Noether currents

By way of introduction to the Noether currents and charges that exist in 0(3) electrodynamics, the inhomogeneous field of Eq. (32) can be considered in the vacuum (source-free space) and split into two particular solutions ... 

The individual terms of the charge current density (Jv) in the vacuum are Noether currents of the type (101)—(106) and we have the following identifications under all conditions ... 

The fact that an invariance of leads to a conserved current is known as Noether s theorem ind the currents are often called Noether currents (see, for example, Ramond, 1981). 

Additive invariants were first studied by Pomeau [pomeau84] and Goles and Vich-niac [golesSb]. Although, as we shall see below, there are some techniques that can be used to extract a few invariants from jjarticular systems, no general methodology currently exists. A fundamental obstacle appears to be that there is no purely discrete analogue of Noether s Theorem. 

Noether charges proportional to the remaining right-hand-side terms do not disappear, leaving one of the Lehnert equations [7-10]. Lehnert introduced the vacuum charge empirically. Lehnert and Roy [10] have given clear empirical evidence for the existence of vacuum charge and current. The latter appears in the 0(3) Ampere-Maxwell law, which in field-matter interaction is... 

In thermodynamic equilibrium, the net result is zero in both cases, but locally, there may be a non-zero rate of doing work by these vacuum charges and currents on a device, creating thermal or mechanical energy. This process is unknown in the received view but conserves energy and is consistent with Noether s theorem [6]. 

Therefore, charge density and current density in the vacuum and in matter take the same form, [see Eqs. (732) and (733)]. This is a general result of assuming an 0(3) vacuum configuration as in Section I. Equations (736) are a form of Noether s theorem and charge/current enters the scene as the result of conservation and topology. Similarly, mass is curvature of the gravitational field. 

This type of transformation is not dependent on spacetime and is purely internal [46] in Noether s theorem. Under a global gauge transformation, Noether s theorem gives the conserved current... 

Because the assumed hypervolume can be reduced to an infinitesimal, stationary or invariant action implies the local form of Noether s theorem, 3/27/2 = 0, an equation of continuity in space-time for the generalized current density determined by the field . 

The conserved current density determined by Noether s theorem is... 

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