Electromagnetic field theory equations

Conventional electromagnetic field theory based on Maxwell s equations and quantum mechanics has been very successful in its application to numerous problems in physics, and has sometimes manifested itself in an extremely good agreement with experimental results. Nevertheless, in certain areas these joint theories do not seem to provide fully adequate descriptions of physical reality. Thus there are unsolved problems leading to difficulties with Maxwell s equations that are not removed by and not directly associated with quantum mechanics [1,2]. 

As a first step, the treatment in this chapter is limited to electromagnetic field theory in orthogonal coordinate systems. Subsequent steps would include more advanced tensor representations and a complete quantization of the extended field equations. 

In the context of Einstein s theory of relativity, we must ask whether Maxwell s expression of the electromagnetic theory is the most general representation consistent with the symmetry requirements of relativity. The answer is negative because the symmetry of Maxwell s equations based on reducible representations of the group of relativity theory. Then there must be additional physical predictions that remain hidden that would not be revealed until the most general 0irreducible) expression of the electromagnetic field theory is used. 

Thus we see that four of the conservation equations in (24) correspond to all four conservation equations of the standard theory one is the conservation of energy (25a) (Poynting s equation), and the other three are the conservation of the three components of momentum (25b) of the standard form of electromagnetic field theory. But since (24) are eight real-number valued equations rather than four, the spinor formalism predicts more facts than the standard vector Maxwell formalism—it is a true generalization. 

Before embarking on the problem of the interaction of the negaton-positon field with the quantized electromagnetic field, we shall first consider the case of the negaton-positon field interacting with an external, classical (prescribed) electromagnetic field. We shall also outline in the present chapter those aspects of the theory of the S-matrix that will be required for the treatment of quantum electrodynamics. Section 10.4 presents a treatment of the Dirac equation in an external field. 

The present theory has been developed in terms of an extended Lorentz invariant form of the electromagnetic field equations, in combination with an addendum of necessary basic quantum conditions. From the results of such a simplified approach, theoretical models have been obtained for a number of physical systems. These models could thus provide some hints and first... 

The electromagnetic field equations on the 0(3) level can be obtained from this purely geometrical theory by using Eq. (631) in the Bianchi identity... 

Traditionally, physics emphasizes the local properties. Indeed, many of its branches are based on partial differential equations, as happens, for instance, with continuum mechanics, field theory, or electromagnetism. In these cases, the corresponding basic equations are constructed by viewing the world locally, since these equations consist in relations between space (and time) derivatives of the coordinates. In consonance, most experiments make measurements in small, simply connected space regions and refer therefore also to local properties. (There are some exceptions the Aharonov-Bohm effect is an interesting example.)... 

The gauge potentials A now have 4x4 traceless representations. The scalar field theory that describes the vacuum will now satisfy field equations that involve all 16 components of the gauge potential. By selectively coupling these fields to the fermions, it should be possible to formulate a theory that recovers a low-energy theory that is the standard model with the 0(3)p gauge theory of electromagnetism. 

Since there is a non-Abelian nature to this theory, we return to the nonrelativistic equation that describes the interaction of a fermion with the electromagnetic field. The Pauli Hamiltonian is modified with the addition of a interaction term [9]... 

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