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68
Chapter 1
independent meaning. Then, the basis elements of an electron are understood in terms of what they do when added in aggregate to constitute an electron. The nomenclature used to describe the elements reflects the analogous macroscopic sources and is adopted for convenience.
GENERATION OF THE ORBITSPHERE CVFS The orbitsphere spin function comprises a constant charge(current)-density function with moving charge confined to a twodimensional spherical shell and comprises a uniform complete coverage. The uniform magnetostatic current-density function Y0 0 ( , ) of the orbitsphere spin function comprises a continuum of correlated orthogonal great-circle current loops wherein each point charge(current)-density element moves time harmonically with constant angular velocity, n , given by Eq. (1.36) and velocity, vn , in the direction of the current given by Eq. (1.35). The current-density function of the orbitsphere is generated from orthogonal great-circle current-density elements (one dimensional “current loops”) that serve as basis elements to form two distributions of an infinite number of great circles wherein each covers one-half of a two-dimensional spherical shell and is defined as a basis element current vector field (“BECVF”) and an orbitsphere current-vector field (“OCVF”). Then, the continuous uniform electron current density function Y0 0 ( , ) (part of Eqs. (1.27-1.29)) that covers the entire spherical surface as a distribution of an infinite number of great circles is generated using the CVFs. First, the generation of the BECVF is achieved by rotation of two great circle basis elements, one in the x’z’-plane and the other in the y’z’-plane, about the i x , i y , 0i z axis by an infinite set of infinitesimal increments of the rotational angle
is stationary on 2 2 this axis. The generation of the OCVF is achieved by rotation of two great circle basis elements, one in the x’y’-plane and the 1 1 other in the plane that bisects the x'y'-quadrant and is parallel to the z'-axis, about the ix , i y , i z axis by an infinite set 2 2 of infinitesimal increments of the rotational angle wherein the current direction is such that the resultant angular momentum having components of L xy and L z is stationary on this axis. The operator to vector of the basis elements of 2 2 2 2 2 form each CVF comprises a convolution of the rotational matrix of great circles basis elements with an infinite series of delta functions of argument of the infinitesimal angular increment. Then, the uniform great-circle distribution Y0 0 ( , ) is exactly generated from the CVFs. The BECVF is convolved with the OCVF over a 2 span that results in the placement of a BECVF at each great circle of the OCVF. Since the angular momentum vector of the BECVF is matched to twice that of one of the OCVF great circle basis elements and the span is over 2 , the resultant angular momentum of the distribution is the same as that of the OCVF, except that coverage of the spherical surface is complete. This current vector distribution is normalized by scaling the constant current of each great circle element resulting in the exact uniformity of the distribution independent of time since K 0 along each great circle. There is no alteration of the angular momentum by normalization since it only affects 1 1 the density parallel to the angular momentum axis of the distribution, the ix , i y , i z -axis. Then, the boundary 2 2 0 conditions of Y0 ( , ) having the desired angular momentum components, coverage, element motion, and uniformity are shown wherein the current direction is such that the resultant angular momentum vector of the basis elements of
1 1 to have been achieved by designating the ix , i y , i z -axis as the z-axis. The resulting exact uniform current distribution 2 2 (Figure 1.22) has the angular momentum components of L xy / and L z (Eqs. (1.127-1.128) and Figure 1.23). 2 4 The z-projection of the angular momentum of a photon given by its orthogonal electric and magnetic fields is 1 m Re r (E B*)dx 4 (Eq. (4.1)). When an electron is formed from a photon as given in the Leptons section, the 8 c angular momentum is conserved in the projections of the orthogonal great circle current loops that serve as the basis elements of the orbitsphere. Special relativity requires that the mathematical equations expressing the laws of nature must be covariant, that is, invariant in form, under the transformations of the Lorentz group. As shown by Eq. (1.37) the angular momentum is invariant of radius or velocity. It is a Lorentz scalar L [7] with respect to the radius of the state. The vector projections of the orbitsphere spin angular momentum relative to the Cartesian coordinates arrived at by summation of the contributions from the electron current elements of the current distribution are given in the Spin Angular Momentum of the Orbitsphere Y0 0 ( , ) with = 0 section. The time-independent current pattern is obtained by defining a basis set for generating the current distribution over the surface of a spherical shell of zero thickness. As such a basis set, consider that the electron current is distributed within the basis elements and then distributed evenly amongst all great circles such the final distribution Y0 0 ( , ) possesses of angular momentum before and after normalization.