Covariant electrodynamics

In section 2.4 a brief summary of Maxwell s theory of electrodynamics has been presented in its classical, three-dimensional form. Since electrodynamics intrinsically is a relativistic gauge field theory, the structure and symmetry properties of this theory become much more apparent in its natural, explicitly 

Electromagnetic theory can be very compactly expressed in Minkowski four-vector notation. Historically, it was this symmetry of Maxwell s equations 

The equation of continuity for electric charge (11.75) has the stmcture of a four-dimensional divergence 

In Lorenz gauge, the function x(r, t) in Eq. (11.114) is chosen such that (= 0. The scalar and vector potentials are then related hy the Lorenz condition 

This can also be expressed as a four-dimensional divergence 

Maxwell s equations in Lorenz gauge can be expressed by a single Minkowski-space equation 

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In refs (Kim,2004 Kim, 2005) we take one step further estimating corrections to the Gaussian effective potential for the U(l) scalar electrodynamics where it represents the standard static GL effective model of superconductivity. Although it was found that, in the covariant pure (f)4 theory in 3 + 1 dimensions,corrections to the GEP are not large (Stancu,1990), we do not expect them to be negligible in three dimensions for high Tc superconductivity, where the system is strongly correlated. 

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. 

These concepts of 0(3) electrodynamics also completely resolve the problem that, in Maxwell-Heaviside electrodynamics, the energy momentum of radiation is defined through an integral over the conventional tensor and for this reason is not manifestly covariant. To make it so requires the use of special hypersurfaces as attempted, for example, by Fermi and Rohrlich [40]. The 0(3) energy momentum (78), in contrast, is generally covariant in 0(3) electrodynamics [11-20]. 

It can be shown straightforwardly, as follows, that there is no holonomy difference if the phase factor (154) is applied to the problem of the Sagnac effect with U(l) covariant derivatives. In other words, the Dirac phase factor [4] of U(l) electrodynamics does not describe the Sagnac effect. For C and A loops, consider the boundary... 

There exist generally covariant four-valued 4-vectors that are components of q, and these can be used to construct the basic structure of 0(3) electrodynamics in terms of single-valued components of the quaternion-valued metric q1. Therefore, the Sachs theory can be reduced to 0(3) electrodynamics, which is a Yang-Mills theory [3,4]. The empirical evidence available for both the Sachs and 0(3) theories is summarized in this review, and discussed more extensively in the individual reviews by Sachs [1] and Evans [2]. In other words, empirical evidence is given of the instances where the Maxwell-Heaviside theory fails and where the Sachs and 0(3) electrodynamics succeed in describing empirical data from various sources. The fusion of the 0(3) and Sachs theories provides proof that the B(3) held [2] is a physical held of curved spacetime, which vanishes in hat spacetime (Maxwell-Heaviside theory [2]). 

In Eq. (5), the product q q is quaternion-valued and non-commutative, but not antisymmetric in the indices p and v. The B<3> held and structure of 0(3) electrodynamics must be found from a special case of Eq. (5) showing that 0(3) electrodynamics is a Yang-Mills theory and also a theory of general relativity [1]. The important conclusion reached is that Yang-Mills theories can be derived from the irreducible representations of the Einstein group. This result is consistent with the fact that all theories of physics must be theories of general relativity in principle. From Eq. (1), it is possible to write four-valued, generally covariant, components such as... 

In 0(3) electrodynamics, the covariant derivative on the classical level is defined by... 

Therefore, it is always possible to write the covariant derivative of the Sachs theory as an 0(3) covariant derivative of 0(3) electrodynamics. Both types of covariant derivative are considered on the classical level. 

The curvature tensor is defined in terms of covariant derivatives of the spin-affine connections fip, and according to Section ( ), has its equivalent in 0(3) electrodynamics. 

To check on the interpretation given in the text of the reduction of Sachs to 0(3) electrodynamics, we can consider generally covariant components such as... 

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