The Mystery Behind the Fine Structure Constant Contracted Radius Ratio Divided by the Mass Ratio? APossibleAtomistInterpretation

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1 The Mystey Behind the Fine Stuctue Constant Contacted Radius Ratio Divided by the Mass Ratio? APossibleAtomistIntepetation Espen Gaade Haug Nowegian Univesity of Life Sciences June 10, 017 Abstact This pape examines vaious altenatives fo what the fine stuctue constant might epesent. In paticula, we look at an altenative whee the fine stuctue constant epesents the adius atio divided by the mass atio of the electon, vesus the poton as newly suggested by Koshy [5], but hee deived and intepeted based on Haug atomism (see [7]). This atio is emakably vey close to the fine stuctue constant, and it is a dimensionless numbe. We also examine othe altenatives such as the poton mass divided by the Higgs mass, which also appeas as a possible candidate fo what the fine stuctue constant might epesent. Key wods: Fine stuctue constant, atomism, electon, poton, adius atio, mass atio, Higgs paticle. 1 The Fine Stuctue Constant In 1916 Anold Sommefeld [1] intoduced the fine stuctue constant in elation to spectal lines. This constant, (014 CODATA ecommended values), plays a vital ole in moden physics. Some have suggested that the fine stuctue constant is elated to the atio of the electon s velocity in the fist cicula obit of the Boh model of the atom to the speed of light in vacuum. An altenative suggestion elates the constant to the Boh adius by a 0 =, whee is the educed Compton wavelength of the electon. Futhemoe, the classical electon adius is given by e = a 0 =. The fine stuctue constant is also elated to the chage of an electon to the Planck chage = e = qp q h c q p p 10 7 h c p. (1) 10 7 The Rydbegs constant is also a function of the fine stuctue constant. We will not comment much on the impotance o elevance of these suggested connections. Still, we ask, why does the fine stuctue constant have exactly this magical value? O, as stated by Richad Feynman: It has been a mystey eve since it was discoveed moe than fifty yeas ago, and all good theoetical physicists put this numbe up on thei wall and woy about it. Immediately you would like to know whee this numbe fo a coupling comes fom: is it elated to o pehaps to the base of natual logaithms? Nobody knows. It s one of the geatest damn mysteies of physics: a magic numbe that comes to us with no undestanding by man. Othes have suggested that atomic stuctues somehow ae linked to the golden atio, which is in tun elated to the fine stuctue atio (see [,, 4]). The golden angle is given by , which is 1 not fa fom one divided by the fine stuctue constant: In this pape, we will suggest othe possible connections to the fine stuctue constant. espenhaug@mac.com. Thanks to Richad Whitehead fo assisting with manuscipt editing. Thanks to Thijs van den Beg fo a useful discussion on the wilmott.com foum in elation to cicle geomety. 1

2 The Contacted Radius Ratio Divided by the Mass Ratio In a ecent woking pape, Koshy [5] inteestingly suggested that the fine stuctue constant could be linked to a adius atio divided by the mass atio. Hee, we build on that idea but in a quite di eent way than Koshy. We assume all matte and enegy consist of indivisible paticles always moving at the speed of light in the void, as assumed by Haug [7, 8]. Haug s newly intoduced atomism theoy gives all the same mathematical end esults as in Einstein s special elativity when using Einstein-Poincaè synchonized clocks. The theoy, moeove, gives uppe bounday conditions such as elativistic mass and how close the speed of mass can be to the speed of light. Each indivisible paticle in the electon moves back and foth at the speed of light ove the educed Compton wavelength of the electon. Only at collision is the electon tuly a mass. Each collision epesents the Planck mass that lasts fo one Planck second. This leads to a mass gap of m pt p times pe second, which gives the well-known electon est-mass (see also [9]). The indivisible paticle has a adius equal to the Planck length [10]. This means that the electon has a adius equal to its educed Compton wavelength when extended 1. Futhemoe, it has only a adius equal to the Planck length when contacted kg. The electon is the mass gap c The poton-electon mass atio is We could assume the mass of a poton consisted of electons (o altenatively 186). We fo a moment assume that each of these electons is a sphee with a adius equal to the Planck length. If we packed these electons into a sphee, how much volume would they take up? In 181, Gauss [11] poved that the most densely one could pack sphees amongst all possible lattice packings was given by p () In 1611, Johannes Keple suggested that this was the maximum possible density fo both egula and iegula aangements; this is known as the Keple conjectue. The Keple conjectue was supposedly poven in 014 by Hale [1]. Based on this, the adius of the lage sphee consisting of lage numbes of densely packed sphees with adius is appoximately given by (see the appendix) R N 6p 18. () This means that the poton s contacted adius is p R (4) Next, we will define the contacted adius atio as q R R = R p = p = 18, (5) which is the poton s contacted adius divided by the contacted adius of the electon. If we then divide this contacted adius atio with the poton s mass divided by the electon s mass, we get a numbe vey close to the fine stuctue constant: Since R (6) = me, we could altenatively have witten this in the following fom: = R (7) Still, this di es somewhat fom the fine stuctue constant ( CODATA 014); the numbe is too lage. Howeve, the appoximation used to calculate the adius of the sphee-packed electons making up the poton mass will actually slightly oveestimate the adius of the sphee-packed sphee. This is because the sphee-packed sphee s oute suface is not smooth but athe jagged. We could measue the aveage adius of the sphee-packed sphees by measuing the adius fom the inside adius and the outside adius and divide by two (see figue 1). Figue 1 illustates how we account fo the sphee-packed sphee s jagged suface, namely the diamete of the aveage of the blue line and the geen line. To find the geen line, we can use Pythagoas theoem to discen the distance, as shown in the lowe pat of figue 1. If one popely adjusts fo the jagged 1 And it is extended c times pe second.

3 Figue 1: The figue illustates the contacted adius of a sphee (hee we only see a coss-section of the sphee). As a sphee-packed sphee s suface must be jagged, a good appoximation fo the adius is found by taking half the aveage of the black-lined diamete and the geen-lined diamete. To find the geen-lined diamete, we need to use Pythagoas theoem as illustated in the subfigue below. The contacted poton adius can in the same way be seen as 186 sphee-packed sphees. The geen-lined diamete is equal to the black-lined diamete minus ( p 1) 0.54, whee is the adius of the small sphees, which, based on ecent developments in mathematical atomism, must be = l p, that is the Planck length. suface of the hypothetical sphee-packed sphee, it then seems that the adius of the lage sphee should be vey close to elative to the electon s adius (the contacted adius atio). q N q 6p 18 + N 6p 18 1+( p 1) q = N 6p p 18 + N = R R 6p =1.401 (8) And fom this, we can calculate the fine stuctue constant by dividing the contacted adius atio by the mass atio: = RR (9) This also means that the fine stuctue constant can be epesented by the contacted adius atio multiplied by the atio of the educed Compton wavelengths. The calculated value is extemely close compaed to c = , which is the fine stuctue constant given by CODATA 014. The di eence between the two numbes is close to c c =0.0161%. We do not claim that this is what the fine stuctue constant must epesent, but again it is inteesting that this is a dimensionless numbe. Altenatively, we could have used the classical electon adius e = 1 e 4 0 c =

4 The classical electon adius divided by the educed Compton wavelength of the poton is: Radius atio = and the fine stuctue constant is given by , (10) = h 1 c h 1 c = (11) In this case it is the electon s adius divided by the poton s adius, while in the above analysis it was the poton s contacted adius divided by the electon s contacted adius (accoding to the atomism model.). We believe that the classical electon adius likely does not exist in a physical sense; it is just an imaginay unit that has the fine stuctue constant embedded. On the othe hand, the contacted adius atio is something that possibly exists if the depth of eality is atomism. When it comes to the elationship between the classical electon adius and the adius of the poton o neuton and thei mass atio, Koshy has in a ecent piece [5] suggested a simila elationship as a possible intepetation of the fine stuctue constant. Howeve, we believe that the Haug atomist model has moe going fo it, vesus what is gained fom the Einstein special elativity mathematical esult when using Einstein-Poincaé synchonized clocks. This theoy also seems to make the most sense when the diamete of the indivisible paticle is the Planck length and its mass is the Planck mass. Moeove, a seies of infinity poblems ae elegantly emoved via the Haug model. On its own this esult could pehaps be seen as nothing moe than numeology. Yet the esult of the Haug model is tuly inteesting when seen in light of ecent developments in mathematical atomism. Haug s mathematical atomism model is vey simple and thus fa has been shown to yield the same mathematical esult as Einstein s special elativity theoy when using Einstein-Poincaé synchonized clocks. Poton Mass Divided by Higgs Mass The 014 CODATA-ecommended poton mass is kg. That is equal to about MeV/c. On 4 July 01, CMS announced the discovey of a peviously unknown boson with mass 15. ± 0.6 GeV/c,see[1]. Thee is still consideable uncetainty about the mass of the Higgs boson [14]. Fo a moment assume the Higgs mass is appoximately MeV/c. In this case the poton mass divided by the Higgs mass would be basically identical to the fine stuctue constant, and it would be a dimensionless constant: mp , (1) m H while a Higgs mass of 15. GeV would give a fine stuctue constant of mp (1) m H 1500 Still, this suggested value of the Higgs boson seems to be too fa away fom what it needs to be to be elated to the fine stuctue constant. Moeove, it is also in elation to electons whee the fine stuctue constant seems to be most impotant. 4 Summay The fine stuctue constant plays an impotant ole in moden physics. Yet it continues to be a mystey as to exactly what it epesents and why it has the mystical value it has. We have in this pape suggested two new possibilities fo what the fine stuctue constant could epesent. It could be elated to what we would call the contacted adius atio of the electon vesus the poton divided by the mass atio, an idea closely elated to the wok of Koshy [5]. The contacted adius atio is given fom sphee packing of Planck diamete sphees and adjusting fo this sphee-packed sphee s jagged suface. This new atio seems to be extemely close to the fine stuctue constant given by CODATA. Altenatively, we have suggested that the fine stuctue constant could be elated to the Higgs mass ove the poton mass, The pevious analysis also means we can wite the classical electon adius as e = R Hee using CODATA (divided by to get the educed fom): and

5 but this late suggestion seems to give a fine stuctue constant consideably o fom the one given by CODATA. We have in this pape not concluded what the fine stuctue constant tuly epesents. But we believe that the speculative idea that spins o fom atomism deseves futhe investigation. Radius of Sphees Constucted fom a Lage Numbe of Small Sphees Assuming small sphees with adius, the volume of such a sphee is V = 4. When we pack the Planck sphees as densely as possible, they will occupy a volume of 4 V t = l p = lpp. p The total volume is then NV t. This means we need a lage sphee with adius NV t = 4 R R = R = s s 4 NVt 4 Np R = N 6p 18. (14) It is impotant to be awae that this fomula will only be a good appoximation fo a vey lage numbe of sphees. In the case of a poton, we will assume it consists of 186 sphees, which is a numbe of sphees whee this fomula should be quite accuate. Refeences [1] A. Sommefeld. On the quantum theoy of spectal lines. Annals of Physics, 51, [] R. Heyovska and S. Naayan. Fine-stuctue constant, anomalous magnetic moment, elativity facto and the golden atio that divides the Boh adius [] M. A. Shebon. Wolfgang Pauli and the fine-stuctue constant. Jounal of Science, (): , 01. [4] R. Heyovska. Golden atio based fine stuctue constant and Rydbeg constant fo hydogen specta. Intenational Jounal of Sciences,,01. [5] J. P. Koshy. Fine stuctue constant a mystey esolved. vixa.og , 017. [6] E. G. Haug. Moden physics incomplete absud elativistic mass intepetation. and the simple solution that saves einstein s fomula [7] E. G. Haug. Unified Revolution, New Fundamental Physics. Oslo, E.G.H. Publishing, 014. [8] E. G. Haug. ThePplanck mass paticle finally discoveed! Good bye to the point paticle hypothesis! [9] E. G. Haug. The mass gap, kg, the planck constant and the gavity gap [10] M. Planck. The Theoy of Radiation. Dove 1959 tanslation, [11] C. F. Gauss. Bespechung des buchs von l.a. seebe: Intesuchungen be die eigenschaften de positiven tenäen quadatischen fomen usw. Gttingsche Gelehte Anzeigen.,

6 [1] T. Hales and et. al. Flyspeck announcingcompletion [1] C. collaboation. Obsevation of a new boson with a mass nea 15 GeV. Cms-Pas-Hig-1-00, 01. [14] G. Aad and et al. Combined measuement of the Higgs boson mass in pp collisions at p s =7and 8 TeV with the atlas and CMS expeiments. Physical Review Lettes, 114,015. 6