The Physical Origin of Electron Spin
- using quantum wave particle structure
 

1.  Introduction

A highly successful mathematical theory of spin has already been developed by P.A.M. Dirac and others (Eisele, 1960).  It predicted the discovery of the positron (Anderson, 1922) and correctly gave the value of the spin, h/4pi angular momentum units.  Prior to this paper, there has been no successful physical description of spin or any suggestion of its origin, although it was recognized to be a quantum phenomenon.  The electron's structure, as well as its spin, had been a mystery.  Providing a physical origin of spin for the first time is the purpose of this paper.

In Dirac's theoretical work the spin of a particle is measured in units of angular momentum, like rotating objects of human size.  But particle spin is uniquely a quantum phenomenon, different than human scale angular momentum.  Its value is fixed and independent of particle mass or angular velocity.  However, spin properties are found to be related to other  properties of the electron's quantum wave function; that is, mirror or parity inversion (P), time inversion (T), and charge inversion (C).  For example, the quantum operation CPT on a particle is found to be invariant,CPT= {C harge inversion} x {Parity inversion} x {Time inversion}  =  an invariant

This study returns to a proposal that was popular sixty years ago among the pioneers of quantum theory:  namely that matter consisted of wave structures in space.  Thus, it was proposed that matter substance, mass and charge, did not exist but were properties of the wave structure.  Wyle, Schroedinger, Clifford, and Einstein were among those who believed that particles were a wave structure.  Their belief was consistent with quantum theory, since the mathematics of quantum theory does not depend on the existence of particle substance or charge substance.  In short, they proposed that quantum waves are real and mass/charge were mere appearances; 'Schaumkommen' in the words of Schroedinger.  The reality of quantum waves, as suggested by Cramer (1986), supports the original concept of W. K. Clifford (1876) that all matter is simply "undulations in the fabric of space."

Wheeler and Feynman (1945) first modeled the electron as spherical inward and outward electromagnetic waves seeking the response of the universe (from other matter) to explain radiation, but encountered difficulties because there are no spherical solutions of the electromagnetic equations using vector fields.  Cramer (1986) discusses the response for real quantum waves. Using a quantum wave equation (scalar fields) and spherical quantum waves, Wolff (1990, '91, '93, '95, '97) found and described a wave structure of matter which successfully predicted the Natural Laws as experimentally measured.  It has  predicted all of the properties of the electron except one - its spin.  Now, this paper completes those predictions with a physical origin of spin that is in accord with quantum theory, the Dirac Equation, and the previous structure of the electron.  

Briefly summarizing Wolff, the electron is comprised of two spherical scalar waves, one inward and one outward.  These waves are superimposed at the origin with opposite amplitudes, as shown in Figure 1in the next section, to form a single resonant standing wave in space centered at the electron's location.  A reversal of the inward wave occurs at the center where r = 0.  Spin appears as a required rotation of the inward wave to become the outward  wave.  The outward wave induces a response of the universe when it encounters other matter in its universe and modulates their outward waves.  The tiny Huygens components of those waves return to the center and become the inward wave.  This simple structure, termed a space resonance (SR), produces all experimental properties of electrons.

This structure, the electron properties, and the laws of nature originate from three basic principles or assumptions.  No other laws are required as these principles are the origin of the laws of nature.  Briefly they are:

Principle I.   A Wave Equation .  Determines the behavior of quantum waves.

Principle II.  Wave Density Principle.  A quantitative generalization of Mach's principle, which determines the density of the quantum wave medium.

Principle III.  Minimum Amplitude Principle (MAP).  The sum of wave amplitudes seeks a minimum at each point.

The following wave equation is the First Principle.

2.  Wave Structure of the Electron

The  structure of the electron consists of solutions of a general wave equation (Wolff, 1990).  This equation governs the behavior of all particle waves in space,  and is:
               (grad)²(AMP) - (1/c²) d ² (AMP)/ dt²= 0                                                [1]

where AMP is a scalar amplitude, c is the velocity of light, and t is the time.  These waves are scalar quantum waves, not electromagnetic waves.  This wave equation has two spherical solutions for the amplitude of the electron: one of them is an inward wave converging to the center; the other is a diverging outward wave.  The two solutions are:
              {IN-amplitude} = (1/r) {AMP-max} exp(iwt + ikr)
             {OUT-amplitude} = (1/r) {AMP-max} exp(iwt - ikr)                                  [2]

              where:
                        w = 2pi mc²/h = the angular frequency
                        k = 2pi/{wave length} = the wave number.

The inward wave converges to its center and rotates to become a diverging outward wave.  The superposition of the continuous inward and outward waves forms the electron, Figure 1, and is termed a space resonance.  To transform the inward wave to an outward wave and obtain constructive interference with proper phase relation requires a rotation and phase shift at the center.  This rotation produces a spin value h/4pi, the same for all charged particles because all particles propagate in the same universal wave medium.

Figure 1.  Electron Structure.  The upper diagram shows a cross-section of the spherical wave structure, something like the layers of an onion.  It is comprised of an inward moving wave and an outward moving wave.  The two waves combine to form a single dynamic standing wave structure with its center as the nominal location of the electron.  Note that the amplitude of a quantum wave is a scalar number, not an electromagnetic vector.  Thus these waves are part of quantum theory, not electric theory.  At the center the quantum wave amplitude (and the electric potential) is finite, not infinite, in agreement with the observed electron (Wolff, 1995).  The lower diagram shows the same quantum wave amplitude plotted along a radius outwards from the electron center. The lower diagram is a 'slice' from the upper diagram.

3.  Spherical Rotation

Rotation of the inward quantum wave at the center to become an outward wave is an absolute requirement to form a particle structure.  Rotation in space has conditions.  Any mechanism that rotates (to creates the quantum "spin") must not destroy the continuity of the space.  The curvilinear coordinates of the space near the particle must participate in the motion of the particle.  Fortunately, nature has provided a way - known as spherical rotation - a unique property of 3-D space.  In mathematical terms this mechanism, according to the group theory of 3-D space, is described by stating that the allowed motions must be represented by the SU(2) group algebra which concerns simply-connected geometries.

Spherical rotation is an astonishing  property of 3-D space.  It permits an object structured of space to rotate about any axis without rupturing the coordinates of space.  After two turns, space regains its original configuration.  This property allows the electron to retain spherical symmetry while imparting a quantized "spin" along an arbitrary axis as the inward waves converge to the center, rotate with a phase shift to become the outward wave, and continually repeat the cycle.

The required phase shift is a 180^o rotation that changes inward wave amplitudes to become those of the outward wave.  There are only two possible directions of rotation, CW or CCW.  One choice is an electron with spin of +h/4pi, and the other is the positron with spin of -h/4pi.

It is an awesome thought that if 3-D space did not have this geometric property of spherical rotation, particles and matter as we know them could not exist.
 

Figure 2.  Radial Plot of the Electron Structure.  When the IN and OUT quantum waves combine they form a standing wave.  This detailed plot, the same as the approximate lower plot of Fig. 1 above, corresponds exactly to the equations below.  The envelope of the wave amplitude matches the Coulomb potential everywhere except at the center, where it is not infinite in agreement with the observations of Lamb and Retherford.  If the electron were moving and observed by another detector atom with relative velocity v, the deBroglie wavelength appears as a Doppler effect on both  waves.  The frequency mc2/h of the waves was first proposed by Schroedinger and deBroglie,  proportional to the mass of the electron.  This frequency is the mass so that mass measurements are actually frequency measurements.  There is no mass 'substance' in nature.

4.  Dirac Theory of Electron Spin

The newly discovered quantum mechanics of the 1920s began to be applied to the physics of particles, seeking to further understand particles.  Nobel Laureate P A.M. Dirac sought to find a relation between quantum theory and the conservation of energy in special relativity given by                         

 E² = p²c² + mo²c⁴           [3]

He speculated that this energy equation might be converted to a quantum equation in the usual way, in which energy E and momentum p are replaced by differential calculus operators:                       

 E = (h/i) {d AMP/dt}   and    
 p = h{d AMP/dx}   +  ..    etc.
                                        [4]

He hoped to find the quantum differential wave equation of the particle.  Unfortunately, Eqn [3] uses squared terms and Eqn [4] cannot.  The road was blocked!  Dirac had a crazy idea, "Let's try to find the  factors of Eqn [3] without squares, by writing a matrix equation"                      
 [Identity]E = [alpha]pc + [beta]m_oc²        
                                                 [5]
where: [Identity] is the identity matrix.
[alpha] and [beta] are new matrix operators of a vector algebra.

Dirac was lucky!  He found that if [alpha] and [beta] were 4-vector matrices then Eqn [5] works okay.  It is the famous Dirac Equation.  Eqns [4] and [5] can now be combined to get [Identity] (ih) d[AMP]/dt = (ch/i) {[alpha-x] d[AMP]/dx + [alpha-y] d[AMP]/dy +
[alpha-z] d[AMP]/dz + [beta]m_oc²[AMP]}

In general, [AMP] is a 4-vector:  [AMP] = [AMP1, AMP2, AMP3, AMP4].

For the electron, this reduces to: [AMP] = [0, 1, AMP3(E,p) , AMP4(E,p)]

Dirac realized that for an electron only two wave functions, AMP3 and AMP4, were needed.  These predicted an electron and a positron of energy E with spin

                           E = ± mc²   and   spin = ± h/4pi

The positron was discovered five years later by Anderson (1931).

Dirac simplified the matrix algebra by introducing 2-vectors (number pairs) which he termed 'spinors.'  Spin matrices, which operate on the vectors, were defined as follows:   

{spin-x} =

0  1 1  0

{spin-y} =

0  i  -i  0

{spin-z} =

1  0 0  -1

{identity} =

1  0 0  1

 Thus Dirac had created a two-number algebra  to describe particles instead of our common single number algebra.  This 'spinor' algebra, while eminently successful, was entirely theoretical and gave no hint of the physical structure of the electron.  Now, in this paper, it is seen that the inward-outward quantum waves are the physical structure which corresponds to the Dirac spinor.  The two waves form a Dirac spinor, as was shown by Battey-Pratt et al (1986).  The two physical spinor elements of the electron, or any charged particle, are as follows:   

[electron amplitude] =

{IN-AMP} {OUT-AMP}

=  (1/r)  {AMP-max}

e(iwt + ikr)            e(iwt - ikr)           [6]

 An easily read description of the algebra of the Dirac Equation is given in Eisele (1960).

5.  Geometric Requirements of Electron Spin

Structuring particles out of space (the continuum) presents a problem if the particles are considered free to spin. If part of the continuum is part of the particle then another part of space would slide past the spinning particle. As a result, the coordinate lines used to map out the whole space would become twisted up and stretched without limit. The structure of space would be torn or ripped so that one part of the continuum would slide past another along a surface of discontinuity.

If you accept the philosophical position that "ripping of space" is unacceptable, then you have to postulate that the mathematical groups of the particle motion are simply connected and compact. In this case the motion in the continuum will be cyclic and the configuration of space can repeatedly return to an earlier initial phase. Does this occur in nature? Yes, nature accommodates this requirement. Mathematicians have long known of the spherical rotation property of 3D space in which a portion of space can rotate and return identically to an earlier state after two turns. This unusual motion was described in Scientific American (Rebbi, 1979) and in the book Gravitation (Misner et al., 1973). It is the basis of spin in this article.

What are the geometric requirements on the motion of a particle which does not destroy the continuity of the space? The curvilinear coordinates of the space near the particle must participate in the motion of the particle. This requirement according to the group theory of 3D space is satisfied by stating that the allowed motions must be represented by a compact simply-connected group. The most elementary such group for the motion of a particle with spherical symmetry is named SU(2). This group provides all the necessary and known properties of spin for charged particles, such as the electron.

4. Understanding Spherical Rotation