*“Scientists are
peeping toms at the keyhole of eternity”*

Arthur Koestler (1905-1983)

*The Roots of Coincidence*

**One of the Greatest Myths of Modern Physics –**

Electron Spin of

**1. Introduction **

The objective of physics is a *study* of nature. However, the former
20th century has passed mainly under the guise of **ascribing**
fictitious attributes to nature directly following Einstein’s approach (where “*was
a great free of the imagination: a great space for self-chosen assertions about
what will ultimately work*” [David Bodanis, E=mc^{2}: *A Biography
of the World’s Most Famous Equation*]). There are a lot of examples which
confirm this statement. Here we present one of them devoted to the notion spin.
The last was introduced in physics in connection with the *theoretical
error *appeared at the mechanical description of Einstein’s-de Haas’s
experiment and based on the thoughtless approach at the derivation of average
current of orbiting electron.

The average current was obtained on the basis of the *mechanical model
*of uniform motion of the electron regarded, in the classical spirit of the
definition of current, as a flow of electric charge (“electron liquid”) in a
conductor. According to such a primitive mechanical model, disregarding
peculiarities of (1) *wave motion* and (2) *closed motion along the
circle*, the average value of orbital current, caused by the orbiting
electron, unfortunately, has been accepted to be equal to the *incorrect
ratio*

, (1.1)

where *T*_{orb} is the period of electron’s revolution along an orbit. This
expression gave rise to all further fittings to the experiment.

By definition of the 1930’s, the magnetic moment of a closed electric circuit
(in the specific case of the electron orbit in the hydrogen atom) is assumed to
be equal to

, (1.2)

where *I* is the average value of current on the orbit, and *S* is the area of the
orbit. The relation
is actually the
*circulation* [1], called in physics the
*current in the magnetic system of units* CGSM.

Resting upon the *incorrect *ratio (1.1) and the definition (1.2), the
*incorrect*
magnetic orbital moment was obtained:

, (1.3)

which in turn led to the *erroneous* formula of the ratio of the electron’s
orbital magnetic moment
to its
**moment of momentum**
on the Bohr orbit:

, (1.4)

The value (1.4) is half as much the **real ratio**

(1.5)

obtained experimentally by A. Einstein and De Haas [2-3]. Eq. (1.4) is
inconsistent also with S.J. Barnett’s experiment [4, 5], *etc*. At that time,
instead of seeking the error in the theoretical derivation of the expression
(1.4), the hypothesis-fitting was accepted. According to this hypothesis, the
proper magnetic moment
equal to the orbital magnetic moment
was attributed to
the electron. Further, naturally, in order to reduce in correspondence with the
proper magnetic moment, the “proper moment of momentum”
(“spin”) of
value was
invented and attributed to the electron as well. Thus, an appearance of
correspondence of the theory to the experiment (Eq. (1.5)) was created as a
result of such a mathematical adjustment:

.
(1.6)

**2. The correct approach**

The erroneous formula of the average current (1.1) is a result of the product of
the frequency of revolution of an electron by the electron charge *e*:
;
*i.e*.,
it represents the ratio of electron charge to the orbital period. Where is the
mistake? Let us turn to the analogy with a mathematical pendulum. Following the
same logic, which led to the formula (1.1), we should to announce the ratio of
amplitude of displacement of a pendulum to the period of its oscillations as the
average speed of its motion, but it is absurdity.

The motion in *inner space* of a circular trajectory, along two successive
half-circumferences, occurs in one direction (clockwise or anticlockwise) (Fig.
2.1a).

**Fig. 2.1.** (a) The amplitudes of displacement, *a*_{m} and
*A*_{m}, in a wave of the
fundamental tone on a circumference; *S*_{p} and *S*_{k} are the potential and kinetic
points (nodes) of the wave; the kinetic node represents the center of a loop of
the wave (b). (c) A mathematical pendulum.

Simultaneously, the same motions *in outer space*, as mutually relative ones, are
opposite-directed; and one revolution of the electron is equivalent to one-half
period (Fig. 2.1b) of pendulum swinging (Fig. 2.1c). This fact shows the
contradictoriness of the circular motion. If *S*_{p} is an arbitrary potential point
of a wave of the fundamental tone (*i.e*., its node), then, the conjugated
diametrically opposite point *S*_{k} will be the kinetic point of the wave (its
loop). The rectilinear amplitude of displacement is equal to the diameter of a
circumference,
. The amplitude of the curvilinear displacement along a
circumference is equal to half-circumference, *i.e*., quarter-wave:
.

Let us consider the transfer of some property *p* (Fig. 2.2a).

**Fig. 2.2.** (a) The exchange between the points A and B at a rectilinear part with
the time interval *T*_{orb} between them and (b) the exchange between the coinciding
points, A and B, at the circular trajectory.

If during a period *T*_{orb}, some property *p* is diminished from point A (analogous
to *S*_{p} in Fig. 2.1a) and the same property is arrived at point B (analogous to
the same point *S*_{p} in Fig. 2.1a), then the average speed of change of the
property is

. (2.1)

Thus, the transfer of any property of some object (a parameter of exchange *p*)
will be characterized by the average rate of exchange *I*, determined by the
expression

. (2.2)

In the case of a circular orbit, when the points of exchange, A and B, coincide
(Fig. 2.2b), an object with mass *m* passes through the cross section *S* with the
average speed of exchange

. (2.3)

This value shows that during a period *T*_{orb}, the mass of an orbiting object on
the *right side* of the cross-section *S decreases* along the value *m* and
*increases*,
along the same value* m*, on the *left side*. Accordingly, the average speed of
exchange of the charge *e* in any section of circular trajectory is defined by the
relation

. (2.4)

In reality, the *orbital motion of the electron is the wave process* represented
by the cylindrical wave field [1]. Therefore, let us show the solution of the
aforementioned problem on the basis of calculation of the wave motion of the
orbiting electron, regarding the electron as a *particle-wave*. The electron
(particle), being the discrete part of the wave, is represented by the *wave
node*. (Other ways, leading to the same result (1.5), are considered in details
in the **book** [1] and partly in the
**paper** placed at this web site)

From the well-known solution of the wave equation for a string with the length
*l*
fixed at both ends,
(where
), follows that
*one half-wave of the fundamental tone
*(where
is the wave speed in the string) is placed on its whole length
*l*:
. If
the ends of the string are joined together, forming a string circle of the
length
with one node, we have

and
. (2.5)

As shown in [1], similar to the case of the wave field of a string, only a
*half-wave of the fundamental tone* is placed on the Bohr first orbit, and the
electron is in the node of the wave. Because of this, the average value of
current *I*, as a harmonic quantity, is determined by the integrals:

or
. (2.6)

The amplitude of the elementary current is defined as

, (2.7)

where
is the frequency of the fundamental tone of the electron orbit, equal to

, (2.8)

is the wave period. Hence, the average current of the electron orbit is

or
. (2.9)

The above derivation, as all other ones described in [1], gives the average
value of the current twice as large (1.1). Two (coinciding in our case) states
of the beginning and the end of the period is the inherent feature of any
periodic functions without which there is no period. And these states cannot be
divided, as states of rest, by two parts belonging to two adjacent periods.

Taking (1.2) into consideration, we find the orbital magnetic moment of the
electron, as the *magnetic moment of harmonic wave of the fundamental tone*,

. (2.10)

From this it follows that the ratio of the orbital magnetic moment of the
electron to the moment of its orbital momentum is

. (2.11)

Just this formula is confirmed experimentally. It undoubtedly proves the
inconsistency of the hypothesis on the electron’s spin of .

**3. Conclusion**

The spin myth gave birth to the theoretical spinomania. Of course, an electron
has its own magnetic field and magnetic moment and moment of momentum. But, as
the calculations show [1], the last is insignificantly small in comparison with
the orbital moment. Let us imagine that the proper moment of momentum of Earth
is equal to one half of its orbital moment of momentum. The Earth cannot endure
such a huge moment and will be destroyed. The same situation will meet an
electron with the “spin” equal to
.

Dirac’s *relativistic wave theory of spin* (1928) [6], created for the proof of
the correctness of an introduction of the spin of such a value, “proved” it.
From that time, the further development has led to the electron being regarded,
not as a real particle (which is unable to endure the huge moment which was
ascribed to it), defined by three spatial coordinates, but as a top-like
structure, possessing an angular momentum of its own. Dirac noted in this
connection [7] that the aim is “*not so much to get a model of the electron as to
get a simple scheme of equations which can be used to calculate all the results
that can be obtained from experiment*”.

As a result, due to the gross fitting, the formal correspondence of the “theory”
with the experiment was realized. We state it resting upon the data [1], which
convincingly show that Eq. (1.1) for the *average value of current* of the
orbiting electron is *erroneous*. Accordingly, all equations obtained on its basis
(including (1.3) and (1.6), *etc*.) are incorrect as well. For this reason, the
Dirac equation is false and has significance only from the point of view of
history of the philosophical and logical errors of the past. With that, in spite
of the leading role of quantum electrodynamics in modern physics, one should not
forget that the correspondence of any theory with the experiment does not quite
mean that the given theory is true and uniquely possible. And what is more, the
possibilities of modern mathematics are so impressive that it can represent any
abstract absurdity as a profound theory (or its development) and fit it to the
experiment. After all, physics must not only “*calculate all the results*” [7],
but also comprehend Nature.

**References**

[1] L. Kreidik and G. Shpenkov, *Atomic Structure of Matter-Space*, Geo. S.,
Bydgoszcz, 2001, 584 p.

[2] A. Einstein, W.J. De Haas, Verch. Deutsch. Phys. Ges. 17, 152 (1915).

[3] A. Einstein, Verch. Deutsch. Phys. Ges. 18, 173 (1916).

[4] S.J. Barnet, Phys. Rev. 6, 171, 239 (1915); 104, 7 (1917).

[5] S.J. Barnett, Rev. Mod. Phys. 7, 129 (1937).

[6] P.A.M. Dirac, *The Principles of Quantum Mechanics *(third edition), Clarendon
Press, 1947.

[7] P.A.M. Dirac, *Classical Theory of Radiating Electrons*, Proc. Roy. Soc., V.
168, p. 148 (1939).

*George Shpenkov*

January 9, 2003

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