac-Yang-Mills model equations for spin 1/2 fermions interacting with two gauge fields simultaneously.…”

“…However the identities (6) can't be fulfilled. Hence we may consider the system of equations ( 7) as a generalization of the system of equations (4).…”

“…this transformation of variables. An equivalent transformation of variables in the system of equation ( 7) is called symmetry if the generators e a and the Hermitian idempotent t do not transform (see [4,8] for details).…”