The Appaent Mystey of the Electon: Thee is no mystey! The electon can be physically undestood! Since the middle of the 9s physicists have been stuggling to undestand the electon. om expeiments it was concluded that the electon is a stuctueless, point-like object, whose entie mass is concentated in its extensionless cente. On the othe hand the electon displays popeties which nomally esult fom an extended stuctue, namely angula momentum (spin), a magnetic moment, and some sot of an intenal oscillation. In 98, when Paul Diac pesented the wave function of the electon (the Diac equation ), it became obvious that thee must be not only an intenal oscillation but also some intenal motion at the speed of light. When Ewin Schödinge found this as a consequence of the Diac equation, he gave the phenomenon the Geman name Zittebewegung, which means some sot of pooly defined oscillation. Subsequently physicists attibuted this intinsic contadiction between the electon s diffeent popeties to the common-sense view that the electon is subject to quantum mechanics and as such not accessible to the human imagination. Howeve, a solution does exist, whose undestanding elies on the application of the classical laws of physics and which is fee of contadictions: if the electon is assumed to be made up of two massless constituent paticles, then this assumption coectly pedicts the expeimental esults. It povides the coect elationship between the paametes of the electon: its mass, its constant angula momentum (spin), and its magnetic moment. The assumption made above has been genealised fo all elementay paticles as a physical model called the Basic Paticle Model.
Intoduction The way to undestand the electon classically is to accept that the electon has an extension, as assumed by the Basic Paticle Model, and is not point-like, as stated by quantum mechanics. Accoding to the Basic Paticle Model evey elementay paticle is made up of massless constituent paticles, which obit each othe at the speed of light c. Thei obital fequency is the de Boglie fequency (igue.). The mass of the entie paticle follows fom the fact that it has extension. Mass, m= Obiting at de Boglie fequency igue.: Stuctue of an elementay paticle Geneal Paticle Popeties. The Mass of the Electon The mass of the electon follows fom the geneal fact that evey extended object necessaily displays inetial behaviou. This is the mass mechanism of elementay paticles. Due to this mechanism, the mass m of an elementay paticle is geneally descibed by the equation m = h (.) c R whee h is the educed Planck constant, c the speed of light and R the adius of the paticle. The above fomula (.) is valid fo all elementay paticles if thei electic chage is neglected. Howeve, as the electon does have an electic chage, the esult given by (.) has to be efined by adding a small coection facto (pesent in the Landé facto). This coection is calculated in Appendix B. In the following we will efe to the geneal deduction of the paticle popeties, paticulaly the mass as given in the oigin of mass. In addition to that deduction, this site will in paticula deal with the popeties of the electon that ae caused by the pesence of an electic chage.. The Magnetic Moment.. The Basic Calculation The assumption of an extended electon helps us to undestand the magnetic moment using a classical view. Accoding to classical electodynamics, the magnetic moment μ of a cuent loop is: μ = i π R. (.) The cuent i flowing in a paticle having one elementay chage e moving at speed c is simply:
c i = e ν = e (.) π R with ν being the obital fequency within the paticle. Combining equations (.) and (.) we get: c e R μ =. (.4) If R is now inseted fom the mass fomula (.), the magnetic moment tuns out to be e μ = h. (.5) m o the electon, this is the 'Boh magneton'. Remak: Please note that this impotant equation has been deduced hee by classical means. Histoically, some attempts to deduce the magnetic moment of the electon by classical means wee made in the fist half of the th centuy. These attempted deductions exclusively used the electomagnetic enegy within the electon in ode to find a elationship between its magnetic moment and its mass. The esult of this calculation was wong by a facto of typically >. Late the coect elationship was povided by the Diac equation of the electon. om this success it was concluded that the electon can be coectly undestood and descibed only though quantum mechanics. - The success of the above classical deduction follows fom the assumption that the foce within the paticle is pimaily the stong foce. Eq. (.) can be used to calculate the size of the electon numeically. If the paametes m e = 9. kg fo the mass of the electon 8 c =.998 m / s h = 4. 546 m kg s ae inseted in eq. (.), the esult fo the electon is R el =.86 m. This is an unfamilia esult because the liteatue states that expeiments ule out any extension of this magnitude. Howeve this appaent conflict does not in fact exist, as is explained below. Remak: Ewin Schödinge analysed the Diac equation fo the electon in 9 []. Among othe esults, this analysis yielded an electon adius of appox. 4* - m, which agees with the above esult. If the mass of the electon is inseted in eq. (.5), it yields the coect magnetic moment of the Boh magneton 4 μ el = 9. 7 Amp m (.6)
.. The Coection fo the Electic Influence As mentioned above, the magnetic moment of the electon is not exactly equal to the Boh magneton but slightly geate, by a facto of appox. -. Conventional physics (QED) attibutes this diffeence to vacuum polaization effects aound the electon. Howeve, if the electic chage of the electon is taken into account, the adius of the electon is slightly lage than that caused by the stong foce. The eason fo this is pesented in detail in Appendix B. - When this coection is used in the above calculation, we aive at a simila coection to that pefomed by Julian Schwinge in 948 using vacuum polaization []. It may be conjectued that any use of vacuum polaization o vitual paticles is unnecessay if the Basic Paticle Model is used; the Landé facto can be deived without esoting to quantum mechanics.. The Angula Momentum (Spin) Equation (.) can be eaanged to give m R c = h (.7) The left side is the fomal definition of the angula momentum fo v = c. The ight side fulfils the expectation towads the spin of an elementay paticle in so fa as it is independent of any specific popeties of the paticle; so that its value is univesal. The facto on the ight side howeve is unsatisfactoy at fist glance as the measued spin coesponds to a facto of ½. It should, howeve, not come as a supise. Eq. (.7) would epesent the angula momentum of two objects obiting each othe, whee each caies half of the classical mass of an electon. The configuation of the Basic Paticle Model is, howeve, diffeent in that the two objects (basic paticles) do not have any classical mass. In addition, the field which enfoces the continuous motion of the basic paticles does not act on the basic paticles tangentially, as would be the case with classical inetial mass, but athe at an angle which lies somewhee between the tangential and the adial diection. The esulting value of the angula inetia is caused by the fact that the speed of the obiting basic paticles and the speed at which the binding field is popagated ae both c. The binding field aiving at a paticle comes fom a etaded position of the othe paticle and so fom a diection, which is not tangential, as illustated in figue.. The effective component fo the angula momentum eflects the facto of ½. igue.: The mechanism of angula momentum So the elation m R c = h (.8) becomes undestandable.
Conflicts with Views of Standad Physics. The "Zittebewegung" The esults descibed above fo the electon also confom - togethe with the othe popeties of the model - to the paametes esulting fom the Diac equation of the electon. Histoically Ewin Schödinge analysed the Diac equation and coined the Geman tem Zittebewegung (meaning tembling motion ) to descibe the oscillation within the electon, which is in fact cicula. This analysis has been giving ise to logical conflicts fo moe than 7 yeas:. Accoding to the Diac equation the electon oscillates at the speed of light c. On the othe hand the electon has a mass. By special elativity an object having mass can neve move at the speed of light. A single object which moves feely in space can neve oscillate, because this would mean a pemanent violation of the law of momentum. In the view of the Basic Paticle Model these discepancies disappea:. Not the electon as a whole is oscillating but its constituents, the basic paticles. These do not have any mass and can theefoe move at the speed of light. As thee ae two constituent paticles, thee is no violation of the law of momentum, if these obit each othe. Howeve, thee is an appaent discepancy between the theoetical pediction and the expeimental data fo the electon: expeiments seem to indicate that the electon does not have any futhe constituents and that its size is seveal odes of magnitude smalle than the value given above. - This discepancy vanishes if the expeiments ae examined fom the pespective of the Basic Paticle Model, fo the following easons. To investigate whethe the electon is made up of constituent paticles, it was bombaded with othe paticles (e.g. electons o potons) at high enegy. The electon did not beak apat, so it was concluded that it does not have constituent pats. Howeve, the Basic Paticle Model states that the constituents of the electon do not have a mass on thei own. So even if one of the constituent paticles is acceleated by any abitay amount, the othe constituent paticle can follow. Thee is not even any foce acting on that constituent. So an electon can neve beak up. igue.: Expeiment to beak up an electon. The Size of the Electon The investigation of the size of the electon is done by scatteing the electon off a poton o anothe electon, fo example, and investigating the angula distibution. If the Basic Paticle Model is assumed fo the electon, then only one of the constituents will be scatteed. Such
expeiments ae pefomed using highly elativistic electons. Due to the time dilation the constituents of the electon will move along a consideably stetched helix. One of the constituents will pass the extenal paticle as a nomal, point-sized paticle. The othe constituent will only paticipate indiectly in this scatteing pocess in that it causes the inetial behaviou of the oveall configuation (i.e. the electon). The scatteing will theefoe be simila to that of a paticle with a point-like size but with the mass of the electon. - om this esult it is conventionally deduced that the electon is point-sized. igue.: Expeiment to detemine the size of an electon om the above consideations it should be obvious that when the Basic Paticle Model is applied to the electon, it does not contadict the expeimental esults. 4 Summay The "Basic Paticle Model" povides a model fo the undestanding of the electon (as well as of othe leptons and also of quaks), which is based on classical physics and agees with expeiment. And this model povides the cause of elativity on a 'mechanistic' basis. Also the efinement of the magnetic moment epesented by the Landé facto can be explained classically. NOTE: The concept of the "Basic Paticle Model" of matte was fist pesented at the Sping Confeence of the Geman Physical Society (Deutsche Physikalische Gesellschaft) on 4 Mach in Desden by Albecht Giese. Comments ae welcome to note@ag-physics.de. Refeences: [] Ewin Schödinge, Übe die käftefeie Bewegung in de elativistischen Quantenmechanik ("On the fee movement in elativistic quantum mechanics"), Beline Be., pp. 48-48 (9). [] J. Schwinge, On Quantum-Electodynamics and the Magnetic Moment of the Electon, Phys.Rev.7, 46L (948) -7-5
Appendix A: The Mass of a Paticle in Geneal The mass equation in oigin of mass was deduced on the assumption that the intenal bond in an elementay paticle is based only on the stong foce. Howeve, in ode to undestand the popeties of the electon, the electic chage has to be taken into account too. Hee we will biefly epeat the calculation using only the stong foce, i.e. we will fist calculate the mass of a paticle without consideing its electic chage. In ode to follow the detailed deivation, please efe to oigin of mass. In the case of a pai of paticles bound puely by the stong inteaction, the Basic Paticle Model indicates that the binding foce is given by the following fomula descibing a multipole configuation: = S (A.) o fo the absolute magnitude Δ = S, (A.) whee is the foce caused by the field acting on the patne paticle; S is the stength of the field which acts hee to bind the two paticles, belonging to the stong foce; is the distance between the two paticles; and the equilibium distance (coesponding to Δ = ) at which the foce disappeas. o the coesponding shape of the potential, efe to fig. A.. igue A.: Potential of the bond between basic paticles making up an elementay paticle If we only conside small acceleations, whee small means that duing the time Δt (see below) the acceleation poduces a change in velocity Δv << c, then we can fo these changes eplace the actual distance in the denominato of eqs. (A.) and (A.) by the equilibium distance. If one paticle (e.g. B) is now acceleated at a constant ate in the diection of the bond, then fo a time Δ = c (A.) t
afte the stat of the acceleation the othe paticle (A) will due to the finite speed of light c - not eceive any infomation about this change in position, and it will be held in its position by the oiginal field. Then, afte the initial peiod of Δt, paticle A will also be acceleated constantly. om now, on the acceleation of paticle A will follow the acceleation of paticle B with this delay of Δt. This delay causes a constant displacement between the paticles, which esults in a constant foce between them, given by (A.). On the othe hand, the cuent field of paticle A will aive at paticle B afte a futhe delay of Δt. Assuming a constant acceleation a, duing this time Δt paticle B will move a distance Δ = a ( Δt ) = a Δt, (A.4) which is now added to the equilibium distance. Due to this distance Δ, the etading foce acting on paticle B in the diection of motion, dm, will have the value dm = S Δ, (A.5) and so, substituting fo Δ fom (A.4) dm = S a Δt. (A.6) Replacing Δt by using the speed of light c (A.) (i.e. with Δt= /c) yields dm = S a. c Accoding to Newton s definition, the quotient of and a is the inetial mass m. So, m dm = dm a whee dm is the foce in the diection of the acceleation and m dm the coesponding inetial mass. And theefoe: m dm = S. c Retuning to eq. (A.5) we note that the full foce dm = S Δ only acts if both the basic paticles ae positioned paallel to the diection of the applied foce. We will now aveage the foce ove one cicuit of the intenal motion of the paticle. This implies the assumption that the electon s mass is measued using an acceleation pependicula to the axis of the electon. The foce changes in the couse of the obital motion and is given by ci = cosϕ. (A.7) dm
whee ϕ is the angle between (a) the diection of the binding foce within the elementay paticle and (b) the diection of acceleation. The aveage foce is detemined by integation: π / π / π [ sinϕ] π = π < eff >= dm cosϕdϕ dϕ = dm. (A.8) dm π / So the aveaged, effective foce is: π / π eff = S Δ. (A.9) π Inseting Δ fom eq. (A.4) now yields eff 4 = S π a Δt (A.) and with the time delay Δ = c : (A.) t eff and theefoe: 4 = S a π c (A.) eff 4 m = = S, (A.) a π c o, in tems of the adius R = / of the electon: m = S. (A.4) π R c We now detemine S; (A.4) can be aanged as: mc R π S =. With the well known E = mc (also deived in Oigin of Mass) this gives π S = E R. (A.5) uthe we use: E = hν fom the assumption that the obital fequency is the de Boglie fequency. om the geomety of the cicula motion we have
ν = c πr fom which it follows that E c = h = h π R c R Substituting this in eq (A.5): π S = h c. (A.6) Appendix B: The Landé facto We shall now detemine the influence of the electic chage, paticulaly its influence on the magnetic moment of the electon. Without electic chage, only consideing the stong inteaction, the binding foce is given by eq. (A.): () = S whee is again the equivalence distance fo =. Now assuming two electic chages, e / fo each basic paticle, the binding foce changes to: () = S ( e ) + 4πε (B.) S e () =. (B.) S 6πε The electic chage causes a diffeent equilibium distance = z, fo which the following must apply: ( ) =. (B.) z So it follows that fo the new equilibium distance: z z = e S 6πε (B.4) z e =. (B.5) S 6πε To abbeviate this, we define
ζ = e S 6πε (B.6) fom which it follows that: z =. (B.7) ζ We can now wite eq. (B.) as S () = ζ S () = + ( ) ζ o with elation to z, using eq. (B.7) S ζ, z () = ( ) z = ζ. (B.8) () S ( ) Next we will detemine ζ. By definition: ζ = e S 6πε and we have (A.5) S π = c h and so e ζ = 6πε. (B.9) π hc We can now abbeviate this by using the definition of the fine stuctue constant: e α =. (B.) πε hc 4 So we get α ζ =. (B.) π
Remembeing that accoding to eq. (B.7) the classical equation fo the magnetic moment of a chage obiting at a velocity c and at a adius of R is given by: c e R μ = (B.) we will now have to eplace the initially detemined adius R by the coected R el (fo the new equilibium distance) to take into account the influence of the electic field: R R Rel =. (B.) ζ This eplacement gives us a coected magnetic moment of c e Rel c e R μco = =. (B.4) ζ o, in tems of the Boh magneton e μ = h Boh, m (efeing to (.4) and (.5)) we get α μco = μboh = μboh μboh + (B.5) ζ α π π the latte being the Julian Schwinge s esult (948). The coection facto in (B.5) has a numeical value of: α π 86. So the deviation of Schwinge s esult fom ou esult above is appox. -6. - It should be noted that the oiginal esult of Schwinge deviates fom the measued value also by appox. -6. Conclusion: The Basic Paticle Model is able to explain the Landé facto in a simila way to the new theoetical appoach pesented by J. Schwinge (a esult fo which he eceived the Nobel pice in 965).