Contents. 1 Introduction 2. 2 Unique and extraordinary properties 2. 3 Equations derived from the model of a particle in a box 4

Transcription

1 Contents 1 Introduction Unique and extraordinary properties 3 Equations derived from the model of a particle in a box Generation of the eight dimensions as dierent states of three and four dimensions toric - dimensions states toric 3- dimensions states toric 4- dimensions states Quaternions and Octonions Introduction to the seven dimensions Kaluza Klein compacted 6 5 Calculating mass of less massive Higgs boson 7 6 Several observations 9 7 conclusion 10 1

2 The God particle: The Higgs boson, extra dimensions and the particle in a box Author: A.Garcés Doz angel @gmail.com Abstract In this paper we show as the lightest Higgs boson, is directly linked to the existence of seven dimensions compacted Kaluza-Klein type, and four extended. The model of a particle in a box allows us to calculate, using the extra dimensions as entries in the well-known equations of this model, the mass of the lightest Higgs boson. This estimate coincides with complete accuracy with that obtained in our previous work, "God and His Creation: The Universe", using the well known quantum mechanical model of a particle in a spherically symmetric potential. Both load match and result in a mass for the lightest Higgs boson of 16.3 Gev Gev 1 Introduction The main feature that comes from getting the value of the mass of the lightest Higgs boson is the uniqueness of the solution. We refer to the uniqueness, as interconnected mathematical properties of some well known groups in unication theories, unique and extraordinary. Ultimately, these are derived from mathematical properties of a single solution that only have certain integers, also well known in the theories of unication. These integers are mainly: 1,, 3, 4, 5, 8, 7, 4 and 71. It includes a special way, the numbers of the Fibonacci series, especially those who are dividers of 40. In our previous work these integers of the Fibonacci series are well explained and used. The properties of octonions and quaternions, which allow the winding in seven dimensions, the toric states derived from the three-dimensional extended Unique and extraordinary properties These properties, interdependent, one could classify as follows: 1. Properties of specic groups. These principal groups are as follows: E8 group, which allows breaking the vacuum by groups E6 and SU(3. God and His Creation: the Universe

3 . The groups SU(11, SU(8, SU(7, SU(6, SU(5, SU(4, SU(3, SU( and U(1 3. Groups of rotations: SO(, SO(3, SO(4, SO(5, SO(6, SO(7, SO(8, SO(9, SO(13 and SO(6 ( 7 / ˆ a dim[so(n] = dim[so(8] = dim(g n=1 ( ˆ b F n/40 dim[su(f n ] = dim[su(10] ; (8! + dim[f 4 ]/48 = 163 = 40 dim[e6] + dim[u(1] 9 ˆ c 5 spins. 5! = dim[su(11] = dim[so(n] = 1/ζ( 3 = 1/ζ( 7 dim[su(4] = n=1 s ˆ d dim[so(6] = dim[su(4] s + 1 ˆ f -7 And -3 are the sum of the squares of the imaginary values of octonions and quaternions. ( 7( 3 = 1 = dim[so(7] = F 8 ; F 8 = eight Fibonacci number = 10D ; = 11D, 8 3 = dim[su(5] = dim[su(3] dim[su(] dim[u(1] = 4! ; = 11D, 7 4 = dim[so(8] 8 octonions, plus the 3 imaginary values of the quaternions. 4 quaternions, plus the 7 imaginary values of the octonions. dim[so(8] + dim[su(5] = ( / dim[f 4 ] = 5 = F n/40 F n ˆ g 6! = dim[su(7] dim[su(4] ; dim[su(7] + dim[su(4] = dim[su(8] ˆ h God and His Creation: the Universe (section, Maximum number of particles in the vacuum. 8! = [ ( 40 dim[su(13] ; 8! + Fn ] = dim[e8] 163 F n/40 ˆ 6 n n=0 F n/40 F n = dim[su(13] = 40 dim[e6] + dim[so(4] ˆ {F n /40} = {1, 1,, 3, 5, 8} 6 n n=0 F n/40 F n = dim[su(13] = 40 K(6D ; K(nD = Kissing Number n Dimensions ˆ I Special factorials. It is known that the amount of factorials who meet the property: n! + 1 = x SU(x = n! has a nite number of solutions, and probably only, the following: SU(5 = 4!, SU(11 = 5! ; SU(71 = 7! ; (K(6D = 7! + 5! + 4!, 7!/dim[SU(7] = 1 + F F n/40 n = ˆ 7!/dim[SO(7] = 40 ˆ K(4D = 7!(dim[SU(5] + dim[su(4] = dim[e6]/ = ˆ j Fundamental angles. Only the partition of the circle by the eight octonions allows a minimum probability of 1/, fully compatible with the Heisenberg uncertainty principle. Similarly, only with these eight octonions and four quaternions, you can generate the toric states windings. These states are generated by toric multistate three dimensions, plus the time dimension. Since, the equations resulting from the model of a particle in a box, are well known by all, will direct these equations for the relevant calculations. 3

4 3 Equations derived from the model of a particle in a box 1. For the particle in a box, the probability density for nding the particle at a given position depends upon its state, and { (/L sin is given by: (1 P (x = ψ(x (nπx/l ; 0 < x < L ; (1 P n (x = 0 ; otherwise. Where, L is the length of the box and n is the number of state. P(x is the probability. 3. The variance in the position = measure of the uncertainty in position of the particle. ( V ar(x = L 1 (1 1/n π 4. Variance in the momentum. (3 V ar(p = ( nπ/l 5. The uncertainties in position and momentum. (4 x p = n π 3 For dimensionless numbers to the lengths of the box, the ratios of lengths used with respect to the Planck length. l p = c/g N ; l p /l p = 1 (5 For the ground state of the vacuum, with n = 1, the only possible solution for 1/ probability of equation (1 is: (6 (/4 sin (π/4 = 1/ ; x = ; n = 1 ; (/4 sin (π/4 = sin (π/8. Where = l p /l p = x, is derived from the zero point energy of vacuum: E 0 = 1 ω ; ω = ω p = c/l p E 0 = 1 ω c / 1 ω p = c/ω p = l p (7 l p /l p = ; (l px + (l py + (l pz + (l pt = 16 ; 16 = 4l p /l p = 4. Really the result obtained by (6, represents the dimensionless ratio of two lengths about planck length, with a trigonometric factor which gives the probability. This probability of 1/ is completely equivalent to the minimum possible value of uncertainty, on the principle of Heisenberg. With the condition of knowing precisely the momentum of the particle Proposition 1. (8 (/4 sin (π/4 = sin (π/8 = 1/ = min( x p/ 3.1 Generation of the eight dimensions as dierent states of three and four dimensions The unique solution with probability 1/, equivalent to the minimum possible uncertainty (Heisenberg principle implies the division of the circle into eight parts. The octonions are what allow this partition, as well as the generation of toric states, using the multistate three and four dimensions, the winding two-dimensional (square of imaginary values of octonions and quaternions. God and His Creation: the Universe ( section, 3.3 The cosmological vacuum, pag

5 3.1.1 toric - dimensions states xy yx xy yx xy yx xz yz zx zy ; xz yz zx zy ; xz yz zx zy 8 3 = 4 states = 8D 3D xx zz xx yy yy zz 3.1. toric 3- dimensions states xyz xzy xyz xzy xyz xzy yzx zxy yxz zyx ; yzx zxy yxz zyx ; yzx zxy yxz zyx 8 3 = 4 states = 8D 3D xxy xxz yyx yyz zzx zzy toric 4- dimensions states P!(xyzt 8 3 = 4 states = 8D 3D Quaternions and Octonions Cuaternions Four and three dimensions ( two solutions : a i + j + k + 1 = 3 torus factor 4π surface ˆ 3 negative curvature, inner surface of the torus ˆ b i 4 + j 4 + k = 4 sphere 4D factor π / = (4π /8 volume Octonions Eight and seven dimensions ( two solutions : ˆ a seven dimensions: 7 e n = 7 negative curvature, inner surface of the torus e 0 ˆ b eight dimensions: ˆ ( 7 e 0 e n ( 7 e 0 e 4 n 7 e 0 e 4 n = 8 / = dim[so(8] ˆ dim[su(5] = 4 = 4! = dim[su(7] ( n + 4 n=1 n = ˆ 4 n=1 ˆ (8D 4D = 70 ( ˆ (8D 7 3D = dim[so(n] n=1 ˆ (10D 7D = 5! = dim[su(11] 5

6 ˆ = 7! ˆ (dim[su(5] + dim[su(10] = dim[su(6] ˆ dim[su(6] = dim[so(9] dim[so(6] ; dim[so(9] + dim[so(6] = dim[su(11]/ ˆ d = dim[su(11]/ d/4 ˆ {d/4} = {1,, 3, 4, 8, 6, 1, 4} ; {d 1 /4, d 1 = n, n 3} = {1,, 4, 8} ˆ {d /4} = {d/4} {d 1 /4} = {3, 6, 1, 4} ; ˆ d 1 = dim[so(6] d 1/4 d = dim[so(9] d /4 4 Introduction to the seven dimensions Kaluza Klein compacted Compactication in circles, for the ratios of the seven dimensions with respect to the Planck length, was based on the well known results. These calculations are presented in our previous work ( God and His Creation: the Universe, section 4 Calculation of dimensions 7D-spherical torus and its possible implications, pag 41, based on Black holes and the existence of extra dimensions, Rosemarie Aben, Milenna van Dijk, Nanne Louw. Proposition. The model of the torus of seven dimensions, assume that the smaller dimension of the torus corresponds to a black hole. With this assumption, the lengths of these seven toric dimensions, with respect to the Planck length, are: l P (D/l P = ( Gd+4 /c 3 1/(d+ ( G N /c 3 1/ = /[ ] 1 ((π d π d/ d+ /Γ(d/ ˆ l P (7D = ˆ The ratio of the length of a black hole d dimensional respect to the Planck length would be: ˆ R H (D = R H (d/l p = ˆ R H (7D = /[ ] 1 (4(π d π d/ d+1 /Γ(d/ (d + 1 Proposition 3. Will be added the third length, depending on the ne structure constant: l γ = α 1 4π = Proposition 4. Using equation (7, we have: + 4 l p (7D + R H (7D 6 < l p (7D + R H (7D = Applying the minimum value of uncertainty, 1/, according to the Heisenberg uncertainty principle, according to equation (8, the minimum value for the Higgs vacuum, its wavelength would be: 6

7 mass λ V H /min(λ H = 1/ ; where λ V H Is the wavelength of the Higgs vacuum. λ H Is the wavelength of Higgs boson lower ˆ m H m(v H(1/ m H 13 Gev 5 Calculating mass of less massive Higgs boson Proposition 5. Since, at higher wavelength corresponds to a mass smaller, then, applying equation (1, a model of particle in a box, using the major length of seven-dimensional torus as a measure of the box with a value for x, the minimum possible, to the vacuum zero point (x =, we have: { (/L sin (1 P (x = ψ(x (nπx/l ; 0 < x < L ; (1 P n (x = 0 ; otherwise P (, l P (7D = (/l P (7D sin (π/l P (7D = λ V H /λ H ˆ (9 λ V H /λ H = (/ sin (π/ = ˆ m H (Gev = (46. Gev = Gev Remark. 1/( sin[π/(4/π] P (, l P (7D m H /m e 7!(dim[SO(7] + dim[so(8] = 7 = m(v H(/ sin (π/ m e (7!5!4!/60/ cos 4 (π/60 [dim real(e8] + 3 = m H /m e God and His Creation: the Universe ( section 4.5, Particle in a spherically symmetric potential, the Higgs vacuum, the spins and the change of scale according to equation (11 Conjecture. sin θ 13 = (l γ R H (7D/l γ ; θ 13 = angle vacuum neutrino oscillation θ 13 = º ˆ sin θ 13 = Observation of Reactor Electron Antineutrino Disappearance in the RENO Experiment, arxiv: v1 [hep-ex] 3 Apr 01 The RENO experiment has observed the disappearance of reactor electron antineutrinos, consis- tent with neutrino oscillations, with a signicance of 6.3 standard deviations. Antineutrinos from six.8 GWth reactors at Yonggwang Nuclear Power Plant in Korea, are detected by two identical detec- tors located at 94 m and 1383 m, respectively, from the reactor array center. In the 9 day data- taking period of 11 August 011 to 6 March 01, the far (near detector observed 1710 ( electron antineutrino candidate events with a background fraction of 4.9% (.7%. A ratio of ob- served to expected number of antineutrinos in the far detector is 0.9±0.010(stat.±0.008(syst.. From the decit, we nd sin θ 13 = 0.103±0.013(stat.±0.011(syst. based on a rate-only analysis. 7

8 Proposition 6. P (, lγ = (/lγ sin (π/l γ = ˆ P (, lγm(v H cos θ 13 m H Remark. ( V ar(x = L 1 (1 1/n π ˆ n = 1 ; V ar(l γ, θ W (GUT = l γ sin θ W (GUT 1 (1 1/n π 1/ Remark. l p(7d/ cos 4 θ 13 π /[1 + 1/{(ln[m H /m e ] 11 ln [m p /m H ]}] Proposition 7. The uncertainties in position and momentum. (4 x p = n π 3 God and His Creation: the Universe ( section 4.5, Particle in a spherically symmetric potential, the Higgs vacuum, the spins and the change of scale according to equation (11 ˆ n = 1 ; ( x p/ = (π 6/1 ˆ V eff (r = V (r + s(s+1 m 0r V(r = 0, or solving the vacuum in the basis of spherical harmonics. V eff (r = s(s+1 m 0r Energy values by the angular momentum: V eff (r = s(s+1 m 0r Isolating the spins: s(s + 1 = Esm0r : E s m 0 r = (E t ; s(s + 1 = E t The Heisenberg uncertainty principle states that: x p s(s + 1 = ( x p x 5,p 5 = E s = s(s+1 m 0r Performing the sum for all spins, we have: s(s + 1 = s The states corresponding to antimatter, or the second solution of Dirac: x 5,p 5 ( x p s(s + 1 ( = ( ; And the total sum: x5,p5 s s(s + 1 = s ; whereby the above equation can be written as: ( x p ( x p + x 5,p 5 ( x p ( Since the Higgs vacuum decays to its minimum energy state with nonzero mass (and stable and electric charge: the electron. It must meet the equation (11 by changing the scale, so you can see the following equation: dx x x5,p5 = ( x p + x 5,p 5 ( x p ( = s s(s + 1 dx x = ˆ s(s + 1 = ln(m X 0 /m X 1 ; s ln(m H /m e = s s(s + 1 C/ the standard model, 1 = 4!/, the constant C / : C/ = 1/ ln(m Z /m e dx x + C = ln(m X 0 /m X 1 = ln(m H /m e = s s(s + 1 ; Using Tables VI and VII, the number of symmetry between fermions and bosons of So they nally holds: ln(m H /m e = s s(s + 1 1/ ln(m Z/m e = m H = m e exp[ s s(s + 1 1/ ln(m Z/m e ] m H (Gev = Gev 8

9 ˆ s s(s + 1/[(π 6/1] = ln[m p /(m H / sin(π/8] ; m p = P lanck mass ˆ ψ(, l p (7D, R H (7D = /l p (7D R H (7D sin[π/l p (7D] sin[π/r H (7D] = 1 + (cos[π/60]/ ln [m p /m H ] Remark. (dim[su(5]/ + [sin (π/l P (7D sin (π/r H (7D sin (π/l γ sin (π/π] ln(m H /m e ( 4 /l P (7D R H (7D l γ π[sin (π/l P (7D sin (π/r H (7D sin (π/l γ sin (π/π] 1/dim[G] ( [ ] ˆ V ar(l γ, n = 1 = l γ π ; V ar(l γ, n = 1 + 7! 7 = m H /m e 6 Several observations Remark. ( / ( / π 3 l p (7D [/l p (7D] sin [π/l p (7D] π 3 l p (7D m H /m(v H ˆ x p/ = 1 π 3 ; n = 1 ˆ (7! + 1! + sin θ W (GUT m H /m e ( 7 /R 7H (7D sin 7 [π/r H (7D] 1/dim[SU(7] ˆ ˆ [ ( / π ] π sin θ 13 sin 1 (π/60 ( 7 ˆ /[55 + (1 π tan θ 13 ] α 1 ; 55 = F 10 = dim[so(n] dim[u(1] ˆ 1/(10 sin θ 13 sin(4/π [ ˆ (/lγ sin (π/l γ ] cos(4/π ; sin(π/4 = cos(π/4 n=1 ˆ [sin (4/π + sin (π/4] + R 8 H (7D ( ˆ Muon mass: m µ /m e = exp V ar[r H (7D, n = 1] + 1 cos(π/5 ˆ V ar(l p (7D, n = 1 = l p (7D 1 ( 1 1 π ; V ar(l p (7D, n = 1/7 l p (7D R H (7D [ ( ]/ ( ˆ exp dim[su(5] l p(7d sin [π/l p (7D] 1 + cos θ W = m H /m e ˆ (l p(7d/1 l p(7d R H (7D 9

10 7 conclusion We think that the results leave little doubt of the existence of compactied dimensions, and its natural connection with the minimum value of the Higgs vacuum: the Higgs boson of mass less. 1 References [1] Observation of Reactor Electron Antineutrino Disappearance in the RENO Experiment, arxiv: v1 [hep-ex] 3 Apr 01 [] Black holes and the existence of extra dimensions, Rosemarie Aben, Milenna van Dijk, Nanne Louw [3] Particle Data Group, [4] God and His Creation: The Universe, A.Garcés Doz 1 I would like to thank to Sir Philip Gibbs for allowing me to expose these humble research 10