The velocity of neutrinos
Fran De Aquino
Maranhao State University, Physics Department, S.Luis/MA, Brazil.
Copyright © 2011 by Fran De Aquino. All Rights Reserved
Recently, the OPERA neutrino experiment at the underground Gran Sasso Laboratory has
measured the velocity of neutrinos from the CERN CNGS beam over a baseline of about 730
km. The experiment shows that neutrinos can have superluminal velocities. This result could, in
principle, be taken as a clear violation of the Special Relativity. However, it will be show here
that neutrinos can actually travel at velocities faster than light speed, without violating Special
Relativity.
Key words: Neutrino mass, Neutrino interactions, Special relativity.
PACS: 14.60.Pq, 13.15.+g, 03.30.+p.
The mass of the electron neutrino ( )eν
is usually measured using the beta decay.
The continuous spectrum of beta decay
electrons terminates at a maximal energy,
which depends on the neutrino mass and on
the emitting nucleus type. Because of the
way that the neutrino mass affects the
electron energy spectrum, the measured
quantity is the square of the neutrino mass.
All recent measurements show that the
neutrino mass squared is negative [1].
However, the square root of a negative
number is an imaginary number. Thus, the
measurements suggest that the electron
neutrino has an imaginary mass. Assuming
that the neutrino has no real mass, and
considering that the imaginary momentum
has a real value, i.e., ( ) ( ) ( ) ( )realimimim SIL ≡= ω
and , we can infer that
the neutrino is an imaginary particle with a
measurable property; the square of its
imaginary mass.
( ) ( ) ( ) ( )realimimgim pVMp ≡=
The OPERA neutrino experiment [2]
at the underground Gran Sasso Laboratory
(LNGS) was designed to perform the first
detection of neutrino oscillations. Recently, it
was reported that the OPERA neutrino
experiment had discovered neutrinos with
velocities greater than the light speed [3].
The neutrinos in question appear to be
reaching the detector 60 nanoseconds faster
than light would take to cover the same
distance. That translates to a speed 0.002%
higher than (the speed
upper limit for real particles in the real
spacetime).
-1m.s8299,792,45=c
The quantization of velocity shows
that there is a speed upper limit, , for
imaginary particles in the real spacetime
(real Universe)
cci >
*. This means that Einstein's
speed limit ( )c not applies to imaginary
particles propagating in the real spacetime.
Theoretical predictions show that
[112 .10 −≈ smci 4]. Consequently, the
imaginary particles, such as the neutrinos,
can reaches velocities faster than light speed.
Therefore, in the case of imaginary particles,
we must replace c in the Lorentz
transformation by in order to
generalize the equations of Special relativity.
Thus, the imaginary kinetic energy of
imaginary particles, for example, is written
in the following form:
( ) icC iim =
( ) ( ) ( )( ) ( )
( )
( )
( ) ( )
( ) ( )20
2
2
2
0
2
2
2
0
1
1
1
1
1
1
imimi
i
imimi
im
im
imimiimiim
Cm
c
V
Cm
C
V
CmmK
⎟⎟
⎟⎟
⎟⎟
⎠
⎞
⎜⎜
⎜⎜
⎜⎜
⎝
⎛
−
−
=
=
⎟⎟
⎟⎟
⎟⎟
⎠
⎞
⎜⎜
⎜⎜
⎜⎜
⎝
⎛
−
−
=−=
where is the imaginary mass of the
particle at rest. The expression above shows
( )imim 0
* The speed upper limit for real particles in the imaginary
spacetime is , because the relativistic expression of the
mass shows that the velocity of real particles cannot be
larger than c in any space-time.
c
2
that the imaginary particle has a real velocity
. This means that imaginary particles
propagating in the real spacetime can be
detected. This is the case, for example, of the
neutrinos with observed in the OPERA
neutrino experiment.
V
cV >
Note that the imaginary kinetic energy
of the particle is what gives to the neutrino
its real velocity ( )( )VK im → . This solves
therefore, the problem of how the neutrino
propagates in the space.
In addition, we can conclude that in the
neutrino-electron reactions, mediated by the
Z particle, the neutrino does not enter as a
real mass but as a real angular momentum
(spin ½). The real mass of the neutrino is
null, but the real angular momentum and the
imaginary angular momentum of the neutrino
are not null. The real angular momentum of
the neutrino, , derives from its imaginary (realS )
angular momentum, according to the
following relation: ( ) ( ) ( ) ( ) ( )h1+=≡= ssSIL realimimim ω .
3
References
[1] Eidelman, S. et al. (Particle Data Group), Phys. Lett. B
592, 1 (2004) and (2005) partial update for edition 2006
(URL: http://pdg.lbl.gov).
[2] OPERA Collaboration, R. Acquafredda et al., JINST 4
(2009) P04018.
[3] Measurement of the neutrino velocity with the OPERA
detector in the CNGS beam (2011) [arXiv:1109.4897v1]
[4] De Aquino, F. (2010) Mathematical Foundations of the
Relativistic Theory of Quantum Gravity, Pacific Journal
of Science and Technology, 11 (1), pp. 173-232.
