Noether

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. 

We recall, from elementary classical mechanics, that symmetry properties of the Lagrangian (or Hamiltonian) generally imply the existence of conserved quantities. If the Lagrangian is invariant under time displacement, for example, then the energy is conserved similarly, translation invariance implies momentum conservation. More generally, Noether s Theorem states that for each continuous N-dimensional group of transformations that commutes with the dynamics, there exist N conserved quantities. 

This work was supported in part by the National Science Foundation under Grant No. DMRO1-35678, and in part by the Deutsche Forschungsgemeinschaft under an Emmy-Noether grant. We thank Dr. Paola Gori-Gorgi for providing the GSB data in Table IV, and the fit parameters for Eq. (24). 

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]. 

In field theory, electric charge [6] is a symmetry of action, because it is a conserved quantity. This requirement leads to the consideration of a complex scalar field . The simplest possibility [U(l)] is that have two components, but in general it may have more than two as in the internal space of 0(3) electrodynamics which consists of the complex basis ((1),(2),(3)). The first two indices denote complex conjugate pairs, and the third is real-valued. These indices superimposed on the 4-vector give a 12-vector. In U(l) theory, the indices (1) and (2) are superimposed on the 4-vector, 4M in free space, so, 4M in U(l) electrodynamics in free space is considered as transverse, that is, determined by (1) and (2) only. These considerations lead to the conclusion that charge is not a point localized on an electron rather, it is a symmetry of action dictated ultimately by the Noether theorem [6]. 

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 ... 

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. 

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 ... 

---