Lehnert equation

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... 

Similarly, Eq. (107) shows that the second Lehnert equation is... 

The latter is therefore related to the concept of the B(3) field through the Lehnert equations, which in the vacuum are... 

The Lehnert equations are consistent [10] with the continuity equation (428) of U(l) electrodynamics. Using the vacuum continuity equation in Lehnert s vacuum Coulomb law, we find... 

To put the 0(3) equations into the form of the Lehnert equations, we use the definitions... 

If the mass of the photon is identically zero, its normalized helicity takes the values +1 and —1 because 7M is proportional to p1 [6]. The 0 component, which usually appears for a boson, is not considered but reappears if the photon has identically nonzero mass. In this case, the Wigner little group becomes 0(3). The 1 held corresponds to, for the photon with a tiny but nonzero mass because, as argued earlier, the structure of the 0(3) held equations is identical with that of the Lehnert equations [Eqs. (612)], which imply photon mass. Therefore p1 and, /M in the laboratory are infinitesimally different from light-like,... 

Therefore the Lehnert equation (253) correctly conserves action under a local U(l) gauge transformation in the vacuum. Such a transformation leads to a vacuum charge current density as the result of gauge theory itself, because U(l) gauge theory has a scalar internal space that supports A and A. These must be complex in order to define the globally conserved charge ... 

The Lehnert equations are a great improvement over the Maxwell-Heaviside equations [45,49] but are unable to describe phenomena such as the Sagnac effect and interferometry [42], for which an 0(3) internal gauge space symmetry is needed. 

M. W. Evans et al., AIAS group paper, Runaway solutions of the Lehnert equations The possibility of extracting energy from the vacuum, Optik (in press). 

The Lehnert field equations in the vacuum also exist in U(l) form, and were originally postulated [7-10] in U(l) gauge field theory. It can be demonstrated as follows, that they originate from the U(l) gauge field equations when matter is not present ... 

The first example of a vacuum current was introduced by Maxwell in order to make the equations of electrostatics and magnetostatics self-consistent. The second examples were introduced in 1979 [7] by Lehnert, and 0(3)... 

More than a century later, Lehnert [7] introduced and developed [7-10] the concept of vacuum charge on the classical level, and showed [7-10] that this concept leads to advantages over the Maxwell-Heaviside equations in the description of empirical data, for example, the problem of an interface with a vacuum [7-10,15]. The introduction of a vacuum charge leads to axisymmetric vacuum solutions akin to the B(3> vacuum component of 0(3) electrodynamics... 

It can be seen that these are U(l) equations, but with the addition of the vacuum charge density pvac and the vacuum current density 7vac. On the 0(3) level, the Lehnert charge density becomes... 

Therefore we reach the important overall conclusion that the structure of the 0(3) equations is a development into 0(3) symmetry of the Lehnert field equations [7-10], which are written in U(l) form. The Lehnert field equations have been extensively developed and tested empirically and theoretically [7-10]. 

On the U(l) level, for example, the structure of the Lehnert [45] and gauge-invariant Proca equations is obtained as follows ... 

The locally gauge-invariant Lehnert field equation corresponding to Eq. (374) was derived as Eq. (350). The photon picks up mass from the vacuum itself, and having derived a locally gauge-invariant Proca equation, canonical quantization can be applied to produce a photon with mass with three space dimensions. 

The Lehnert field equation is obtained from this Lagrangian using the Euler-Lagrange equation... 

From a pragmatic viewpoint, there is no need for a model of the photon. One may be content with a description of the particle based entirely on the equations that it obeys. This is a very respectable scientific stance. There is another equally respectable scientific position—try to understand the mathematical equations in relation to a physical model. In previous paragraph we mentioned the attempts of several investigators [30-33]. More recent trials are those of Warburton [34], Fox [35], Scully and Sargent [36], Hunter and Wadlinger [37,38], Evans and Vigier [39], Barbosa and Gonzalez [40], and Lehnert [41]. For additional contemporary models see Hunter et al. [42]. 

Lehnert and Roy [10] found axial magnetic field component in the direction of propagation of photon considering the axisymmetric wave modes in Maxwell s equations with space charge in vacuo. 

Munera and Guzman [56] obtained new explicit noncyclic solutions for the three-dimensional time-dependent wave equation in spherical coordinates. Their solutions constitute a new solution for the classical Maxwell equations. It is shown that the class of Lorenz-invariant inductive phenomena may have longitudinal fields as solution. But here, these solutions correspond to massless particles. Hence, in this framework a photon with zero rest mass may be compatible with a longitudinal field in contrast to that Lehnert, Evans, and Roscoe frameworks. But the extra degrees of freedom associated with this kind of longitudinal solution without nonzero photon mass is not clear, at least at the present state of development of the theory. More efforts are needed to clarify this situation. 

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