Nonzero divergence

The present chapter is devoted mainly to one of these new theories, in particular to its possible applications to photon physics and optics. This theory is based on the hypothesis of a nonzero divergence of the electric field in vacuo, in combination with the condition of Lorentz invariance. The nonzero electric field divergence, with an associated space-charge current density, introduces an extra degree of freedom that leads to new possible states of the electromagnetic field. This concept originated from some ideas by the author in the late 1960s, the first of which was published in a series of separate papers [10,12], and later in more complete forms and in reviews [13-20]. 

In Eq. (4.11), Eo is the external field coupled with the order parameter ij. is depolarization or demagnetization field that appears due to the nonzero divergence (rfiv(n) 0) of order parameter n in confined systems [85, 86]. 

A feature of a critical point, line, or surface is that it is located where divergences of various properties, in particular correlation lengths, occur. Moreover it is reasonable to assume that at such a point there is always an order parameter that is zero on one side of the transition and tliat becomes nonzero on the other side. Nothing of this sort occurs at a first-order transition, even the gradual liquid-gas transition shown in figure A2.5.3 and figure A2.5.4. 

C. Present Nonzero Electric Field Divergence Theory... 

Present Case of a Nonzero Electrical Field Divergence... 

The introduction of the current density (3) in 3-space is, in fact, less intuitive than what could appear at first glance. As soon as the charge density (4) is permitted to exist as the result of a nonzero electric field divergence, the Lorentz invariance of a 4-current (7) with the time part namely requires the associated space part to adopt the form (3), that is, by necessity. 

The degree of freedom introduced by a nonzero electric field divergence leads both to new features of the electromagnetic field and to the possibility of... 

In principle, this nonzero conductivity effect could also be included in the present theory of a nonzero electric field divergence. 

The extra degree of freedom introduced into the present theory by the nonzero electric field divergence gives rise to new classes of phenomena such as bound steady electromagnetic equilibria and free dynamic states, including wave phenomena. These possibilities are demonstrated by Fig. 1. 

Figure 1. New features introduced by the concept of nonzero electric field divergence in vacuum space. The arrows point to possible areas of application.

Figure 2. The three fundamental wave types of an extended electromagnetic theory with nonzero electric field divergence in the vacuum, as demonstrated by the simple case of plane waves.

The analysis of plane waves is straightforward in several respects. As soon as we begin to consider waves varying in more than one space dimension, however, we will encounter new phenomena that further complicate the analysis. This also applies to the superposition of elementary modes to form wavepackets. In this section an attempt is made to investigate dissipation-free axially symmetric modes in presence of a nonzero electric field divergence... 

When D2 r = DiE- = 0, Eqs. (77) and (79)-(81) can all be satisfied when div E / 0. This branch represents an electromagnetic space-charge (EMS) mode with nonzero electric field divergence in vacuo. 

The nonzero solutions of these held components either diverge at the origin or become divergent at large distances from the axis of symmetry. Such solutions are therefore not physically relevant to conhgurations that are extended over the entire vacuum space. The introduction of artificial internal boundaries within the vacuum region would also become irrelevant from the physical point of view, nor would it remove the difficulties with the boundary conditions. 

Present Case of a Nonzero Electrical Field Divergence From now on we therefore consider branch 2 of the axisymmetric EMS mode. 

In the case of the EMS mode of Eq. (45), the limit of zero rest mass corresponds to cos a = 0. In this limit where cos a and mo are exactly equal to zero, the result is like that of a conventional axisymmetric EM mode that either diverges at the axis or at infinity, and must be discarded as pointed out in Section VII.A.l on branch 1 of solutions. Therefore the present results hold only for a nonzero rest mass, but this mass can be allowed to become very small. This implies that the quantum conditions me2 = hv for the total energy and 5 = h/2n for the angular momentum are satisfied for a whole class of small values of cos a and the corresponding rest mass. 

For the present theory to result in electrically charged particle states it therefore becomes necessary to look into radial functions R that are divergent at the origin. This leads to the subsequent question whether the corresponding integrals (B.9) would then be able to form the basis of an equilibrium having finite and nonzero values of all the quantities qit, Mo, mo, and, v(1. In the next section we will shown how this question can be answered. 

It is physically obvious that for P to be stationary the divergence of J must vanish. One cannot conclude, however, that J itself must vanish, for it may be that the stationary state contains a circulating flow with nonzero curl. Hence, in contrast with the one-variable case, even the stationary solution of (4.1) cannot always be found. ... 

Exercise. For L-> — oo the numerator and the denominator of (2.9) either both diverge or both converge. In the former case nR m - 1. In the latter case nRtm < 1, so that there is a nonzero probability for the particle to disappear into — oo. 

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