Field tensors electrodynamics

The development just given illustrates the fact that the topology of the vacuum determines the nature of the gauge transformation, field tensor, and field equations, as inferred in Section (I). The covariant derivative plays a central role in each case for example, the homogeneous field equation of 0(3) electrodynamics is a Jacobi identity made up of covariant derivatives in an internal 0(3) symmetry gauge group. The equivalent of the Jacobi identity in general relativity is the Bianchi identity. 

Therefore, the vacuum charge and current densities of Panofsky and Phillips [86], or of Lehnert and Roy [10], are given a topological meaning in 0(3) electrodynamics. In this condensed notation, the vacuum 0(3) field tensor is given by... 

The handedness, or chirality, inherent in foundational electrodynamics at the U(l) level manifests itself clearly in the Beltrami form (903). The chiral nature of the field is inherent in left- and right-handed circular polarization, and the distinction between axial and polar vector is lost. This result is seen in Eq. (901), where , is a tensor form that contains axial and polar components of the potential. This is precisely analogous with the fact that the field tensor F, contains polar (electric) and axial (magnetic) components intermixed. Therefore, in propagating electromagnetic radiation, there is no distinction between polar and axial. In the received view, however, it is almost always asserted that E and A are polar vectors and that is an axial vector. 

To consider magnetic flux density components of IAIV, Q must have the units of weber and R, the scalar curvature, must have units of inverse square meters. In the flat spacetime limit, R 0, so it is clear that the non-Abelian part of the field tensor, Eq. (6), vanishes in special relativity. The complete field tensor F vanishes [1] in flat spacetime because the curvature tensor vanishes. These considerations refute the Maxwell-Heaviside theory, which is developed in flat spacetime, and show that 0(3) electrodynamics is a theory of conformally curved spacetime. Most generally, the Sachs theory is a closed field theory that, in principle, unifies all four fields gravitational, electromagnetic, weak, and strong. 

There are well known similarities between the Riemann curvature tensor of general relativity and the field tensor in non-Abelian electrodynamics. The Riemann tensor is... 

Therefore, the definition of the field tensor in 0(3) electrodynamics gives the first two components of the B cyclic theorem [47-62]... 

In electrodynamics, we define the strength tensor of electromagnetic field... 

In classical electrodynamics, the field equations for the Maxwell field A/( depend only on the antisymmetric tensor which is invariant under a gauge transformation A/l A/l + ticduxix), where x is an arbitrary scalar field in space-time. Thus the vector field A/( is not completely determined by the theory. It is customary to impose an auxiliary gauge condition, such as 9/x/Fx = 0, in order to simplify the field equations. In the presence of an externally determined electric current density 4-vector j11, the Maxwell Lagrangian density is... 

This tensor is less general than the dielectric tensor of classical electrodynamics (1.69), since it contains the interaction with only the retarded transverse fields. For each wave vector K, (1.78) provides two solutions whose eigenpolarizations are orthogonal. The principal dielectric constants are obtained by the evaluation of the 2 x 2 determinants of (1.78) (i =1,2) ... 

The coupling factor between electrodynamics and translational mechanics is not classically used as such but as a piezoelectric voltage coefficient g (in m C ) divided by a characteristic length. In an anisotropic three-dimensional material, this coefficient is a tensor that links the stress F a to the electric field E and is equivalent to the multiplication of the coupling factor with the spatial integration of the stress (i.e., the lineic density of the force) ... 

The expression derived in Eq. (7.38) is defined as the Hall conductivity an it is the ratio of the current in the x direction to the effective electric field Ey in the perpendicular direction, as usually defined in the classical Hall effect in electrodynamics. In more general terms, this ratio can be viewed as one of the off-diagonal components of the conductivity tensor cr, (f, j = x, y), which relates the current to the electric field ... 

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