Tensors Field

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

The method of Thole was developed with the help of the induced dipole formulation, when all dipoles interact through the dipole field tensor. The modification introduced by Thole consisted in changing the dipole field tensor ... 

However, we have previously seen in Chapter 5 that the number of elements in the zero-field tensor can be reduced to two (Equation 5.25) by making D traceless, and so the spin Hamiltonian can be written as... 

The tensorial product of local field tensors is not averaged, but replaced by an arithmetic product of the average magnitude of four local field factors. 

An invariant scalar density can be constructed from the field tensor as the double sum... 

The second-order particle-field tensor Gh(X, t) has the same form as G (x, t) (i.e., (6.57)), but with the particle fields appearing in place of the mean fields ... 

In general gauge field theory [6], the field tensor is proportional to the commutator of variant derivatives. This is the result of a round trip or closed... 

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. 

Recall that in general gauge field theory, for any gauge group, the field tensor is defined through the commutator of covariant derivatives. In condensed notation [6]... 

Potential differences are primary in gauge theory, because they define both the covariant derivative and the field tensor. In Whittaker s theory [27,28], potentials can exist without the presence of fields, but the converse is not true. This conclusion can be demonstrated as follows. Equation (479b) is invariant under... 

In contravariant covariant notation, the field tensors are defined by [101]... 

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... 

Therefore the fact that 9 is arbitrary in U(l) theory compels that theory to assert that photon mass is zero. This is an unphysical result based on the Lorentz group. When we come to consider the Poincare group, as in section XIII, we find that the Wigner little group for a particle with identically zero mass is E(2), and this is unphysical. Since 9 in the U(l) gauge transform is entirely arbitrary, it is also unphysical. On the U(l) level, the Euler-Lagrange equation (825) seems to contain four unknowns, the four components of , and the field tensor H v seems to contain six unknowns. This situation is simply the result of the term 7/MV in the initial Lagrangian (824) from which Eq. (826) is obtained. However, the fundamental field tensor is defined by the 4-curl ... 

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. 

The non-Abelian component of the field tensor is defined through a metric that is a set of four quaternion-valued components of a 4-vector, a 4-vector each of whose components can be represented by a 2 x 2 matrix. In condensed notation ... 

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. 

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... 

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