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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. 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