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  The Physical Origin of Electron Spin

               A study using quantum-wave particle structure

                                 Milo Wolff

                              Technotran Press
                1124 Third Street, Manhattan Beach, CA 90266
                         mwolff@beachnet.gen.ca.us

     Abstract:

     It is shown how the spin of the electron and other charged
particles
     arises out of the quantum wave structure of matter. Spin is a
result
     of spherical rotation in quantum space of the inward (advanced)
     quantum waves of an electron at the electron center in order to
become
     the outward (retarded) waves. Rotation is required to maintain
proper
     phase relations of the wave amplitudes, similar to mirror
reflection
     of e-m waves. The spherical rotation, which is a unique property
of 3D
     space, can be described using SU(2) group theory algebra. In the
SU(2)
     algebra, the IN and OUT waves of the charged particle are the
elements
     of a Dirac spinor wave function. Thus all charged particles
satisfy
     the Dirac Equation.


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

In Dirac's theoretical work the spin of a particle is measured in
units of
angular momentum, just like rotating objects of human size. But
particle
spin is uniquely a quantum phenomenon because it is not related to the
mass
of the particle. It is different than human-scale angular momentum.
Its
value is fixed and cannot be calculated from simple mass or angular
velocity. Spin properties are found to be related to other quantum
properties of the electron wave function: mirror inversion(P), time
inversion(T) and charge inversion (C). For example, the quantum
operation
CPT is found to be invariant,


     CPT = {Charge inversion} x {Parity operation} x {Time inversion}
�>
     invariant

This study returns to a proposal that was popular sixty years ago
among the
pioneers of quantum theory. Namely that matter consisted entirely of
perturbations or undulations in the fabric of space. Thus, the
speculated
matter substance, mass and charge, did not exist. Wyle, Shroedinger,
Clifford, and Einstein were among those who believed that particles
were
structured out of a continuum of space. Their belief is consistent
with the
quantum theory of matter since quantum theory does not depend on the
existence of particle substance or charge substance. A belief in 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."

The wave structure of matter has been intensively studied by by Wolff
(1990, 91, 93, 95, 97). It has successfully predicted all of the
properties
of the electron, except one - spin. This paper completes those
predictions
with a physical origin of spin that accords with quantum theory, the
Dirac
Equation, and the Wolff structure of the electron. Spin appears as a
required rotation of the advanced (IN) and retarded (OUT) quantum
waves
that comprise the electron structure.


1.1 Wave Structure of the Electron.



Figure 1. Electron structure.                   [Electron's wave
structure]

     The upper diagram shows a cross-section of the spherical wave
     structure, something like the layers of an onion. There are two
sets
     of layers, one set is inward moving waves and the other set is
outward
     moving waves. The IN-going and OUT-going waves combine to form a
     standing wave. That is the amplitude rises and falls but there is
no
     in or out motion. 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. Indeed, electric
waves
     are transitions between various energy states of atoms and
molecules
     which are determined by the quantum wave structure. However the
     electric potential is proportional to the quantum wave amplitude.
At
     the center the quantum wave amplitude (and electric potential) is
     finite, not infinite, in agreement with the observed electron.
From
     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.







[Image]

     Figure 2. Radial plot of a Space Resonance.

     When the IN and OUT quantum waves combine, they form a standing
wave
     as shown here. This is a detailed plot  - the same as the
approximate
     lower plot of Figure 1 above but it corresponds exactly to the
     equations below.  The envelope of the wave amplitude matches the
     Coulomb potential everywhere except at the center where it is
finite
     in agreement with the observations of Lamb and Retherford. The
wave
     frequency is that of the mass energy of the electron. If the
electron
     were moving (say it is in an H atom), and observed by another
detector
     atom with relative velocity v,  a deBroglie wavelength would
appear to
     modulate the waves.  The frequency of the waves mc2/h 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.

Equations of the waves.
The Wolff structure of the electron consists of two spherical waves,
one
advanced (IN) and one retarded (OUT) which are solutions of a general
wave
equation. This equation, which governs the behavior of electron waves
in
space, is


     (gradient)2{amplitude} -1/c2 (deriv) 2{amplitude} / (deriv) t2 =
0

where (amplitude) is a scalar amplitude, c is the velocity of light,
and t
is the time. These waves are scalar waves not directional vector
waves.
This wave equation has two spherical solutions for the amplitude: one
of
them is an IN wave converging to the center and the other is a
diverging
OUT wave. The two solutions are:


     {IN-amplitude} = (1/r) {Max-amplitude} exp(iwt + ikr)
     {OUT-amplitude} = (1/r) {Max-amplitude} exp(iwt - ikr)

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

The inward wave converges to a center, rotates to become the outward
wave
which diverges from the center. The superposition of these two, a
continuous IN wave and the OUT wave forms the electron and is termed a
space resonance. To super-impose the IN and OUT waves and obtain
constructive interference with proper phase relation requires a
rotation
and phase shift at the center. This rotation leads to a constant value
of
spin, h/4pi, for all charged particles. This structure can be further
reviewed in Wolff (1993 and/or 1995).
   The structure, the electron properties, and the laws of nature all
originate from three basic principles or assumptions. No other laws
are
required. The wave equation is the First principle.

1.2. Spherical Rotation

Rotation of the inward wave at the center to become an outward wave is
an
absolute requirement to form a particle. Rotation in space also has
requirements. Any mechanism that rotates (and 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 3D space. In mathematical terms, this
mechanism, according to the group theory of 3D space, is described by
stating that the allowed motions must be represented by the SU(2)
group
algebra which concerns simply-connected geometries.
    Rotation of the inward waves involves this astonishing
spherical-rotation property of 3D 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 spherical waves
continually
move inward and outward.
    It is an awesome thought that if 3D space did not have this
geometric
property of spherical rotation, particles and matter as we know them
could
not exist.

2. Dirac Theory of Electron Spin

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


                               E2 = p2c2 +
     mo2c4                                       (1)

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)((deriv).../(deriv)t) and px = h((deriv).../(deriv)x)
..
     etc.                               (2)

He hoped to find the quantum differential wave equation of the
particle.
Unfortunately, Equation (1) uses squared terms and Equation (2)
cannot.
Dirac had a crazy idea, "Let's try to find the  factors of Equation
(1)
without squares, by writing a matrix equation,


     [Identity]E = [alpha]pc +
     [beta]moc2                                                  (3)

     where:
          [Identity] is the identity operator
          [alpha] and [beta] are new operators of a vector algebra.

Dirac was lucky! He found that if [alpha] and [beta] were 4-vector
matrices
then Equation (3) works okay. It is the famous Dirac Equation.
Equations
(2) and (3) can now be combined to get,


     [Identity](ih)(deriv)[Amp]/(deriv)t = (ch/i)
     ([alpha-x](deriv)[Amp]/(deriv)x + [alpha-y](deriv)[Amp]/(deriv)y
+
     [alpha]z(deriv)[Amp]/(deriv)z + [alpha]moc2[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,

     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 which he
termed "spinors." Spin matrices, which operate on the vectors, were
defined
as follows:




      {spin-x} =0  1  {spin-y} = 0  i  {spin-z} =  1  0 {identity} = 1
0
                1  0             -i  0            0  -1              0
1


To be viable, a physical structure must accord with quantum theory.
The
Wolff electron structure of spherical inward and outward quantum waves
satisfies quantum theory, and indeed is its origin. The two waves form
a
Dirac spinor, as was shown by Battey-Pratt et al (1986). Thus the two
elements of the Dirac spinor of the electron are as follows,



                                                                e(iwt
+
     [electron         {IN-amplitude} =  (1/r)  x               ikr)
     amplitude] =     {OUT-amplitude} {Max-amplitude}
                                                                e(iwt
- ikr)




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

Many popular theories of the electron regard the Dirac wave functions
of
the electron to be unreal. The functions are only regarded as
mathematical
devices for calculating probabilities. However Wolff (1993), in
agreement
with Cramer (1986), finds that spherical wave functions are real. In
this
paper the discovery that spin (certainly real) occurs as a result of
rotation of the spherical waves, shows that quantum waves are also
real.
This is not the only evidence. It is shown elsewhere (Wolff, 1997A, B)
that
quantum waves underlie the structure of all matter and are the origin
of
the laws of Nature.

3. 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

This seldom studied motion can be modeled by a ball held by threads
attached to a frame. The threads represent the coordinates of the
space and
the rotating ball represents a property of the space at the center of
a
charged particle composed of converging and diverging quantum waves.
The
ball can be turned about any given axis starting from any initial
position.
If the ball is rotated indefinitely it will be found that after every
two
rotations the system returns to its original configuration.

In the traditional analysis of rotating objects, it is usual to assume
that
the process of inverting the axis of spin is identical to reversing
the
spin. However, if the object is an electron which is continuously
connected
to its environment as part of the space around it, this ceases to be
true.
A careful distinction must be made between the inversion and the
reversal
of particle spin. This distinction provides insight to one of the most
fundamental properties of particles.

To reverse the spin axis, one can reverse time ( t --> -t) or reverse
the
angular velocity (w --> -w). Either are equivalent to exchanging the
outgoing spherical wave of an electron with the incoming wave. Then
the
spinor becomes,



                                            eiwt
e-iwt
     {amplitude} =                                    -->
                                             0                      0



To invert the spin axis of the structure of the particle, it is
necessary
to turn the structure about one of the axes perpendicular to the z
spin
axis, for example the y axis. Then the inverted spin state is given by
the
inversion matrix operation,




                                       0 -1      eiwt
0
     {amplitude} =                                           -->
eiwt
                                       1  0      0


Thus, inversion and reversal are not the same. The difference between
these
operations is characteristic of the quantum nature of the electron.
They
are distinct from our human-sized view of rotating objects and are
important to understand particle structure.


5. The Group Mathematics of Spherical Rotation

Each configuration of the spherically rotating ball (or the electron
center) can be represented by a point on a Euclidean 4-D hypersphere
which
is also the space of the SU(2) mathematics group. A rotation in the
spherical mode can be represented by any operator that will transform
one
vector into another position. It is usual to assign the hypersphere a
unit
radius. Then the rotations of the ball can be described by the
mathematics
of the SU(2) group. It is also convenient to place the center of the
unit
hypersphere at the origin and let the vector (1,0,0,0) represent an
initial
configuration of the ball or electron. Any other configuration is
often
chosen with the symbols (a,b,c,d). Then a2+ b2 + c2 + d2 = 1.

A common representation for the hypersphere vectors is the quaternion
notation


                                 {amplitude} = a + ib + jc + kd

It can be shown (Battey-Pratt and Racey, 1986) that the 4x4 quaternion
operator is equivalent to a 2x2 operator as follows:



                                                   a + id   c  -
                                  {amplitude} =         ib
                                                  c + i    a - id


where the matrix elements (often just 1, i, or 0) are now complex
numbers.
You can see that the determinant of this is also a2 + b2 + c2 + d2 =
1, as
above. The spinor (operand) form of (amplitude) is:




     {amplitude}  =                               a + id
                                                   c + ib



This is the notation of the Spinors invented by Dirac to represent the
electron configuration, as shown in TABLE I. They also represent
rotations
in the spherical mode which are members of the closed uni-modular
SU(2)
group.

TABLE I: Properties of Spherical Rotation for

an electron in the SU(2) Representation



     OPERATION        SU(2)      initial   Final SU(2)      Equivalent
   (Dirac symbol)    Operator SU(2) Spinor    Spinor        Quaternion
                                                            Operator
 Leaves space as it
        is.            1  0         1           1               1
     [Identity]        0  1         0           0
 Rotates space 180o
                       0  i         1           0
 about the x-axis.     i  0         0           i               i
      (spin-x)
 Rotates space 180o
                       0  -1        1           0
  about the y-axis     1  0         0           1               j
      (spin-y)
 Rotates space 180o
                       i  0         1           i
  about the z-axis     0  -i        0           0               k
      (spin-y)


For example, the spherical quantum waves in space can be rotated 180o
about
the z axis by the operator (spin-z). If there is continuous rotation
of the
quantum wave in space with angular velocity w, the spinor is
represented by


                                                            eiwt
                                 {amplitude} =
                                                              0


6. How Spin Arises from the Wave Structure of the Electron

The wave structure of the electron is composed of a spherical inward
quantum wave and an outward wave traveling at light speed c (Wolff,
1990,
1993, 1995). Figures 1 and 2 show the wave structure of an electron
termed
a space resonance. The outward (OUT) wave of an electron travels to
and
communicates with other matter in its universe. When these waves
arrive at
other matter, a signature is modulated into their outward waves. These
outward-wave signatures are the response (Wheeler & Feynman, 1945;
Cramer,
1986, Ryazanov, 1991) from the other matter. The total of response
waves
from other matter in the universe, as a Fourier combination, becomes
the
inward (IN) wave of the initial electron. The returned inward waves
converge to the initial wave center and reflect with a phase shift
rotating
them to become the outward wave and repeating the cycle again.

The central phase shift is similar to the phase shift of light when it
reflects at a mirror. The required phase shift is a 180o rotation of
the
wave, either CW or CCW. There are only two possible combinations of
the
rotating inward and outward waves. One choice of rotation becomes an
electron, the other becomes a positron. The angular momentum change
upon
rotation is either +h/4 or -h/4 . This is the origin of spin. One wave
set
is the mirror image of the other set producing the CPT invariance
rule.

7. Conclusions

7.1. A COMPLETE SET OF ELECTRON PROPERTIES

The origin of spin completes the properties of the electron. All
properties
can now be derived from the space-resonance structure and match all
experimental observations of the electron. There is now little doubt
that
matter is composed of spherical quantum wave structures that obey the
three
principles of wave structure of matter. But note that spin, and other
properties, are attributes of the underlying quantum space rather than
of
the individual particle. This is why spin, like charge, has only one
value
for all particles. The properties depend on the structure of space.


7.2. A UNIVERSE OF QUANTUM WAVES AND SPACE

Although the origin of spin has been a fascinating problem of physics
for
sixty years, spin itself is not the important result. Instead, the
most
extraordinary conclusion of the wave electron structure is that the
laws of
physics and the structure of matter ultimately depend upon the waves
from
the total of matter in a universe. Every particle communicates its
wave
state with all other matter so that the particle structure, energy
exchange, and the laws of physics are properties of the entire
ensemble.
This is the origin of Mach's Principle. The universal properties of
the
quantum space waves are also found to underlie the universal clock and
the
constants of nature.

This structure settles a century old paradox of whether particles are
waves
or point-like bits of matter. They are wave structures in space. There
is
nothing but space. As Clifford speculated 100 years ago, matter is
simply,
"undulations in the fabric of space".


7.3 THE SIMPLE ELECTRON

The elegance of the electron structure is its basic simplicity. It is
only
two spherical waves gracefully undulating around a center, each
transforming into the other. Its spherical wave structure combines
with the
waves of other charged particles to create myriads of standing wave
structures. These structures become the crystalline matter of the
solid
state. If you could see its wave structure, a crystal might appear
like
many shimmering bubbles neatly joined in geometric arrays.  The arrays
are
held together with immense rigidity -  a property of space.

The next frontier science of the future is to understand the meaning
and
structure of space.

8. References and Further Reading

   * Apeiron, V2, No. 4, Oct. 1995. This contains eight articles
discussing
     various interpretations of quantum theory.
   * E. Battey-Pratt, and T. Racey (1980) "Geometric Model for
Fundamental
     Particles," Intl. J. Theor. Phys. 19, 437-475. They recognized
that
     electron spin was a geometric property of space and could exist
in a
     spherical structure.
   * Louis Duc de Broglie (1924), PhD thesis "Recherch� sur la Theorie
des
     Quanta," U. of Paris. He proposed a wavelength l=h/p for the
quantum
     waves of an electron containing an oscillator of frequency,
mc2/h. as
     in the space resonance.
   * William Clifford (1956), "On the Space Theory of Matter" The
World of
     Mathematics , p 568, Simon & Schuster, NY. An English
mathematician at
     the Royal Philosophical Society, he first suggested (1876) that
matter
     was composed of pure waves.
   * John Cramer (1986), "The Transactional Interpretation of Quantum
     Mechanics", Rev. Mod. Phys 58, 647-687. He interpreted the waves
of
     quantum mechanics as real, in contrast to the unreal but popular
     "probability wave." He named an offer-wave (outward) and a
     response-wave (inward).
   * John A. Eisele (1960), Modern Quantum Mechanics with Applications
to
     Elementary Particle Physics . Wiley- Interscience, (John Wiley &
Sons,
     NY, London). This book discusses the Dirac Equation in detail.
   * C.W. Misner, K. Thorne, and J.A. Wheeler (1973), Gravitation ,
W.H.
     Freeman Co. p1149. This book covers many pioneering ideas
including
     the 3D space property of spherical rotation .
   * C. Rebbi (1979) "Solitons" Scientific American, Feb., 92, 168. He
     discusses spherical rotation applied to Solitons.
   * Giorgi Ryazanov (1991), Proc. 1st Int'l Sakharov Conf. Phys.,
Moscow,
     May 21-31, pp. 331-375, Nova Sci. Publ., NY. He used a
Wheeler-Feynman
     method to deduce that Natural Laws are the response of the
universe.
   * J. Wheeler and R. Feynman (1945), "Interaction with the Absorber
as
     the Mechanism of Radiation," Rev. Mod. Phys. 17, 157. They
modeled the
     electron with inward and outward waves to investigate the energy
     transfer mechanism to an absorber.
   * Milo Wolff (1990), Exploring the Physics of the Unknown Universe
,
     ISBN 0-9627787-0-2. (Technotran Press, CA) A reader-friendly
     investigation of the natural laws with applications to particles
and
     cosmology.
   * Milo Wolff (1991), "Microphysics, Fundamental Laws and
Cosmology".
     Proc. 1st Int'l Sakharov Conf. Phys., Moscow, May 21-31, pp.
     1131-1150, Nova Sci. Publ., NY.
   * Milo Wolff (1993), "Fundamental Laws, Microphysics and
Cosmology,"
     Physics Essays 6, pp 181-203.
   * Milo Wolff (1995), "Beyond the Point Particle - A Wave Structure
for
     the Electron," Galilean Electrodynamics 6, No. 5, pp. 83-91.
   * Milo Wolff (1997A) "Exploring the Universe and the Origin of it
Laws,"
     Temple University Frontier Perspectives 6, No 2, pp. 44-56.
   * Milo Wolff (1997B) "The Eight-Fold Way of the Universe," Apeiron
4,
     no. 4. Oct (1997).