1

Theory of the muon neutrino velocity

Zoran B.Todorvić

Faculty of Science, Pristina University, K.Mitrovica, Serbia

Abstract

One approach of the theory of the neutrino velocity has been proposed, which is based on the modified Lorentz

contraction factor which is well known in theory of relativity. The eigenvalues of the neutrino velocity have been

obtained from the matrix velocities states Using derived formulae the numerical calculations have been

performed and has been shown that obtained result is in well agreement with experimental data.

PACS numbers: 14.60.-z, 14.60.Lm, 14.60.St

I, INTRODUCTION

At the date 22.Sept.2011. the experimental data of the measuring the muon neutrino velocity

in OPERA laboratory at CERN Institute announced and obtained results show the possibility

of travelling neutrino faster than light 1 .But, the other seite of the physics is theoretical

physics and its task is to construct such mathematical model which need to give the

explanation of that natural phenomenon.We think that results performed in OPERA

laboratory are valuable and we will attempt to confirm it by our mathematical model of the

neutrino velocities in the next theorettical consideration.

II. THEORETICAL CONSIDERATION OF THE NEUTRINO VELOCITY

In theory of relativity there is well known the Lorentz factor

2

2

1

1

cv

1

where v is particle velocity and c is the speed of light.

We think in nature there are particles which can travel faster than speed of light. For such

particles, especially for the muon neutrino, we introduce a hypothesis that the modified

Lorentz contraction factor designated by is in the relation with the neutrino unchanged

velocity v , if we simple change the sign under the square root in the relation 1 , then we

obtain the following expression:

1

1

2

2

c

v

2

In order to construct the theory of the neutrino velocity, we will equalize the relation 2 by

the reciprocal value of the fine structure constant which written down

2

1

1

1

2

2

c

v 3

From relation 3 one finds the neutrino velocity in explicit form:

21 cv 4

II.1 THE COMPLEX FORM OF THE NEUTRINO VELOCITY

We introduce a supposition that the neutrino velocity may be shown as a complex number

consisted of the sum of a really and an imaginary part:

vivv ~Im~Re~ 5

Where the real part we equalize with the speed of light

cv ~Re 6

Who’s the best experimental value is smc 299792458 2

We think that the imaginary part v~Im produces electromagnetic field of the physical

vacuum and it may be expressed by the formula

ciciv 2exp~Im 7

where is the fine structure constant 37599240072973525.0

or 035999679.1371 2 .

Using 6 and 7 the resultant velocity of the neutrino in the complex form will be:

cicv ~ 8

Apparently, the absolute value of the complex number 8 is equal to:

21~

cV 9

III. THE MATRIX DESCRIPTION OF THE NEUTRINO VELOCITY

The state of the neutrino velocity we may describe by the following matrix:

01

10

ci

ciM

10

3

The characteristic equation of the matrix 10 one finds by the relation:

0det IM ,

Or, in the explicit form it reads:

0

1

1det

ci

ci 11

Calculating the determinant 11 , one obtains eigenvalues equation:

011 22 cii a11

whose roots are:

2

12 1 c b11

and the eigenvalues of the matrix 10 are: 2

2

2

1 1,1 cc

The eigenvectors for these eigenvalues are determined by using appropriate eigenvalue

equations which give the matrix of the eigenvectors P and its appropriate inverse matrix 1P ,

which read:

111

111

112

1

1111

11

1

ii

ii

iiP

iiiiP

So, using standard procedure for the matrix diagonalization we obtain the diagonal matrix of

the matrix 10 in the form:

2

2

1

10

01

1111

11

01

10

111

111

112

1

c

c

iiiiic

ic

ii

ii

iiPMP

12

Where diagonal elements are eigenvalues which represent measurable physical quantities and

in our case the neutrino velocities are equal to:

2

22

2

11

1

1

cv

cv 13

4

IV. NUMERICAL CALCULATION

First at all, we may calculate the neutrino velocity by formula 5 :

12 2998004401~ mscv

The relative difference of the neutrino velocity with respect to the light velocity we find by

the following relation:

51066.2

c

cv

Using the experimental data of the OPERA laboratory we may find the possible difference

between theoretical result and experimental one, and it amounts to:

1

exp

1

exp 853.299799892147.547 msvmsvvV

and relative difference is:

6

exp

10825.1

v

vr

V. THE CONCEQUENCE OF THE PROPOSED THEORY

The first consequence of the proposed theory may be in the possibility for determining the

limit velocity of the electron in nature. If we take into account the Lorentz factor ant it

equalize with the reciprocal amount of the fine structure constant we will get:

1

1

1

2

2

max

c

ve

e

From this relation we may obtain the possible limit in nature of the electron velocity:

2

max 1 cve

Thus, we may use this relation to form a connection between muon neutrino and the electron,

by their velocities:

2

2

1

1

emavv

5

The numerical calculation for the electron gives its possible limit velocity:

1

max 716.299784475 msve

The difference between both, of the experimental and theoretical values, and the electron limit

velocity we get, respectively:

1

exp 137.15417716.299784475853.299799892 msv

1264.15984716.299784475299800440 msvth

And appropriate relative differences are:

5

max

max

5

max

maxexp

exp

10325.5

10142.5

e

eth

e

e

v

vv

v

vv

Looking at the formulae for the neutrino and the electron we my construct the formula for the

photon velocity in complex form in the dependence with the fine structure constant , as:

sincosexp~ icicvc

and the modulo of cv~ is

constccvc 22 sincos~

what tell us that the photon velocity is independent of the fine structure constant and in the

vacuum it is equal to c .

VI. CONCLUSION

On the basis of the derived theory for the neutrino velocity and especially because of the

agreement of the theoretical data with the experimental ones, we may claim that there are

particles in nature which may travel faster than light. It means that experimental data which

are made at CERN Institute in its OPERA laboratory are possible and exact with the minor

difference with respect to the real value of the muon neutrino velocity, as a consequence of

the measuring method. So, we may conclude that the limit velocities defined by theory of

relativity are broken and muon neutrino travels faster than light.

6

References:

1 N.Agafanova, arxiv.org/pdf/1109.4997, 22.Sept. 2011 Measurement of the neutrino velocity with the

OPERA detector in the CNGS beams

2 Peter J.Mohr, Barry N.Taylor, and David B. Newell, Rev.Mod.Phys. 80, 633-730 (2008)

CODATA recommended values of the fundamental physical constants: 2006