The masses of the first family of fermions and of the Higgs boson are equal to integer powers of 2 Nathalie Olivi-Tran To cite this version: Nathalie Olivi-Tran. The masses of the first family of fermions and of the Higgs boson are equal to integer powers of 2. QCD14, S.Narison, Jun 2014, MONTPELLIER, France. pp.272-275, 10.1016/j.nuclphysbps.2015.01.057. hal-01186623 HAL Id: hal-01186623 https://hal.archives-ouvertes.fr/hal-01186623 Submitted on 27 Aug 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. See discussions, stats, and author profiles for this publication at: http://www.researchgate.net/publication/273127325 The masses of the first family of fermions and of the Higgs boson are equal to integer powers of 2 ARTICLE · JANUARY 2015 DOI: 10.1016/j.nuclphysbps.2015.01.057 1 AUTHOR: Nathalie Olivi-Tran Université de Montpellier 77 PUBLICATIONS 174 CITATIONS SEE PROFILE Available from: Nathalie Olivi-Tran Retrieved on: 27 August 2015 The masses of the first family of fermions and of the Higgs boson are equal to integer powers of 2 a, Nathalie Olivi-Tran ∗ aLaboratoire Charles Coulomb, UMR CNRS 5221, cc. 074, Place Eug`ene Bataillon, 34095 - Montpellier Cedex 05, France. Laboratoire Charles Coulomb, Universit´eMontpellier 2, cc. 074, Place Eug`ene Bataillon, 34095 - Montpellier Cedex 05, France. Abstract We noticed that the first family of fermions and the Higgs boson have masses which are equal to integer powers of 2 in eV/c2 units (i.e. in the Planck length units). We made the hypothesis that, if spacetime is composed of small hypercubes of one Planck length edge, it exists elementary wavefunctions which are equal to √2 exp(ikxi) if it corresponds to a space dimension or equal to √2 exp(iωt) if it corresponds to a time dimension. By using the Dirac propagation equation and combinatorics we showed that the electron has a mass of 219eV/c2, the quark has a mass of 221eV/c2 and the electron neutrino has a mass of 2eV/c2. Finally, the Higgs boson is showed to have a mass of 237eV/c2 Keywords: theoretical masses of elementary particles, fourdimensional space, real space theory 1. Introduction the electron is fourdimensional (t, x, y, z) • I make the assumption that our threedimensional uni- the quark is three dimensional (t, x, y). verse is embedded in a four dimensional euclidean • space. Time is a function of the fourth dimension of this space [1–3]. If we apply this hypothesis to particle finally, the electronic neutrino is twodimensional • physics, we may say that elementary particles are four- (t, x) and x, y and z are equivalent. It is only dimensional, threedimensional and twodimensional. due to an equivalent propagation equation to the The coordinates (x, y, z, t) are not orthonormal. In- Fermi Dirac equation that when this neutrino prop- deed, time t evolves as log(r) where r is the comoving agates, there are infinitesimal rotations between the distance in cosmology [1]. characteristic coordinates (leading to flavor oscilla- Let us make the additional assumption that for each of tions). these four dimensions there are functions like exp(ikxi) with (xi = x, y, z, t) which vibrate (like in string theory). To obtain the remaining fermions (elementary parti- So, our reasoning is simply the description of how to cles), one has to modify the quantum number n (similar distribute these functions in the fourdimensional space. to the quantum number during oscillation in a hypercu- In the following, the reasoning applies in real space. bic box). Thus, the remaining fermions of the Standard A previous paper of mine (see [5]) predicts that the Model may be seen as excited states of the first fermion Higgs potential in real space is a hypercubic box in our family. fourdimensional space. So, we made a classification of elementary particles To obtain the first family of fermions from the Stan- with respect to their dimensions [4]. To verify these dard Model ( i.e. quark up, electron, electron neutrino) conclusions, we analyze here the masses of the first fam- one may say that: ily of fermions. One may remark that the masses of the first family of fermions (elementary particles) and of the Higgs boson ∗Speaker 2 Email address: [email protected] are integer powers of 2 if expressed in eV/c . For details (Nathalie Olivi-Tran) on these masses, see table I. Preprint submitted to Nuc. Phys. (Proc. Suppl.) September 3, 2014 Name measured mass 2n A hint is to write equation (6) with the use of one unique in eV/c2 matrix M containing all Dirac matrices. Moreover, one Electron 511.103 219 524.103 has to take into account the fact that this large matrix has ≈ ≈ Quark up 2.3.106 221 2.1.106 to contain all possible combinations of Dirac matrices . ≈ ≈ Electron neutrino 2 21 Hence, we modify the Dirac equation by putting more ≈ Higgs boson 126.109 237 137.109 constraints: all space dimensions have to be equivalent ≈ ≈ and time cannot go backwards. Table 1: Masses of the first family of fermions and of the Higgs boson iM∂µψ = mψ (7) In order to explain this property, let us analyze the Dirac equation which is the propagation equation of part Thus there are 3 possibilities of arranging γ1, γ2, γ3; all of the first family of fermions (except the electron neu- space dimensions are thus equivalent. Because time trino). cannot go backwards, we take into account half of the The Dirac equation may be written: matrix γ0 which is equal to the Pauli matrix σ0. The large matrix M containing all combinations of γµ matri- iγµ∂ ψ mψ = 0 (1) µ − ces over x, y and z is then: with ψ the wavefunction, m the mass of the fermion and γ 000000000 with the Dirac matrices: 1 0 γ2 00000000 0 = I2 0 0 0 γ3 0000000 γ , (2) 0 I2 ! 0 0 0 γ3 000000 − 0 0 0 0 γ1 0000 0 0 σ (8) γ1 = x , (3) 00000 γ2 0 0 0 0 σx 0 ! 000000 γ2 0 0 0 − 0000000 γ3 0 0 0 σ 2 = y 00000000 γ1 0 γ , (4) σy 0 ! 000000000 σ0 − 0 σ We see that, if we work with the coordinate vectors γ3 = z , (5) σz 0 ! ( √2 exp(ikxi) , √2 exp(iωt)), we have to multiply the − l.h.s of the modified Dirac equation (7) by the Jacobian where σν are the Pauli matrices. corresponding to these new coordinates. This Jacobian On a mathematical point of view, the Dirac matrices is equal to √2N where N is the dimension of the large are representative of infinitesimal rotations within the matrix M. wavefunction of a given elementary particle. The modified dirac equation (7) may be reduced to Let us make two hypotheses: first, the elementary the dirac equation (6). So, the eigenfunctions of the wave function corresponds to the eigenfunction of a modified Dirac equation may be found by making per- square potential with dimensions corresponding to the mutations of x, y and z in the eigenfunctions of the Dirac = Planck length (we take ~ 1 here). This first hypothesis equation: this means that x, y, z are equivalent. leads to elementary eigenfunctions which are equal to So the large matrix containing all combinations of γµ √2 exp(ikxi) if the eigenfunction corresponds to a space has a dimension of 4X3X3 + 2 = 38 :i.e. 38X38. The dimension or equal to √2 exp(iωt) if the eigenfunction eigenvalue of the Dirac equation is equal to 1, so the corresponds to a temporal dimension. eigenvalue m of the modified Dirac equation is equal to m = √238 = 219eV/c2. 2 2. Mass of the electron Finally, if one deals with eV/c units, the theoretical mass of the electron is equal to 219eV/c2 524keV ( the Following Olivi-Tran and Gottiniaux [4], the electron measured mass of the electron is 511keV≈). is fourdimensional. So, the wavefunction has four sub- components which may be seen also as wavefunctions. In order to get the mass of the electron, let us analyze 3. Mass of the quark up the Dirac equation which may be rewritten: Following Olivi-Tran and Gottiniaux [4], the quark µ iγ ∂µψ = mψ (6) up is threedimensional. So, the wavefunction has three 2 subcomponents which may be seen also as wavefunc- as the electron neutrino has only one spatial dimension, tions. In order to get the mass of the quark up, let us there is no ’rotations’ between different spatial dimen- analyze the Dirac equation which may be rewritten: sions (as for the electron and the quark). In the prop- agation equation for neutrino, there would be no Dirac µ = iγ ∂µψ mcψ (9) matrices and only two subeigenfunctions would come A hint is to write equation (9) with the use of one unique into account: one for space the other for time. matrix M containing all Dirac matrices. Moreover, one A theoretical propagation function for the neu- √ has to take into account the fact that this large matrix has trino would then use 2 subfunctions 2 exp(iωt) and √ to contain all possible combinations of Dirac matrices 2 exp(ikx).