Download An Alternative View of the Fine Structure Constant and its Drift

An Alternative View of the Fine Structure Constant and its Drift:
Bringing the Flux Quanta into the Definition
Michael Edmund Tobar
Frequency Standards and Metrology Research Group, School of Physics, University of
Western Australia Crawley, 6009, WA, Australia
2
The usual definition of the fine structure constant in MKS units is written as α = e 4πε ch . In
0
actual fact the constant could be represented in many ways depending on which fundamental
constants and units are chosen. Because α is an electromagnetic constant, I choose to represent
it in terms of only electromagnetic constants. For example, in MKS units it may be represented
in terms of the quantum of electric charge e, the quantum of magnetic flux (fluxoid) φ0, and the
permittivity and permeability of free space, ε0 and µ 0. While in CGS units, it is given by the
ratio of only the electric and magnetic flux quanta. Introducing the concept of the electric and
magnetic flux of force associated with the quanta of electric and magnetic flux, a
representation of α is presented, which is independent of the unit system. I show that α may be
understood as the ratio of the quanta of electric and magnetic flux of force, while h c is
proportional to the product. Thus, a drift in α is shown to manifest due to a differential change
in the fraction of electric and magnetic flux of force associated with e and φ 0. Conversely a
drift in hc would occur from a common mode drift of the same variables. It is shown that it is
impossible to determine a measurement of the common mode component. This leads to the
necessary conclusion that it is impossible to determine which fundamental constant is varying
if α varies as well as the physical process.
I INTRODUCTION
Two of the most important fundamental constants in Quantum Electrodynamics (QED) are the
velocity of light, c, and Plank’s constant, h. The speed of light occurs naturally in Classical
Electrodynamics as the speed of electromagnetic radiation, which is invariant with respect to
any moving frame of reference. It is also regarded in Relativity as a limiting velocity attainable
for a nonzero rest mass particle. In contrast, Plank’s constant occurs naturally in Quantum
Mechanics as twice the minimum allowed quanta of the angular momentum of an electron. It is
ironic that these two constants that actually describe mechanical quantities (i.e. a maximum
velocity and a minimum angular momentum), are considered fundamental to QED. The fine
structure constant has also been labeled a fundamental constant in its own right
1
and maybe
for this reason has been labeled the fundamental constant of QED. It is also commonly referred
as the electromagnetic coupling constant as it is considered as a measure of the strength of the
electromagnetic field. It was first introduced by Sommerfield 2 to explain the fine structure in
hydrogen atoms, as the ratio of the electron velocity to the speed of light; the meaning of the
constant has now evolved to be a measure of the strength of the electromagnetic field. Recent
experimental evidence that the fine structure constant may be drifting
3, 4
and has triggered
much interest in theories that account for the drift in fundamental constants
5-10
. Also, it has
provided stimulus to laboratory tests, which aim to improve the precision of measurements of
the constancy of the fine structure constant11-13. In this paper I choose to represent the fine
structure constant in terms of only electromagnetic quantities. For the MKS representation, this
includes the quantum of electric charge e, the quantum of magnetic flux φ0, and the permittivity
and permeability of free space, ε0 and µ0. In light of this new definition the supposed drift is
analyzed.
To solve systems in Classical Electrodynamics it is common to represent the problem in terms
of the co-ordinates of charge or magnetic flux. They are considered dual variables, and one
may formulate a problem in terms of either of these quantities and get the same solution. This
fact may lead one to consider solving quantum systems in terms of magnetic flux. This has
been attempted in the past. Jehle spent a large part of his life developing a theory of the
electron and elementary particles based on quantized magnetic flux loops14. Also, Dirac
suggested the existence of the magnetic monopole to describe the charge of the electron15. Both
these theories rely on a physical relationship between flux and charge. In the case of the Jehle
model, he suggested that the electron was made of quantized flux loops, spinning at the
Zitterbewegung frequency. Since it is well known the magnetic flux is quantized, it seems
plausible that the quantized value of flux should play a role in the description of particle
physics. Jehle developed a series of papers, which attempted to do this based on the quantized
flux loop14,16-18. The question of the relation between the electric and magnetic properties is
fundamental to electrodynamics. One expects a relationship as moving charge produces
magnetic flux.
II ALTERNATIVE REPRESENTATION OF THE FINE STRUCTURE CONSTANT
MKS Unit Representation
At first glance it is not obvious that the fine structure constant is a measure of the strength of
the electromagnetic interaction. If one thinks about the force of the electromagnetic field it is
not dimensionless and depends on the separation and velocity of charges, as described by the
Lorentz force equation. Moreover electrons also have a magnetic dipole moment; thus the
magnetic component of the electron must be important as well as the electric. So why is it
regarded as such, surely a dimensionless number can not tell us about the strength of an
interaction if we think of the strength as a force? In fact by itself the fine structure constant (or
coupling constant) can not do this, as what is the strength referred to? To answer this question
we look at the static Coulomb force between two charges. The force, Fe, is given by;
e2
Fe =
4πε o r 2
(1)
Here e is the electric charge in Coulombs, r is the separation between the charges in meters,
and ε0 is the permittivity of free space in Farads per meter. To consider the strength between
two static charges in terms of fundamental constants, one must ignore the inverse square nature
2
of the force and just consider the constant of proportionality e 4πε . Because this
o
proportionality constant has the dimensions of energy times distance, the dimensionless
constant can be constructed by dividing by hc, which also has the same units. Thus, it is usual
to write the fine structure constant in MKS units as the following;
α=
e2
4πε o hc
(2)
Does it make sense to define the electromagnetic coupling constant as such? Similar coupling
constants are also written for the strong, weak and gravitational forces. If one analyses these
coupling constants it may be seen that they too are represented as a dimensionless constant by
comparing the constants of proportionality of the force with hc 1. Thus, when we discuss these
dimensionless constants relative to one another they represent comparative strengths of the
different forces. For example, the strong force coupling constant, is approximately one, the
electromagnetic is 1/137 the weak is 10-6, and gravity 10-39.
At first glance one may state that α must be proportional to the strength of the Coulomb Force
as it is proportional to e 2. But is it? This constant could actually be written in many ways
depending on which fundamental constants you wish to choose. The fine structure constant is
touted as an electromagnetic constant. If this is so, should it not incorporate the key quantum
and classical constants of electromagnetic origin, rather than a velocity and an angular
momentum? The fact that it incorporates the Coulomb term does indeed suggest the electric
nature of the constant, but what about the magnetic? Atomic systems are not statically charged
states. The electron also has a magnetic moment defined by its spin, so the magnetic nature is
clear. This is actually hidden in the hc term that we divided by. For example, all transitions
between electron orbit and spin states, when they interact with electromagnetic radiation, are
governed by the following equation
E ph =
hc
λ ph
(3)
where Eph is the energy of the absorbed or emitted photon, λph is the wavelength and hc is the
constant of proportionality. This simple equation describes the electromagnetic interaction
between orbiting or spinning electrons and the energy of the interacting photon. Maybe this
answers the question? The product of Plank’s constant and the speed of light seem to be more
fundamental to QED than the isolated variables themselves. Another pointer to this fact is the
relationship of hc to the Casimir force. For instance, two capacitor plates of area A separated
by a distance L has a neutral force due to vacuum fluctuations give by;
 π2A 
Fcas = hc
4
 240 L 
(4)
it is evident that the constant of proportionality is hc.
As mentioned previously, I choose to represent the fine structure constant in terms of the
fundamental electric and magnetic variables. To start the analysis I present the two key
relations between the fundamental constants for classical and quantum electromagnetism in
MKS units:
1
ε o µo
(5)
φ e
h= o
2π
(6)
c=
Here µ0 is the permeability of free space and φ0 is the quantum of flux, or fluxoid. Equation (5)
and (6) basically give a simple description of the relation between electric and magnetic
quantities. Equation (5) represents the classical description, where ε0 and µ 0 defines the
electromagnetic properties of vacuum and also determine the speed of light. Likewise φ0 and e
define the quanta of electric charge and magnetic flux, which also determine Plank’s constant.
Now given that the fine structure constant is an electromagnetic constant it would be
instructive to substitute (5) and (6) into (2) to actually express it in terms of the independent
electric and magnetic constants, φ0, e, ε0 and µ0. If we do this (2) becomes:
α=
1 e
2 φo
µo
εo
(7)
This representation of the fine structure constant is now expressed as ratios of the classical and
quantum electromagnetic constants, with c and h eliminated. In actual fact
µo
ε o is the
impedance of free space and e φ is the Quantum Hall conductance. For this arrangement the
o
fine structure constant is proportional to the impedance of free space and the quantum Hall
conductance. The former can be considered the classical part of the constant and the latter the
quantum part.
CGS Unit Representation
In CGS units the quanta of electric charge and magnetic flux are given by;
qcgs =
e
; φ0 cgs =
4πε o
4π
φ0
µ0
(8)
Substituting (8) into (7) gives the equivalent CGS representation;
α = 2π
qcgs
φ0 cgs
(9)
Thus, in CGS units the Fine Structure Constant is proportional to the ratio of the elementary
charge to flux quanta, and has no classical part.
It is clear if one tries to interpret the Fine Structure Constant in terms of dimensional constants
there is an obvious problem, as the interpretation depends on the units and the dimensional
constants. In the next sub-section I introduce a representation, which is independent of the unit
representation.
Representation Independent of Units
In classical electrodynamics it is common to solve systems using either flux or charge as the
Lagrangian variable. The descriptions are considered dual and end up with the same solution.
If charge and angular momentum are quantized, it makes sense that flux is quantized as well.
This was predicted by
19,20
and experimentally discovered by21. It is intriguing to consider, as
for classical physics, a dual description of quantum systems using quantized flux, as was
discussed earlier in terms of the Jehle model of the electron14. The Jehle model of the electron
succeeds in modeling the electron as a bundle of rotating quantized flux loops of φ0.
Consequently, an alternative description of (2) and (7) in terms of the static Coulomb electric
force as well as the static magnetic loop force will be developed here. To achieve this it would
be appropriate to write the fine structure in terms of the Coulomb force in (1) and the magnetic
force that holds the north and south pole of a quantized magnetic loop together. The magnetic
force in MKS units is given by;
Fφ =
φo2
2 µo A
(10)
where A is the effective cross section area of the magnetic loop as shown in figure 1.
N
S
φo
φo
S
N
A
Figure 1. Schematic of a magnetic flux loop with effective cross section area A.
The problem with the magnetic and Coulomb force equations given by (1) and (10) is that they
are not a constant, (1) follows the inverse square law and (10) depends on the cross section
area over the path of the magnetic loop. To solve this problem we may consider the flux of
electric and magnetic force, as the force field multiplied by the cross sectional area
perpendicular to the field lines. The flux of electric force due to the point charge at distance r
from the electron is simply given by the Coulomb force in (1) times the area of the enclosed
sphere (4πr2). Consequently the electric and magnetic flux of force in MKS units is given by
Φe =
e2
εo
and Φφ =
φo2
2 µo
(11)
Here, Φe and Φφ may be considered as the flux of electric and magnetic force generated by a
static quantized charge, e, and a static quantized magnetic loop of quantized magnetic flux, φ0,
respectively.
Considering equation (5), (6), (7) and (11) one can then show the following relations;
α=
hc =
1
2 2
Φe
Φφ
(12)
2
Φ e Φφ
2π
(13)
Equations (12) and (13) describe a more general situation involving the most important
dimensionless constant of QED (12), and the most important dimensioned constant of QED
(13), in terms of the fundamental quanta of electric and magnetic flux of force.
It is possible to show that the representations given by (12) and (13) are independent of the unit
system at hand. For example in CGS units (11) and (6) will become;
2
Φ e = 4πqcgs
, Φφ =
φ02cgs
8π
and h =
φ0 cgs qcgs
2πc
(14)
Combining (14) with (9) to solve for α and hc in terms of Φe and Φφ , the same solutions given
by (12) and (13) are obtained. The author has also verified the relation for the other common
unit representations. The point here is that Φe and Φφ have the same dimensions, and the ratio
and product become independent of unit representation.
III DESCRIBING THE VARIATION OF THE FINE STRUCTURE CONSTANT IN
TERMS OF ELECTRIC AND MAGNETIC QUANTITIES
It is interesting to consider the meaning of (12) and (13) if the Fine Structure Constant in fact
drifts. So how should the fine structure constant be considered with respect to the definition
above? If (12) and (13) are combined to eliminate Φφ, then without considering the magnetic
energy the original definition remains, i.e. the fine structure constant is a measure of the flux of
Coulomb force with respect to hc. However, considering (12) and (13) together to include the
magnetic force term, some more general conclusions can be made.
It is apparent that the fine structure constant is proportional to the square root of the ratio of the
strength of the electric and magnetic quanta of force. In contrast, h c is proportional to the
product. It is not necessary to differentiate between unit representation as (12) and (13) are
independent of this. Thus, by implicitly differentiating (13) and (14) above we obtain;
∆α 1  ∆Φ e ∆Φφ 
= 
−

2  Φe
α
Φφ 
∆(hc) 1  ∆Φ e ∆Φφ 
= 
+

2  Φe
hc
Φφ 
(14)
Thus, we have succeeded in representing drift in α and hc in terms of the change of the same
electric and magnetic quantities. Here a drift in α is due to a differential variation in the
strength of the electric and magnetic force, while a drift in hc is due to a common mode drift.
The current question: Is it e or hc that drifts?
Recently there has been much discussion on whether or not e or h c drifts if a drifts7.10,22-25.
Whatever the process, that causes the drift, if the total electromagnetic energy of an electron
was conserved then ∆Φ e Φ e = − ∆Φφ Φφ . For this case a drift in α may be written as;
∆Φφ
∆α ∆Φ e
∆e
∆(hc)
and
=
=−
=2
=0
α
Φe
Φφ
e
hc
(15)
Physically this means that the electric energy would be converted to magnetic energy or vice
versa. If this occurred one would then get a drift in α independent of hc. Likewise, if only a
common mode drift occurred, hc would drift independent of α. In this case another energy
process would need to be involved. For example, if mass was converted to electromagnetic
energy. This would also result in a corresponding drift in the Casimir force, as it is clearly
proportional to h c (see eq. 4). For α and hc drift to be related, the differential and common
mode components of (14) must be correlated. This would occur if, for example, mass was
converted to only magnetic or electric energy but not both, such that either
∆Φ e Φ e = 0 or ∆Φφ Φφ = 0 . In this case, from (14) it is evident that;
∆Φφ
∆α ∆(hc)
∆α
∆ ( hc )
∆Φ e
=
if
= 0, and
=−
if
=0
α
Φφ
α
Φe
hc
hc
(16)
Measurement of drift in hc has been attempted previously, for a review on these measurements
see 1. This translates to the question of whether or not it makes sense to define drift of the
common mode term given in (14). Bekenstein26 showed these attempts generated null results as
the constancy was actually implied in the analysis. Maybe other ways to determine whether or
not this quantity drifts can be determined, i.e., by directly measuring the Casimir force as a
function of time. However, if one attempts such a measurement, the Casimir force must be
measured with respect to a reference. The reference must be another force independent of the
Casimir Force. For example, if a readout of the force is constructed based on an electrical or
magnetic circuit then the reference would be the Coulomb or Magnetic force respectively. This
would then lead to a calibrated measurement of hc/ Φ e or hc/ Φφ . From (12) and (13) it is easy
to show;
α=
1 Φ e π hc
=
4π hc
2 Φφ
(17)
Thus, an attempt to measure a drift in h c would lead to a measurement of drift in α. If one
chose another force for the read out, like strong, weak or gravitational, then this could be
interpreted as a measurement of drift of one of the other corresponding coupling constant. This
is the basic dilemma, the measurement process does not allow us to determine the common
mode component of (14). This is the reason it is impossible to determine, which fundamental
constant is drifting if α drifts. This also means it is impossible to determine the physical
process, which causes the drift, as to determine this unequivocally both common mode and
differential components of (14) must be known.
IV CONCLUSION
An alternative representation of the fine structure constant has been presented, which only
includes electromagnetic quantities. This has enabled the introduction of the quantized
magnetic loop flux of force into the definition along with the Coulomb flux of force. Thus, I
was able to show that the fine structure constant is proportional to the square root of the ratio
of the quantized electric and magnetic flux of force independent of unit representation.
Conversely, hc was shown to be proportional to the product. This definition has enabled any
drift in the fine structure constant to be described as a differential change in the energy
associated with the quanta of electric charge and the energy associated with a quantum of loop
flux. In contrast, a drift in hc (and hence Casimir Force) can be described as a common mode
change in the same variables. From this definition it was shown that it is not possible to
determine which fundamental constant drifts (and hence the physical process), if a drift in α
occurs because it is impossible to measure the drift in the common mode component.
ACKNOWLEDGEMENTS
The author has had many interesting discussions with Dr. Ian Mc Arthur and Dr. Paul Abbott
regarding this work. This work was funded by the Australian Research Council.
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