Electromagnetic field tensor

If we introduce the electromagnetic field tensor operator F ix), which can be decomposed as follows ... 

The quaternion-valued vector potential and the 4-current J both depend directly on the curvature tensor. The electromagnetic field tensor in the Sachs theory has the form... 

The metric coefficient in the theory of gravitation [110] is locally diagonal, but in order to develop a metric for vacuum electromagnetism, the antisymmetry of the field must be considered. The electromagnetic field tensor on the U(l) level is an angular momentum tensor in four dimensions, made up of rotation and boost generators of the Poincare group. An ordinary axial vector in three-dimensional space can always be expressed as the sum of cross-products of unit vectors... 

We have argued here and elsewhere [44] that the plane-wave representation of classical electromagnetism is far from complete. In tensor language, this incompleteness means that the antisymmetric electromagnetic field tensor on the 0(3) level must be proportional to an antisymmetric frame tensor of spacetime, 7 , derived from the Riemannian tensor by contraction on two indices ... 

In Vol. 114, part 1, Sachs has shown that the most general form of the electromagnetic field tensor is... 

Despite the clarification offered by this notation, the notation can also cause confusion, because in the present literature, the electromagnetic field tensor is always referred to as F, whether F is defined with respect to U(l) or SU(2) or other symmetry situations. Therefore, although we prefer this notation, we shall not proceed with it. However, it is important to note that the A field in the U(l) situation is a vector or a number, but in the SU(2) or non-Abelian situation, it is a tensor or a matrix-valued function. 

In Section V it will be shown that the quaternion structure of the fields that correspond to the electromagnetic field tensor and its current density source, implies a very important consequence for electromagnetism. It is that the local limit of the time component of the four-current density yields a derived normalization. The latter is the condition that was imposed (originally by Max Bom) to interpret quantum mechanics as a probability calculus. Here, it is a derived result that is an asymptotic feature (in the flat spacetime limit) of a field theory that may not generally be interpreted in terms of probabilities. Thus, the derivation of the electromagnetic field equations in general relativity reveals, as a bonus, a natural normalization condition that is conventionally imposed in quantum mechanics. 

From the above result, it could be inferred exactly that such irreducible subspaces of the state space establish the proper mathematical domain of the classical physical field quantities. In fact, the demonstration was undertaken by using a relativistic electromagnetic field tensor, F,y, and its antisymmetric property ... 

At this point it is crucial to note that this latter form was chosen to obey the requirement of Lorentz invariance. Under a gauge transformation only the interaction term is modified as the electromagnetic field tensor is gauge invariant... 

The Maxwell equations can be written most easily with the use of the electromagnetic field tensor ... 

The gauge field = (p,A) contains the scalar and vector potentials and defines the antisymmetric electromagnetic field tensor f already introduced in section 3.4. A comprises, as before, external, i.e., non-dynamical, electromagnetic potentials in addition to the radiation field. In accordance with section 2.4 the dynamical electric and magnetic fields can be obtained as... 

The expressions involved in continua, such as the electromagnetic field tensors and that of the electromagnetic momentum-energy, are presented in the case of media with a single component. The Maxwell equations are written, as are the balances of the electrical charge and the electrical momentum-energy in a polarized or non-polarized medium. 

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