Planck unit theory: Fine structure constant alpha and sqrt of Planck momentum

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1 Plank unit theory: Fine struture onstant alpha and sqrt of Plank momentum Malolm Maleod The primary onstants; G,, h, e, α, k B, m e... range in preision from low G (4-digits) to exat values (, µ 0 ). A major problem in onstruting a Plank unit theory is that the Plank units are limited to the preision of G and so to 4-digits. By postulating the sqrt of Plank momentum Q as the link between mass and harge, a Plank ampere A Q is onstruted as a geometrial shape; the volume of veloity/mass. From this Plank Ampere an then be derived µ 0 (permeability of vauum) whih in turn gives a formula for Plank length l p and for a magneti monopole (A.l p ). From the monopole an be formed an eletron whih is then used to solve the Rydberg onstant R. Consequently G, h, e, m e... an then be solved in terms of the 4 most aurate onstants, µ 0, Rydberg onstant (12 digit preision) and the fine struture onstant alpha α (10 digit preision). Plank temperature T P and so Boltzmanns onstant k B are funtions of the ampere and veloity A.. The eletron formula suggests a Plank unit theory whereby partiles are dimensionless formulas ditating the frequeny of Plank events via a periodi (analog) eletri wave-state to digital (integer) Plank-time-mass point-state osillation. This wave-partile duality (osillation) suggests a MUH Mathematial Universe Hypothesis where partiles and photons modulate magneti monopoles. The dimensions of our universe then redue to the 3 dimensions of motion; sqrt of Plank momentum, Plank time and. 1 Introdution J. Barrow et al noted in a Sientifi Amerian artile... Some things never hange. Physiists all them the onstants of nature. Suh quantities as the veloity of light,, Newton s onstant of gravitation, G, and the mass of the eletron, m e, are assumed to be the same at all plaes and times in the universe. They form the saffolding around whih the theories of physis are ereted, and they define the fabri of our universe. Physis has progressed by making ever more aurate measurements of their values. And yet, remarkably, no one has ever suessfully predited or explained any of the onstants. Physiists have no idea why they take the speial numerial values that they do. In SI units, is 299,792,458; G is 6.673e-11; and m e is e-31 -numbers that follow no disernible pattern. The only thread running through the values is that if many of them were even slightly different, omplex atomi strutures suh as living beings would not be possible. The desire to explain the onstants has been one of the driving fores behind efforts to develop a omplete unified desription of nature, or theory of everything. Physiists have hoped that suh a theory would show that eah of the onstants of nature ould have only one logially possible value. It would reveal an underlying order to the seeming arbitrariness of nature. [1] 3 Mass onstants Defining in terms of Plank momentum (instead of Plank mass), the mass onstants as Plank units beome; 4 Ampere A Q m P = 2.π.Q2 (2) G = l p. 3 2.π.Q 2 (3) h = 2.π.Q 2.2.π.l p (4) F p = E p l p t p = 2.l p (Proposed) Ampere A Q = veloity/mass [19] (5) = 2.π.Q2 t p (6) 2 Quintessene momentum Plank momentum (veloity*mass) = 2.π.Q 2 [19] kg.m Q = units = s (1) A Q = 8.3 α.q, units = m 2 m 3 kg.s 2. (kg.m/s) = ( kg.s )3 (7) Where: Plank Temperature = A Q. Elementary harge = A Q. t p Magneti monopole = A Q. l p Eletron = t p. (A Q. l p ) 3 Magneton = A Q. l 2 p Malolm Maleod. Alpha and sqrt of Plank momentum 1

2 5 Elementary harge e = 8.3 α.q.2.l p 3 e = A.s = A Q.t p = 16.l p. 2 α.q 3, units = m 2 kg.s. (kg.m/s) (8) E σ = t x.σ 3 e (16) nb. the onversion of Plank time t p, elementary harge e and speed of light to SI units 1s, 1C, 1m/s requires dimensionless numbers whih are numerially equivalent (t x, e x, x ). 6 Vauum permeability The ampere is that onstant urrent whih, if maintained in two straight parallel ondutors of infinite length, of negligible irular ross setion, and plaed 1 meter apart in vauum, would produe between these ondutors a fore equal to exatly newton per meter of length. gives: F eletri A 2 Q = 2.π.Q2.( α.q3 α.t p 8. 3 )2 = π.α.q8 64.l p. = 2 (9) µ 0 = π2.α.q 8 32.l p. 5 = 4.π 10 7 (10) ɛ 0 = 32.l p. 3 π 2.α.Q 8 (11) k e = π.α.q8 128.l p. 3 (12) Plank mass: Compton wavelength: Frequeny: t p = e 44 s t x e = 1s 44 e = e 19 C = 1C e x e 19 = m/s = 1m/s x m e = m P.E σ (17) λ e = 2.π.l p E σ (18) T e = 2.π.l p E σ. = t p = 1. t p (19) E σ σ 3 e t x Gravitation oupling onstant: 7 Plank length l p l p in terms of Q, α,. The magneti onstant µ 0 has a fixed value. From eqn.10 l p = π2.α.q µ 0. 5 (13) µ 0 = 4.π.10 7 N/A 2 l p = 57.π.α.Q 8 5 (14) para-positronium lifetime: ortho-positronium lifetime: α G = ( m P.E σ m P ) 2 = E 2 σ (20) t 1 = t 0 = α5. t p (21) σ 3 e t x 9.π.α 6 2.σ 3 e.(π 2 9).t p t x (22) 8 Eletron as magneti monopole m e in terms of m P, t p, α, e,. [19] Up-quark Down quark σ 2 σ 1 The ampere-meter is the SI unit for pole strength (the produt of harge and veloity) in a magnet (A.m = e.). A Magneti monopole [4] is a hypothetial partile that is a magnet with only 1 pole. A dimensionless geometrial formula for a magneti monopole σ e and eletron E σ is proposed. σ e = 2.π 2 3.α 2.e x. x = A.(m/s).s = A.m (15) 9 Bohr magneton µ = e.h.n = 8.m P.l 2 p. 3 = A Q.l 2 p 4.π.m e α.q 3.m e µ = A.m 2 E σ = A Q.l p. σ 3 e. t p t x (23) 2 9 Bohr magneton

3 10 Redued formulas Replaing l p with eqn.14, the natural onstants an be redued to Q, α, h = π 3.α.Q 10 5 (24) e = π.Q 5 3 (25) 13 Temperature When an Ampere travels at the speed of light we have Plank temperature; defining a Kelvin -Q K Q temperature sale with mksa units. T P = 8.4 π.α.q 3 = A Q. π ; units = K Q (32) T P = A.m/s The Rydberg onstant R π 4 m e = m P α 5.Q 7 x (26) Boltzmann s onstant k B k B = E p T P = π2.α.q ; units = J/K Q (33) R = m e.e 4.µ h 3 = 11 Quintessene momentum π α 8.Q 8.Q 7 x (27) The Rydberg onstant, with a 12-digit preision R = (55) [5] is the most aurate of the natural onstants. onsequently we may re-define Q in terms of this onstant. Q 15 = 12 von Klitzing onstant π α 8.R (28) π 2. 5 Q = ( ) α 8 15 (29).R The von Klitzing onstant redues to α and. R K = h e 2 = R K = (18) [17] π.α α = (96) 1. ( )/(1 + 5) = α = ( )/(.φ.µ 0 ) = As example: Eletron magneti moment = (25) (30) g = π.α 1 3.π 2.α π.α 1 3 π 2.α (31) 4 π.α5 2. James Gilson [18] g = α = Where 1K Q x = 1K. By onvention the water boiling point has been set: 100C at kpa (standard pressure). If we set standard temperature - pressure where 100C at kpa, then we an replae K with an mksa unit K Q [20]. 14 Numerial solutions CODATA 2010 values α = (44) [7] R = (55) [5] h = (29) e 34 [6] e = (35) e 19 [9] m e = (40) e 31 [10] G = (80) e 11 [12] µ 0 = 4.π/10 7 k B = (13) e 23 [13] Using α = gives h = e 34 e = e 19 m e = e 31 G = e 11 R = µ 0 = 4.π/10 7 All results agree with experimental CODATA values exept for G. However the same alulated values were used to solve Plank onstant, the eletron wavelength and eletron mass and so by extension the Rydberg onstant; and these 4 values all have the requisite preision. We may also note that the alulated G agrees with the Sandia National Laboratories G; Parks et al G = (21) e 11 [14] Refer to the website [19] for further details and a omplete list of alulated onstants. Malolm Maleod. Alpha and sqrt of Plank momentum 3

4 15 Summary The 3 units of motion; Plank momentum, Plank time and formed the mass domain. From the sqrt of Plank momentum Q, and alpha was formed an ampere. From this ampere, Plank time and was formed the harge domain whih inludes partiles and partile properties. As only the ampere formula was hypothesised, i.e.: the other formulas were all derived, and as the ampere has a simple ubi geometry, and as all results are within CODATA preision, I argue that the signifiane of this approah annot be easily dismissed as numerology but rather deserves futher analysis. A universe whose dimensions are motion also suggests a Plank unit theory and a Mathematial Universe Hypothesis MUH for partiles and photons redue to modulated units of momentum that ditate the frequeny of Plank events. Wave-partile duality then beomes a analog eletri wavestate (the partile frequeny) to digital Plank-mass pointstate osillation, both formulas, mass m2 and energy hv, are thereby funtions of partile frequeny, and so by altering this frequeny [19], we may freely adjust partile mass and wavelength with respet to the 2 universal onstants Plank time and. Relativity then redues to geometry [21]. 2.π 2 π 2.α 2.Q l 2 p. m l 4 p e α 4.Q 12 4.π 2.Q 4.4.π 2.l 2 p q p = E n = m e. 2 2.α 2.n 2 (37) q p = 4.π.ɛ π 32.l p. 3 π 2.α.Q 8 2.π.Q2.l p = α.e (38) r e = r e = 256.l2 p. 4 α 2.Q π e 2 4.π.ɛ 0.m e. 2 π 2.α.Q 8 32.l p. 3 1 m e. 2 = l p.m P α.m e (39) 16 Referene formulas These formulas are ross referened with ommon formulas α = 2.h µ 0.e π.Q 2 32.l p. 5 α 2.Q π.l p π 2.α.Q l 2 p. 4 = α = α (34) 1 µ0.ɛ 0 µ 0.ɛ 0 = π2.α.q 8 32.l p l p. 3 π 2.α.Q 8 = 1 2 = (35) R = m e.e 4.µ h 3 m e l 4 p. 8 α 4.Q 12 π 4.α 2.Q l 2 p R = 1 8.π 3.Q 6.8.π 3.l 3 p m e 4.π.l p.α 2.m P (36) E n = 2.π2.k 2 e.m e.e 4 h 2.n 2 m e = B2.r 2.e 2.V V p = E p e B 2.r 2.e 2 = π2.α 2.Q l 2 p. 4 l 2 1 E p 64.l 4 p. 4 p α 2.Q 6 2.π.Q 2. B 2.r 2.e 2 Maple ode: pi := : := : a := : R := : E p = m P (40) Q := (pi 2 5 /( a 8 R)) (1/15) : lp := (5 7 pi a Q 8 / 5 ) : mp := (2 pi Q 2 /) : tp := 2 lp/ : G = 2 lp/mp e = 16 lp 2 /(a Q 3 ) h = 2 pi Q 2 2 pi lp m e = mp tp (pi 2 Q 3 /(24 a lp 3 )) Referene formulas

5 Referenes 1. SiAm 06/05, P57: Constants, J Barrow, J Webb 2. hyperphysis.phy-astr.gsu.edu/hbase/eletri/elefie.html en.wikipedia.org/wiki/magneti-monopole Parks, H. V, Faller, J. E. Phys. Rev. Lett. (2010) 15. A. M. Jeffery, R. E. Elmquist, L. H. Lee, J. Q. Shields, and R. F. Dziuba, IEEE Trans. Instrum. Meas. 46, 264, (1997) full/nature10104.html 17. A. Tzalenhuk et al, Towards a quantum resistane standard based on epitaxial graphene, Nature Nanotehnology 5, (2010) private orrespondene (Marian Gheorghe) 21. M. J. Maleod, Plato s Code: the geometry of momentum (2013 edition) Malolm Maleod. Alpha and sqrt of Plank momentum 5