AETHER SCIENCE PAPERS: NO. 7

THE THEORY OF THE PROTON CONSTANTS

Copyright © Harold Aspden, 1988

ABSTRACT

The proton/electron mass ratio and the proton magnetic moment
measured in nuclear magnetons have values now known to a very
high degree of precision. The only theory of record that gives
physical reasoning which can account for these dimensionless
numerical ratios is one reported in Hadronic Journal in its
extended application to the neutron and the deuteron. In view
of the recent CODATA specification of the proton constants,
which is of higher precision than was known when the proton
theory was first published, it is appropriate to review that
theory and present it in a simplified and updated form. It is
found that the theory remains viable, nothwithstanding the
stringent demands imposed by the measurement data.

INTRODUCTION

The proton is the most important charged particle in the
universe, accounting for almost all of the mass of electrical
matter. Yet, physicists accept that all that is known about the
proton is the fact that it has a charge equal in magnitude but
opposite in polarity to that of the electron, a mass that is
1836.152701(37) times that of the electron, a magnetic moment in
spin that is 2.791847386(63) times that expected from the
classical theory of its response in orbital charge motion in a
magnetic field. In addition there is some indication from
experiment that it has a three-part composition.

Surprisingly, all the interest centres on the latter aspect
and the related quark theories in the context of what is known
as ‘quantum chromodynamics’ (QCD) and little, if any, independent
interest is shown in the direct physical justification for the
numerical ratios, which are so clearly the personal ‘signature’
of this fundamental particle guiding enquiry as to its nature.

Equally surprising is the fact that those scientists engaged
on precision measurement of particle properties can actually
achieve such remarkable accuracy in a quest that surely has to
be seen as probing the consistency and universality of the
physical world. Something determines the ratio 1836.1527 and
physics cannot claim to be on track until we determine a causal
basis.

The author has proposed such a theory [1] and has subsequently
reported on developments [2] but this theory is seen as outside
the mainstream interest, which wills that the answers must emerge
from QCD. Accordingly, there is little that can be done but to
wait for QCD to run its course, except to refine the presentation
of the theory and test and update it as new precision
measurements are reported. The recent CODATA revision [3] is an
event bearing upon this task. The proton data are now so
precise that some very fine features of the proton theory come
under scrutiny. This paper reports on this examination and shows
that the theory holds up well.

It is stressed at the outset that the methods used by the
author in this proton theory do not stand in isolation. They
have connection with the whole spectrum of particle physics and
the quantum properties of the vacuum field, as evidenced by the
nature of the photon. To endorse this reference is made to
several recent papers concerning hadron mass determination [4,
5, 6]. It is the object of this paper to confine attention to the
proton per se, though opportunity will be taken to develop the
analysis that leads to the formulation and derivation of the
fine-structure constant.

THE PROTON MASS

We will proceed directly to the calculation of the
proton-electron mass ratio.

The scenario under discussion is that of an extremely active
field with surplus energy that could not be accommodated into a
quiescent ‘equilibrium’ background of the vacuum state. The
protons are stable and the creation process relates essentially
to events in parts of the universe and at periods when the proton
could be created. In quiescent space and matter as we know it
locally we may suppose that there are processes occurring that
involve Nature’s attempts to create new protons, but that these
actions fail to reach the creation threshold. Obviously, in the
high energy particle experiments conditions may be produced that
result in transformations and could conceivably produce protons
from particles of lower rest-mass energy, but pending such
discoveries the interest centres on the process which could have
been effective in the early period of creation of the universe.

The key factor in this enquiry is the synthesis of the proton
from the merging of n virtual muons, a virtual muon in this
context being a feature of an active vacuum state in which pairs
of virtual muons of opposite charges populate unit cells of space
and are mutually annihilated and recreated at random positions
at the Compton electron frequency.

The virtual muon signifies an energy quantum that is slightly
smaller than the mass-energy of the real muon. It is calculable
from first principles in terms of the electron rest-mass energy,
but before discussing the rigorous determination of this quantity
we will proceed on the tentative basis that the virtual
muon/electron mass ratio is approximately 206.333.

The major step, outlined in reference [4], which accounts for
proton creation is then the process described by the equation:

This equation declares that, if an even number k of virtual
muons μ of both polarities come together to create a neutral
complex, this complex can restructure in a minimal energy
configuration in which a single charge (say -e) of energy kμ has
close association with a single charge (+e) of energy z, such
that z has a value corresponding to least energy of the total
complex. Then, suppose enough (n) virtual muons impact this
complex to drive out z and leave a residual complex in which a
new positive charge of energy P is paired with kμ in a minimal
energy configuration. The condition specified by the equation
is that z can then merge with this complex to destroy itself and
the kμ unit to leave P in isolation.

The conditions just specified are somewhat unique, because the
techniques used to formulate the energy of a charge pair complex
are very precise. The energy of the complex (kμ:z) is simply the
sum of kμ and z plus the negative potential of the Coulomb
interaction between the charges nucleated by kμ and z. We use
the classical J. J. Thomson formula for energy as 2e²/3a, where
a is charge radius to find that two such charges of different
energy in surface contact (touching radii) would have a mutual
Coulomb potential of -3kμz/2(kμ+z). Adding kμ and z to obtain
an overall energy expression and differentiating this energy to
find its minimum as z varies and kμ is constant then leads to a
minimal energy which can be verified as being (6^1/2-3/2)kμ,
because z/kμ becomes (3/2)^1/2 – l.

On this basis equation (1) assumes the form:

It may be verified that there are integer solutions for n and
k satisfying this double equation and that their minimal values
are n = 7 and k = 2.

This, in itself, is surprising and it makes the proton
equation (1) rather special. It gives physical basis for the
belief that a natural stable particle form P would be created if
there were surplus energy in the virtual muon field. The
particle would develop from a merging of 9 virtual muons and
would have a rest mass-energy in electron units equal to the
value of P given by equation (9), namely:

In order to determine P it is next necessary to establish how
Nature decides the value of μ.

THE VIRTUAL MUON

It is inappropriate to think in terms of the mass of the
virtual muon. The quantum involved is an energy of μ times the
rest-mass energy of the electron, but, as the virtual muons
define an inertial rest frame in the space field we shall now
analyze, there is no reason for attributing a mass property and
particularly a gravitational property to that hidden field which
characterizes the space medium. The evidence that these virtual
muons do exist in the vacuum field is really the success of
theory which builds on that supposition. The virtual muon mass
has to be a calculable quantity from the theory, since it cannot
be directly measured and it would be inappropriate to assume that
it is identical to the mass of the real muon. Indeed, the only
reason for calling the virtual muon a ‘muon’ is that we know that
P is approximately 1836 and equation (3) tells us that μ will be
of the order of 206, which is close to the real muon/electron
mass ratio.

In spite of this it is convenient to think in terms of an
energy density in the virtual muon field and use an equivalent
mass density ρ, even though ρ could never be sensed as mass.
This represents the statistical incidence of a virtual muon
population in the equilibrium field of vacuum space. Take a
characteristic cubic cell of such space of side d, corresponding
to the presence of a virtual muon pair of notional mass 2μm,
where m denotes electron rest mass. Then:

Now, from the photon theory that will be summarized in a later
section, it is known that:

where a is 2/3 times the classical electron radius e²/mc². Therefore, the chance of there being one or more virtual muons within a sphere bounded by the radius a is calculable. Indeed, it is relevant on a cosmological scale to the determination of
the Hubble constant [7]. However, our interest centres here on
the ‘equilibrium’ condition which is that the density ρ assigns
a notional mass m’ to a characteristic volume of space that we
relate to a sphere of radius b:

An intuitive step that we now take assigns the electron charge
e to this space quantum of notional mass m’ and identifies the
electric energy of such a quantum with the energy represented by
this mass term. The applicable formula relating the radius
bounding charge and the energy is that already used for the
electron, so that:

By now writing N as (b/a)³ it is possible to combine equations (4) to (7) so as to eliminate all terms except N and μ. The
resulting equation is:

To find N we next take the result already deduced. The
threshold for proton creation depends upon 9 virtual muons
merging in their statistical migration in the space region that
they populate. Imagine that this process involves initial
electron-positron creation governed by the threshold set by the
value of N in the system just described. N is really the number
of electron charge volumes that can be matched to the
characteristic charge volume of the m’ quantum. Given the charge
e of this quantum unit, the creation of charge pairs as electrons
and positrons and the creation of a residual electron implies
that N must be odd. Otherwise, charge parity would be violated.

The condition for “equilibrium” space requires that N must be
the odd number just below the threshold at which 9 virtual muons
can initiate the proton creation process. Then there will be
attempts at creation of protons in a space occupied by these
below-threshold quanta m’ but the action will depend upon vacuum
fluctuations which lead to instabilities and preclude excitation
of stable proton states.

Note that the proton equation (1) requires that μ should be
the same for all 9 virtual muons involved in a successful
transition to a proton.

Before developing this theme further, we will first estimate
μ by writing the approximate relationship:

Combining this with (8)leads to the approximation:

from which N is approximately 1949. Remember then that N must
be an odd integer, but equally we would like μ to be an odd
integer if the virtual muons are to create electron-positron
components in their attack on the m’ system. Equation (8) does
not permit both N and μ to be integers and N has to be unique.
Therefore, there is no alternative to accepting that the most
expedient physical solution to the system requirements is that
μ occurs as a mix of two types of virtual muon for which the μ
values are adjacent odd integers. To satisfy (9) with N having
a value near 1949 there must be a mix of μ = 205 and μ = 207 in
the virtual muon population of the equilibrium space field.

Then, bringing nine such virtual muons together, four of one
polarity and five of opposite polarity, the latter being opposite
to the charge polarity of the m’ quantum, we have a basis for
initiating the process according to equation (1) and producing
a proton plus an electron or an antiproton plus a positron.

Considering the physical effects in the intense energy fields
accompanying proton creation, one could argue that it is unlikely
that the constants of physics as we now known them for
equilibrium space could be affected. Thus, supposing that N has
a certain value for equilibrium space, it would seem that N could
be marginally different at times when space is violently
disturbed. A spread of N values including those conducive to a
process involving nine virtual muons of the same μ value could
occur in such very energetic fields. This is the basis on which
this argument now proceeds.

The essential point, so far as the quantitative aspects of the
theory are concerned, is that the lower μ value of 205 sets the
threshold for the chaotic state permitting proton creation. The
equilibrium space condition sets in when N is the odd number just
below that of 9 times 205. It follows that N is uniquely
determined as 1843.

From equation (8) we then know the precise composition of the
mix of the μ=205 and μ = 207 virtual muons, because the mean
value of μ is shown to be 206.3329, meaning that there are very
nearly two μ=207 virtual muons for every such muon with μ=205.

THE PRECISE MASS OF THE PROTON

It is now possible to deduce the precise mass of the proton,
assuming that protons were created in a very energetic phase and
that, as a stable particle family, the two types of proton that
would be generated from the two virtual muon types would exchange
energy to adjust to equal mass values. Thus the mean value of
μ as given by equation (8) with N = 1843 is determining.

The proton/electron mass ratio is rigorously formulated by
combining equations (3) and (8) with N = 1843. Accordingly:

which gives:

To compare this with the measured value it is necessary to
make allowance for the fact that the theory assigns a finite
volume to the electron charge. If the quanta m’ have a charge
-e then the remainder of a unit cell of the space field must have
a distributed charge totalling +e, because the vacuum is
electrically neutral overall. It follows that the motion of an
electron is accompanied by the counter motion of a small amount
of positive charge given by:

Therefore:

In effect, the electron manifests an electrodynamic charge
effect that is not the universal charge quantum e but is larger
than this by 1.07 parts in ten million. It follows that a
measurement of the proton/electron mass ratio that assumes that
proton charge and electron charge are of equal magnitude will be
in error by amounts of this order.

This is a vital consideration because measurements now exceed
this degree of precision. The proton itself is known to exhibit
a finite form nucleated on more than one charge and attributed
to a quark structure. The dimensions of this finite form are
known to be of an order measured in fermi, which applies also to
the charge volume assigned above to the electron by use of the
radius a. Evidently the measured proton/electron mass ratio will
be affected by both the finite form of the electron charge and
that of the proton. Our theory estimates a few parts in ten
million as the expected discrepancy between the true mass ratio
as given by (l2) and the mass ratio expected from measurement.

On this basis it is interesting to take the latest CODATA
value for this measurement, namely 1836.152701(37) and compare
it with the true mass ratio given by equation (12). The
discrepancy is 2.09 parts in ten million, which allowing for the
uncertainty in the measurement is equivalent to a δe/e effect of
twice that given by equation (14).~

The empirical indication is that the proton itself involves
the finite volume of an electron-positron pair which
underestimates the effective charge of the proton by 2.l4 parts
in ten million in a measurement effectively referenced on a point
electron charge. Alternatively, the proton has on average a
single finite electron volume associated with it and, being
positive, this means a reduced effective charge by one of the
units under discussion, with the ratio measured against an
electron which has its electric charge effectively increased by
one such unit.

The calculated estimate of the measured proton/electron mass
ratio on this basis gives the value:

Before commenting further on this, it is appropriate to look
now at the other precision constant relating to the proton. This
is its magnetic moment measured in nuclear magnetons.

THE PROTON MAGNETIC MOMENT

Here it is not proposed to give the full analysis of the
formulation of the proton magnetic moment. It is based on
similar principles to those outlined above and a summary analysis
is presented in a reference of recent record [8]. Suffice it
here to note that, consistent with the above principles, the
proton is involved in a resonant interaction with the virtual
muon population. This resonance is governed by the μ=205
quantity for one third of any period of time and by the μ=207
quantity for two thirds of the time. However, the prior
reference did not develop that theme. It merely gave the two
criteria for either one such resonance or the other and noted
that some intermediate effect would probably apply in fact.

The new CODATA value of the measured quantity is, however, so
precise that it becomes necessary to take this argument to its
limits and check the theory at its extreme prediction.

The reason that very high precision is expected from the
technique adopted is the success of related theory for the
electron magnetic moment [9] which rivals quantum electrodynamics
(QED) by giving what is a precise theoretical correlation, as
judged against the CODATA value of the electron g-factor in
relation to the fine-structure constant.

The theory for the proton magnetic moment matches up to this
test and has no rival theory in QED or QCD that can warrant
comparison.

The formula deduced in reference [8] has the form:

where:

and θ is an angle of approximately 120 degrees determined in a
resonant wave field centred on the proton by interactions
involving the virtual-μ system. The angle θ complements two other angles φ and β summing to 180 degrees, where φ and β are related to the trigonometry of a triangle of sides R, 2r and r, so that:

and 2φ is determined as the number X of such double triangles
that represent a complete resonant wave pattern set up by making
Y complete revolutions around the proton. This really says that:

where φ is in degrees of angle.

The optimum resonant states are determined when the Y/X values
for each of the two μ-state resonances give very nearly the same
value of γ. These are found to be Y/X=615/5765=123/1153 for
the μ=205 state and 1035/9702=115/1078 for the μ=207 state.
Note that 615=3(205) and 1035=5(207). Also 123 and 205 have
the common prime factor 41, whereas 115 and 207 share the prime
factor 23. The proton characteristic enters via the approximation of 5765 to π times 1836, the proton/electron mass ratio. By working through the equations just presented to determine γ for either resonance one can then obtain, for the μ=205 resonance:

and, for the μ=207 resonance:

The credibility of the theory depends, in the main, upon the
degree of assurance that the Y/X ratios quoted are determined by
optimum physically-resonant processes. The crux of the analysis
is to find an X/Y ratio close to a value which makes θ equal to
120 degrees. The ratio should be about 9.42. However, for
resonance with the two muon states and near equality of the γ-values to assure minimum fluctuation during transitions, the reader can verify that the resonances specified are the only ones that can apply on optimum physical grounds.

Since the above argument was developed for reference [8], the
new CODATA measurements have appeared and γ is quoted to the
number of digits given in (20) and (21) with a standard deviation
of 63. It is therefore appropriate to extend the calculation by
first allowing for the mix of interacting muon states in the 2
to 1 ratio of μ=207 to μ=205. This gives a operative value
of:

Also, just as has been seen for the proton mass, this might
be the true gyromagnetic factor but there could be some aspect
of the measurement that causes the finite form of the electron
to get involved. The actual measurement of γ is indirect and is
a composite of three other measurements, one being the electron
g-factor. The measurement of the spin magnetic moment ratio of
electron and proton is involved, as is the connecting
measurement, namely that of the proton-electron mass ratio.

When we look at the empirical indications there is a case for
allowing γ to be incremented by one unit of the factor of 1.07
parts in ten million. This puts γ as deduced from the
theoretical value as:

which compares with a CODATA measured value of 2.792847386(63).

THE ELECTRON-POSITRON FIELD

The determination of γ of the proton is based on a direct
estimate of the proton-electron mass ratio. The double unit
electron charge volume affecting this term equally affects the
measurement of γ. The extreme accuracy of the computation of the
electron g-factor shows that no such charge volume term has any
net effect upon that measurement. This then requires the spin
magnetic moment ratio of electron to proton to be subject to a
single unit electron charge volume increment factor. This is a
valid theoretical proposition because the electron is a discrete
charge having the relevant volume and the proton component in
spin will be the core proton component in which the mass is
concentrated and which must have negligible charge volume.

The clear indications are, therefore, that the proton in
translational motion exhibits motion of a charge that has, on
average, displaced the unit volume of the classical Thomson
electron. For such a positive charge this reduces its effective
charge by one unit, whereas that of the electron is effectively
increased. Hence the double unit factor in the measured
proton-electron mass ratio.

The model of the proton which this conjures in the author’s
mind is one of a concentrated single unit of charge which in
translational motion depends upon association with an
electron-positron field. To obtain the single unit electron
charge volume, there then has to be a two state condition with
energy conservation as the electron-positron pair cycles between
mutual annihilation and recreation so as to spend effectively
half the time in existence. Possibly the exchanges could involve
the concentrated core of the proton and account for its
quark-like properties. In the latter case, so far as the
derivation of γ is concerned, this did not depend upon the
particular mass or charge of the charge core involved. The ratio
calculation is based on pure geometry of the cavity resonant
field system. Hence the same factor applies to all the quarks
in which the proton has a three-part quark composition.

This, of course, is speculation and is a matter for further
development. The extremely precise way in which the earlier
analysis has led to the derivation of a γ value in perfect accord
with the measured value is indicative that the fundamental
resonance contemplated has a sound basis. Certainly, there is
little scope left for developing the calculation further.

THE PHOTON DATA

Equation (5) needs to be justified to make the above analysis
self-contained. This comes from a derivation of an expression
involving Planck’s constant h in two separate ways based on the
dynamics of a synchronous electrical lattice. The theory is of
prior record [10] but see also [11]. The key steps in the
analysis involve deducing the two-dimensional harmonic oscillator
condition by which energy E stored is equal to angular momentum
added H times the natural angular frequency ω=2πmc²/h of the system. Then a 3x3x3 cubic lattice group of the m’ quanta is
taken as the basic spin unit. From considerations analogous to
hydrodynamics the effective inertial mass of a sphere moving in
a fluid having the same mass density is exactly half that of the
quantum m’ involved. The spin unit has an angular momentum H
when spinning at an angular speed of 2π(f/4), where f is the
perturbation frequency resulting from the four corners of the
cubic spin unit. The angular momentum of such a spin unit is 36d
(m’/2) times its angular speed. Therefore:

Replacing H by the expression involving E:

Now note that E is hf so that this equation gives:

A corresponding equation is deduced from the natural frequency
ω of the system by matching restoring force rate against the
centrifugal effects of m’. Allowing for dynamic balance so that
motion is about a radius x with displacement 2x:

Substituting 2πmc²/h for ω then gives:

Comparing (26) and (28):

which is the equation (5) that we sought to verify.

Putting this value of d in equation (26) and using the Thonson electron formula gives:

which, from our early derivation of m’/m via the N expression is
108π(8/1843)^1/6, or 137.0359148.

The latest CODATA value of this constant is 137.0359895(61)
and the extension of the theory to deduce this constant with
equally high precision is a task to be covered elsewhere. It can
be recorded here, however, that because the charges e associated
with m’ are at rest in the electrodynamic reference frame, in
that they define that frame, whereas the virtual muons defined
the inertial frame, there is no charge volume effect modifying
the value of the fine-structure constant deduced above. However,
when the constant is measured by the behaviour of electrons the
measured value overestimates e by two units of the 1.07 in ten
million term. This means that the discrepancy we must explain
is 7.59 parts in ten million. For reasons which will be
discussed elsewhere [12] and which have already been outlined in
a recent Hadronic Journal paper [13], the author sees this as
corresponding to a discrepancy matching a term (v/c)²/2. Thus
v, which represents cosmic motion through of the space field
relative to a preferred frame, has a mean value of about 37O km/s
subject to the uncertainty of the measurements involved
(equivalent to about 10 km/s).

One important point which will have occurred to the reader is
the question of reconciling the mixed μ=205 and μ=207 with
the actual measurement of the real muon/electron mass ratio.
This has been done and is already of record [14, 15]. The real
muon is a changing condition of a μ=205 and μ=207 system
which involves an electron-positron field. Field cavity
resonance gives a precise formula for the real muon/electron mass
ratio (ignoring the electron charge volume factor introduced in
this paper). The value obtained is 206.7683078, which compares
with the CODATA value of 206.768262(30). The discrepancy happens
to be 2.22 parts in ten million or within a few percent of two
of the units discussed in this paper. However, until even more
precise data is available or data available which might highlight
differences as between negative and positive muons, there is
little value in debating this subject. It is further noted that
the theory as applied to the real muon affords a precise estimate
of the muon lifetime [15] and also explains the muon g-factor
[16].

The above theory for the proton shows that there is very good
reason for believing that a virtual muon population in an
energetic space field accounts for the existence of the proton
and causes surplus energy to materialize in the matter form seen
in the hydrogen atom. This gives us a genuine reason for
understanding why physics needs the muon. Hitherto, physicists
have wondered where the muon fits into the general scheme. Note
also that the tau lepton has a role in the gravitational
interaction, as already reported [6].

This paper has presented the author’s theory in a new context,
showing that the proton creation process can, via the muon
interaction, determine the 1843 parameter which is crucial to the
determination of the m’/m quantity which fixes the precision
value of the fine-structure constant. The author’s early theory
[l] had approached this topic from the viewpoint of electric
potential thresholds which, though still relevant, may have
appeared arbitrary in the denial of negative potential sites for
m’. The present paper therefore breaks new ground by providing
a much more direct approach to the determination of m’/m, with
consequent simplication and containment of the argument to
physical events involving field structures associated with
particle concentrations. Proton creation can, therefore, be
explained without first developing what is, in effect, a new
aether theory. Evenso, a full understanding of photon behaviour,
especially in the light of the latest experimental developments
in quantum mechanics does appear to lead inevitably to a revival
of some modified form of aether theory with a preferred frame.

It is concluded that the author’s proton theory remains
viable in spite of the very stringent demands placed by the high
precision measurements reported in the new CODATA values.

REFERENCES

[1] H. Aspden and D. M. Eagles, Physics Letters, 4lA, 423
(1972).

[2] H. Aspden, Lett. Nuovo Cimento, 38, 423 (1983).

[3] CODATA Bulletin No. 63, November l986.

[4] H. Aspden, Hadronic J., 9, 129 (1986).

[5] H. Aspden, Hadronic J., 9, 137 (1986).

[6] H. Aspden, Hadronic J., 9, 153 (1986).

[7] H. Aspden, Lett. Nuovo Cimento, 41, 252 (1984).

[8] H. Aspden, Spec. Sc. Tech., 9, 315 (1986).

[9] H. Aspden, Physics Letters, 119, 109 (1986).

[10] H. Aspden, ‘Physics without Einstein’, (Sabberton,
Southampton, England) p. 120 (l969)

[11] H. Aspden, Lett. Nuovo Cimento, 40, 53 (1984).

[12] H. Aspden, ‘Tests of photon theory in terms of precision
measurement’, Conference on Problems in Quantum Physics;
Gdansk ’87, 21-25 September 1987. (Proceedings later published by World Scientific of Singapore, New Jersey and Hong Kong under title “Problems in quantum phgysics; Gdansk ’87”); see also shorter paper in “Sparawozdania Gdanskiego Towarzystwa Naukowego” (Transactions of Gdansk Scientific society) 14, 167-172 (1988).

[13] H. Aspden, Hadronic J., 10, 187 (1988)

[14] H. Aspden, Lett. Nuovo Cimento, 38, 342 (1983).

[15] H. Aspden, ‘Physics Unified’, (Sabberton, Southampton,
England), p. 146 (l980).

[16] H. Aspden, Lett. Nuovo Cimento, 39, 271 (1984).

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