News shocked the physicist society that neutrinos may move faster than light. Most physicists affirm that this is impossible, according to the well-established relativity. Some try to find a new theory other than relativity to explain the phenomena. In the following we show it may be explained within relativity, if the quantum theory is also taken into account

Qi-Ren Zhang

English

arXiv

Nature Precedings

QUANTIZATION MAKES RELATIVITY COMPATIBLE WITH SUPERLUMINAL
PHENOMENA
Qi-Ren Zhang
Department of Technical Physics, Peking University , Beijing,100871,China
A news[1, 2] shocked physicist society, neutrinos may
move fast than light. Most physicists affirm that it is
impossible according to the well established relativity.
Some people try to find new theory other than relativity
to explain the phenomena. In the following we would
show, it may be explained within relativity, if the quan-
tum theory is also taken into account.
The precise determination of the neutrino velocity v,
defined as the ratio of the precisely measured distance
from CERN to OPERA to the time of flight of neu-
trinos traveling through the Earths crust, was carried
out by the experiment group OPERA[1], and found
(v − c)/c = (2.37 ± 0.32(stat.)+0.34−0.24(sys.)) × 10−5. Since
neutrinos participate only in weak interaction, they are
always practically free. If they are classical, the above
definition of their velocity v is precise, and the measured
value shows their motion is superluminal. But this is im-
possible, since the mass-velocity relation for a classical
particle given by relativity shows the mass of the parti-
cle with a velocity v > c will be imaginary, and therefore
is un-physical[3, 4]. Suppose a particle initially has a ve-
locity v < c and then accelerated by certain mechanism,
when the velocity of the particle approaches light speed
c, the mass of the particle will approach infinity, and it
in turn prevents the velocity reaching and crossing the
limit value c. We see the contradiction appears when
one assumes the particle being classical. The way out
of this problem is to remember that neutrinos are not
classical particles, they are quanta of the neutrino field.
According to quantum theory[5], they do not move along
trajectories, their space distribution at a given time is
statistically described by a wave function, and its time
evolution is governed by the wave equation. The posi-
tion and the velocity of a neutrino could not be defined
simultaneously at any given time. Moreover, one is even
not able to trace a single neutrino, to identify a neutrino
at time t and position x being the one originally at time
t0 and position x0. The measured quantity v is not the
neutrino velocity. One has to reconsider what is the true
meaning of the observed superluminal phenomena.
The wave function describing the neutrino motion also
describes a wave, it is the neutrino wave. In this experi-
ment it is a wave packet produced at CERN and moves
to Gran Sasso Laboratory(LNGS). Since the dimension
of the wave packet is negligibly small comparing with the
distance between CERN and LNGS, one may consider it
as a point, its position and velocity are well defined at
each time. The velocity of the wave packet as a whole
is its group velocity vg = kc2/ω = pc2/E, in which
k and ω are wave vector and circular frequency of the
wave and p and E are momentum and energy of the
neutrino described by the wave respectively. There are
relativistic relations p = h¯k,E = h¯ω,ω =
√
k2 + κ2 c,
and E =
√
p2 + m2c2 c, with m and κ ≡ mc/h¯ to be the
mass and the reciprocal of Compton wavelength for the
neutrino respectively. The dimension of the wave packet
here is assumed to be quite large compare to its wave-
length so that its wave vector and therefore the circular
frequency are quite well defined. Momentum p and en-
ergy E are good quantum numbers of the neutrino. From
above relations we see that the group velocity of the neu-
trino wave packet is exactly the classical velocity of neu-
trinos described by it, and is less than the light speed c
for massive neutrinos. It seems nothing is superluminal.
One thing is crucial. Waves associated to massive par-
ticles are dispersive. Wave packets not only displace but
also expand. It means different points in the wave packet
move with different velocities. Especially, the wave front
may move with a velocity larger than the group velocity.
Could it be superluminal?
The neutrino wave Ψ is governed by Dirac equation
(γμ∂μ + κ)Ψ = 0 , (1)
in which γμ with μ = 1, 2, 3, 4, are covariant Dirac ma-
trices satisfying
γμγν + γνγμ = 2δμν . (2)
However, the wave function satisfying Dirac equation (1)
also satisfies Klein-Gordon equation
∂μ∂μΨ− κ2Ψ = 0 . (3)
On the other hand, for free particles, from a wave func-
tion Ψ0 with four components satisfying Klein-Gordon
equation (3) one may construct a solution of Dirac equa-
tion. Take Ψ1 ≡ (γμ∂μ − κ)Ψ0. If Ψ1 = 0, Ψ ≡ Ψ1
is a solution of the Dirac equation (1), because of the
relation (2). If Ψ1 = 0, one may construct Ψ ≡ γ5Ψ0
with γ5 ≡ γ1γ2γ3γ4, which is a solution of (1), because of
γ5γμ = −γμγ5 for μ = 1, 2, 3, 4. It means, if one is inter-
ested in the space distribution of neutrinos only, he may
ignore the complexity introduced by the spinor property
of neutrino wave function, and consider a one component
wave function Φ instead.
The motion of wave packet is one dimensional along
the straight line from CERN to LNGS. Our problem is
therefore one dimensional along this direction x. To con-
sider the wave packet expansion, let us use a coordinate
system moving with the packet, in which the center of the
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2wave packet does not move. This is the wave packet co-
ordinate system S0. Suppose initially the wave function
is
Φ(x, 0) =
{
cos(πxa ) if −0.5a ≤ x ≤ 0.5a
0 otherwise , (4)
which is centered at the origin of this coordinate system,
and is non-zero only in the interval −0.5a < x < 0.5a.
The statistical interpretation of wave function is some-
what complicated in relativistic quantum theory. But
the qualitative aspect is clear. Anyway, the probability
of finding a particle in a region with zero wave func-
tion is zero. Contrarily, in the region with nonzero wave
function, the probability of finding a particle is nonzero.
Therefore, equation (4) shows that at t = 0 one cannot
find any neutrino outside the region 0.5a ≤ x ≤ 0.5a.
A solution of Klein-Gordon equation (3) satisfying the
initial condition (4) is
Φ(x, t) =
∫ ∞
−∞
a cos(ka/2)
π2 − k2a2 cos(kx−
√
k2 + κ2 ct)dk ,(5)
at time t. It is not difficult to work out the integral in
this solution to a required precision. Take a = 10λc, it
means the initial wave packet length is 10 times Compton
wave length λc ≡ h¯/mc. The calculated wave function at
time t = 10a/c is shown in Fig.1. We see the wave front
        
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FIG. 1: Wave function in the case a = 10λc at time t = 10a/c
.
initially at x = 0.5a has moved to x = 10.5a in the time
interval (0, t), and therefore moves with light speed c. If
the wave front keeps sharp, so that the wave function at
time t keeps being zero in the region x ≥ 10.5a, and there-
fore the probability of finding a neutrino in this region is
zero, everything would seem normal. Any particle found
in the region x < 10.5a at time t may be thought to come
from somewhere in the initial region 0.5a ≤ x ≤ 0.5a
with a velocity not exceed the light velocity c. But this
is not the case. During propagation the wave front dif-
fuses. The wave function near x = 10.5a is shown in
Fig.2 with large scale. It is clearly non-zero in the region
x > 10.5a at time t = 10a/c, though it decays rapidly
           
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FIG. 2: Wave function in the vicinity of x = 10.5a at time
t = 10a/c .
with the increase of distance x − 10.5a. One therefore
may find neutrino in this region. When this happens, the
superluminal problem appears. Classically, this means a
particle initially in the region 0.5a ≤ x ≤ 0.5a moved to
a point with x > 10.5a in a time duration Δt = 10a/c,
the velocity exceeds light speed c. But this is not the
true picture. According to above relativistic quantum
theoretical analysis the neutrino does not move along a
trajectory, its velocity is not defined. The superlumi-
nal problem for the neutrino itself is not justified. What
is superluminal is the neutrino wave propagation. It is
solved from relativistic wave equation, therefore is a re-
sult of relativity, rather than its opposite. Superluminal
phenomena is therefore explained by relativity itself, if it
is quantized.
The superluminal effect is something like the barrier
penetration effect. Barrier is a space in which the poten-
tial energy of a particle is higher than its total energy.
Since the kinetic energy is non-negative, a particle cannot
penetrate into the barrier because of the energy conser-
vation in classical mechanics. But according to quantum
mechanics, the potential energy, the kinetic energy and
the total energy of a particle cannot be determined simul-
taneously, the above classical restriction does not work.
Since the wave function solved from Schro¨dinger equation
is non-zero somewhere in the barrier, but decays rapidly
with the increase of distance from the barrier surface,
the probability of finding particle in the barrier is not
zero. The barrier penetration is therefore possible. Of
course, correspondence principle requires that quantum
mechanics should approach classical mechanics when the
planck constant becomes unimportant. There is a pen-
etration depth, one can find the particle in barrier only
within this depth from the barrier surface. Superlumi-
nal effect may be viewed as a kind of penetration effect.
The quantum essence makes a particle be able to appear
in a zone which is forbidden by classical mechanics. In
supperluminal effect this zone is v > c in velocity space.
There is also a microscopic depth of penetration, one can
find particle in the zone only within this depth from the
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3surface v = c.
Let us return to the laboratory coordinate system S,
in which CERN and LNGS are at rest, but the wave
packet moves from CERN to LNGS with its group ve-
locity. For two events happened at a pair of space-time
points (x1, t1) and (x2, t2) one may define a velocity
v ≡ (x2 − x1)/(t2 − t1) , (6)
irrelevant to the picture of motion being classical or not.
For a given pair of events, the defined velocity is different
in different inertial coordinate systems. They are related
by relativistic theorem for velocity addition. In our case,
the velocity v0 defined in the system S0 and the velocity
v defined in the system S are related by
v =
v0 + vg
1 + v0vgc2
. (7)
Since for vg < c, dv/dv0 > 0, and for v0 = c, v = c, we see
for v0 > c one has v > c. It means a phenomenon appears
to be superluminal in system S0 must be superluminal
too in the system S. The above analysis also shows, when
the superluminal phenomenon happens nothing needs to
be radiated. It explains the superluminal phenomena
seen in the OPERA experiment, at least qualitatively.
A quantitative comparison between experimental obser-
vation and theoretical calculation needs more details of
experimental conditions.
Anyway, the relativity allows something moves fast
than light. Although it could not be a classical parti-
cle but may be waves. For fermions, including neutri-
nos, it must be quantum waves and no classical limits.
However, there are bosons, their quantum waves may ap-
proach classical limits, by boson condensations. Above
analysis shows these classical waves may propagate fast
than light. Therefore the superluminal communication is
possible. But, it is a custom to believe any superluminal
process violates causality. Consider a superluminal con-
nection between two space like points (x1, t1) and (x2, t2),
satisfying (x1−x2)2 > c2(t1− t2)2. Under Lorentz trans-
formations the sign of their time difference t1 − t2 may
be different in different coordinate systems. It means,
one sees t1 > t2 in a coordinate system, but t1 < t2 in
another. If one defines the earlier event to be the cause,
and the later event to be its result, he has to assert the
event 1(2) to be the cause(result) when he observes in one
coordinate system, but assert the same event to be the
result(cause) when he observes in another coordinate sys-
tem. The definite causal relation seems absent. However
this point has to be checked carefully. The above inter-
pretation on causality is observational. The cause should
appear earlier than the result. But there are always some
subjective elements in observations, for an example the
choice of the coordinate system. On the other hand, the
causality should be totally objective. The cause makes
the result in an object.To observe this relation, one had
better choose a coordinate system, in which the object is
at rest. The laboratory system S, in which laboratories
in CERN and LNGS are at rest, is suitable to observe the
creation, displacement and detection of neutrinos. The
wave packet system, in which the center of packet is at
rest, is suitable to observe the expansion of neutrino wave
packet. In these systems the causality is clear. Neutrinos
created in CERN move to LNGS and absorbed there by
detectors. No contradiction appears. One may of course
choose another coordinate system S′ moving from CERN
to LNGS with a velocity 0 < v′ < c satisfying v′v > c2,
in which v is the velocity observed in this experiment. In
this system S′, the observer would see neutrinos emerge
from detectors in LNGS, move to CERN and disappear
there. The causality seems violated. However, in our
view, this is only a mistake made by the observer. He
took a wrong coordinate system. It makes him see a
distorted scenery. The objective causality is therefore
hidden.
Beside superluminal effect, the structure of diffused
wave front may contain some other useful information,
including the mass of its quantum, the shape and dimen-
sion of the initial wave, and so on. A further research of
their possible applications is interesting. For example, it
may offer a new way for measuring the neutrino mass.
[1] T. Adam et. al. http : //arxiv.org/abs/1109.4897 (2011)
[2] E. S. Reich Nature 477, 520 (2011)
[3] P. G. Bergmann Introduction to the Theory of Relativity,
(Prentice-Hall Inc, Englewood Cliffs, N.J. 1942)
[4] W. Pauli, Theory of Relativity, (Pergamon Press, London,
1958)
[5] P. A. M. Dirac The Principle of Quantum Mechanics,
(Clarendon Press, Oxford, 1958)
Acknowledgements The work is supported by the
National Nature Science Foundation of China (Grant
No.10875003). Some high precision computation was car-
ried out at National Supercomputer Center in Tianjin,
and the calculations were performed on TianHe-1A. The
author also thanks Dr. X. Chen for his help in how to
run quadruple precision Fortran programs on PC.
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Quantization makes relativity compatible with superluminal phenomena

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Quantization makes relativity compatible with superluminal phenomena

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