THE PHYSICAL ORIGIN OF SPIN OF THE ELECTRON
The Physical Origin of Electron Spin
A study using quantum-wave particle structure
Milo Wolff
Technotran Press
1124 Third Street, Manhattan Beach, CA 90266.
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,
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CPT = {Charge inversion} x {Parity operation} x {Time inversion}Ê Ñ>Ê 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 that description 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.
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1.1 WAVE STRUCTURE OF THE ELECTRON
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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
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(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 (amplitude): one of them is an IN wave converging to the center and the other is a diverging OUT wave. The two solutions are:
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{amplitude-IN} = (1/r) {amplitude-0} exp(iwt + ikr)
{amplitude-OUT} = (1/r) {amplitude-0} exp(iwt - ikr)
Êwhere:
w = 2 mc2/h = the angular frequency
k = 2 / {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 termed a space resonance , forms the electron. To super-impose the IN and OUT waves and obtain constructive interference with proper phase relation requires a unique rotation and phase shift
at the center. This rotation leads to a constant value of spin, h/4 , for all charged particles. To fully appreciate the origin of spin from the wave structure of the electron, the structure should 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 . Figures 1 and 2 summarize the structure. The spin properties are discussed below.
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1.2. SPHERICAL ROTATION
Rotation in space has requirements. Any proposal to describe the quantum property of "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. This requirement, according to the group theory of 3D space, is satisfied by stating that the allowed motions must be represented by the SU(2) group algebra which describes simply-connected geometries.
Rotation of the inward waves involves the astonishing property of 3D space called spherical rotation . 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 then outward.
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 Adrianne Maurice Dirac sought to find a relation between quantum theory and the conservation of energy in special relativity given by
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E2 = p2c2 + mo2c4ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ (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 :
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E = (h/i)((deriv).../(deriv)t) and px = h((deriv).../(deriv)x) ... etc.ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ (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 non-squared factors of Equation (1) by writing,
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[Identity]E = [alpha] pc + [beta]moc2ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ (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,
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[Identity](ih)(deriv)[Amp Vec]/(deriv)t = (ch/i) {[alpha-x](deriv)(psi)/(deriv)x + [alpha-y](deriv)(psi)/(deriv)y +
[alpha]z(deriv)(psi)/(deriv)z + [alpha]moc2(psi)}
In general, (psi) is a 4-vector, (psi) = [(psi-1), (psi-2), (psi-3), (psi-4)]
For the electron, this reduces to (psi) = [0, 1, (psi-3)(E,p) , (psi-4)(E,p)]
Dirac realized that for an electron only two wave functions, (psi)3 and (psi)4, were needed. These predicted an electron and a positron of energy E,
E = mc2 and spin = h/4 .
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:
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{spin-x} = 0Ê 1
1Ê 0 {spin-y} =Ê 0Ê iÊ
-iÊ 0 {spin-z} = 1Ê 0
0Ê -1 {????} = 1Ê 0
0Ê 1 ÊTo be viable, a physical structure must accord with quantum theory. The Wolff electron structure of spherical inward and outward quantum waves satisfies, and indeed, is the origin of quantum theory. The two waves form a Dirac spinor, as was shown by Battey-Pratt et al (1986). Thus the spinor of the electron is as follows,
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[amplitude] = {amplitude-IN}
{amplitude-OUT} =Ê (1/r)Ê xÊ {amplitude-o}Ê x e(iwt + ikr)Ê
e(iwt - ikr)ÊÊ
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 indeed 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 it slides 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,
Ê
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{amplitude} = eiwt
0 --> e-iwt
0 ÊÊ
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,
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Ê
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{amplitude} = 0 -1Ê eiwt
1Ê 0Ê 0Ê --> 0Ê
eiwt ÊThus, 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.
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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 Euclidian 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
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{amplitude} = a + ib + jc + kdIt can be shown (Battey-Pratt and Racey, 1986) that the 4x4 quaternion operator is equivalent to a 2x2 operator as follows:
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{amplitude} =Ê a + idÊÊÊ -c + ib
c + ibÊÊÊÊ a - id Êwhere 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:
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{amplitude}Ê = a + idÊÊ
c + ibÊ Ê
ÊThis 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
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OPERATIONÊ
(Dirac symbol) SU(2)Ê
OperatorÊ initial SU(2) Spinor Final SU(2) Spinor Equivalent Quaternion OperatorÊ
Leaves space as it is.Ê
[Identity] 1Ê 0
0Ê 1 1
0 1
0 1
Rotates space 180o about the x-axis.Ê
(spin-x) 0Ê i
iÊ 0 1
0 0
i i
Rotates space 180oÊ
about the y-axis
(spin-y) 0Ê -1
1Ê 0 1
0 0
1 j
Rotates space 180oÊ
about the z-axis
(spin-y) iÊ 0
0Ê -i 1
0 i
0 k Ê
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
Ê
{amplitude} = eiwtÊ
0 ÊÊ
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.
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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".
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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, transforming into each another. 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 their wave structure, a crystal might appear like many shimmering bubbles neatly joined in geometric arrays held together with immense rigidity - also a property of space.
The next frontier science of the future is to understand the meaning and structure of space .