Besides, the electron thermal potential, [[phi].sub.e] = [E.sub.te]/e, is 26 mV, and the electron Debye length, [[lambda].sub.e] = [square root of ([[epsilon].sub.0][E.sub.t]e/([n.sub.e][e.sup.2]))], is 1.2 [Angstrom] (the same order of the radii of isolated neutral atoms; note that the classical electron radius
The extent of this static field configuration essentially matches the classical electron radius
. There are two more topological quantum numbers which relate to spin and photon number.
This, besides agreeing with the classical electron radius
as well as with the canonical momentum of a charged particle in an external field, in the sense that P = e[A.sub.tot] = eA + e[A.sub.ext] = mv + e[A.sub.ext], implies that the mass E/[c.sup.2] is of electromagnetic origin .
Laser wakefield accelerators are operated in underdense plasma of which the electron plasma frequency is given by [w.sub.p] = [(4[pi][e.sup.2][n.sub.e]/[m.sub.e]).sup.1/2] with electron density [n.sub.e] [much less than to] [n.sub.c], where [n.sub.c] = [pi]/(re[[lambda].sup.2.sub.L]) is the critical plasma density for laser wavelength 6L and [r.sub.e] = [e.sup.2]/[m.sub.e][c.sup.2] the classical electron radius
. In this condition, laser pulses propagates through plasma at the linear group velocity given by [v.sub.g] = c[(1 - [w.sup.2.sub.p]/ [w.sup.2.sub.L]).sup.1/2] for [a.sub.0] [much less than to] 1.
The computed value of [??] corresponds with the classical electron radius
, approximately [10.sup.-15] m.
It is customary that a single element of such a tube is an element with the size of a classical electron radius
[r.sub.e] and its mass [m.sub.e].
In paper  proceeding from conditions of conservation of charge and constancy parameters [[mu].sub.0] and [[epsilon].sub.0], parameters of the vortex thread [m.sub.y], v, r, for an arbitrary [p.sup.+] - [e.sup.-] - contour defined as a proportion of the speed of light and electron radius
Eventually the electron radius
[([u.sup.2.sub.e,2] + [v.sup.2.sub.e,2]).sup.1/2] becomes larger than the weak interaction threshold 0.002 everywhere on the trajectory.
In classical physics the idea that the electron radius
r0 is purely electromagnetic leads to the calculation
The smallest elliptic radius is a little bit greater than the classical electron radius
The relation is analogous to famous Dirac's ratio [R.sub.U]/[r.sub.e] ~ [10.sup.40] which relates the Universe radius with classical electron radius
. However, Dirac's ratio is actually valid only approximately (with precision of "the same order of magnitude"), in opposite, the suggested replacement is an exact equation given as follows:
The Lorentz invariance of the electron radius
can be formally satisfied, in the case where this radius is allowed to shrink to that of a point charge.