Wave Packet for Massless Fermions and its Implication to the Superluminal Velocity Statistics of Neutrino

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by  Kelin Wang

Wave Packet for Massless Fermions and its Implication to the Superluminal Velocity
Statistics of Neutrino
Kelin Wang
Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
Zexian Cao
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
Non-dispersivewavepacketformassless fermions isformulated onthebasisof squeezedcoherent
statesthatareputinaformofcommoneigenfunctionfortheHamiltonianandthehelicityoperator,
starting from the Dirac equation. The wave packet thus constructed is demonstrated to propagate
at a constant velocity as that of light. This explicit expression of wave packet for the massless
fermionscanfacilitatetheoreticalanalysisofproblemswhereawavepacketisofformalsignificance.
2
1 Furthermore, extensive wave packet may result in a superluminal velocity statistics if determined
0 fromthetime-of-flightmeasurement,asrecentlydoneonmuonneutrinos,whenathresholdparticle
2 flux or energy transfer, which is eventually referred to the propagation of wave packet, to invoke a
detection event is assumed.
n
a PACSnumbers: 03. 65. Pm;05. 30. Fk;06. 30. Gv;14. 60. Lm
J

6
In fundamental quantum theory, a free particle is rep- we know that light comes in indivisible unit of photons,

] resented by a plane wave ei(kx ωt) which is the eigen- our description of relationship between light and photon
h −
p functionofthe momentumoperator. However,the plane is incomplete since, for example, the light can be po-
- wave cannot be normalized, thus it does not fit our pic- larized but we don’t know how each photon can keep a
p
tureofparticleswhichatleastshouldmaketheirpresence record of the light’s polarity. This would take us away
e
h only in a limited region of microscopic scale [1]. Packet from the traditional ideal of elementary particles, and
[ waveshavebeenformulatedforlightcominginindivisible more significance may be attributed to the concept of

unitofphotons,akindofmasslessboson,andformassive wave pack. Put this fact aside, we notice that the pho-
1
v fermions such as electron, based on a dispersion relation tons now can be brought into light pulses of enormously
1 derived from the electromagnetic wave equation for the differing durations. It is reasonably conceivable that the
4 formerandfromtheSchro¨dingerequationorDiracequa- wave packets describing light can in principle assume a
3
tion for the latter, respectively. Remarkably, however, macroscopic extension, noting that a temporal duration
1
there is no mathematical formulation of wave packet for of 1.0 ns corresponds to a spatial extension of 30 cm for
.
1
massless fermions, to the best knowledge of the authors. a light pulse subject to none of the decoherence mecha-
0
With the absence of massless fermions, the picture of nisms.
2
particles as a wave packet is obviously incomplete.
1 A quite natural question may be posed about mass-
:
v With the introduction of wavepacket,various particle less fermions: how couldthey be representedas a packet
i behaviorsareoftendiscussedasamatteroffactinterms wave, and what the extensive wave packet may imply
X
of this concept. Yet in many circumstances, for instance for the investigationof behaviors of such particles? This
r
a when talking about the collisionor detection by a detec- may be a question of some relevance,say,with regardto
tor,the particlesaresimplythoughtofasapointobject. the recent report of superluminal velocity measurement
Thehandlingofparticlesasapointobjectisavalidcon- on muon neutrinos [2]. Neutrino is a massless fermion
venience when the size of the particles is much smaller in the standardmodel. Some experimentalobservations,
than the scale of the experimentalsetup, or to be nearer e.g., neutrino oscillation, point to the probability that
tothereality,lessthanthesizeofatomsoratomicclusters neutrinos may have mass. It is believed that the upper
that constitute the responsive agent of a detector. This limit for heavier neutrinos is at the order of magnitude
is not a challenging requirement for electrons or protons of 2 eV, and the lightest neutrino could in principle be
since these massive fermions have a very short Compton massless [3]. We shall always presuppose m = 0 with-
ν
wavelength and they just behave as distinct identities out distinguish the flavors of the neutrinos in the cur-
though formally they can also be represented as a wave rent work. Now, a massless neutrino, following relativis-
packet. For such massivefermions,the concepts ofpoint tic quantummechanics,shouldpropagateatthe velocity
particle and wave packet do not cause much confusion, of light in vacuum, c, and the detection of its velocity,
and the output of measurement for one single particle if the neutrino behaves as a point particle, is expected
strikingamacroscopicdetectorshouldgiveasharpvalue to give a sharp value of c within the precision of the
withinthe precisionofthedetectionmethod. Inthecase method. However,Adam andcoworkersrecently showed
of massless bosons, things become complicated. Though that the velocity of muon neutrino with the average en-
2
ergy of 17 GeV, by time-of-flight measurement, displays which correspond to the eigenvalue +1 and -1, respec-
surprisinglya skeweddistributionbiasedatthe >cside. tively. The eigenstates in Eq.(5) are, however, not
This has been even interpreted as the proof of superlu- normalizable since the integral arising from the term
minalexistence. Adetailedinspectionofthedataforthe eβb†b† 0 is unlimited for β 1. Consequently, such
| i | | ≥ 2
experimentperformedattheaverageof28.1GeVfurther eigenstatesofauniquehelicityisnotproperforthephys-
indicates that the data taking only eventsat energy>20 ical representation of a massless fermon though it was
GeV give to a more severe superluminal velocity statis- claimedthatthestate ofamasslessneutrinocouldnever
tics [2]. Could it be possible that the massless neutrinos be a superposition of two different helicities and the ex-
propagateas a trainof waveata groupvelocity of c, yet perimentgaveadirectindicationthatthehelicityofneu-
the behaviorof the wavepacketcould leadto anenergy- trinoisnegative[5]. Inordertoconstructanormalizable
dependent velocity statistics deviating from c? physical state in the form of wave packet for a massless
In this article, we report the formulation of wave fermion, the mixing of helicity seems unavoidable. Let’s
packet, starting from the Dirac equation, for massless take the wave packet given below as the initial state of
fermions onthe basis of the squeezedcoherentstates [4], the massless fermion
and the wave packet thus constructed is non-dispersive
abnydaspsruompianggataestharteshthoeldvpealortciictyle oflfuxligohrte.neWrgey tsreaentshfeart, |t = 0i=Z e−γ(α−α0)2 ee−−1122bb††bb††++aabb††||00ii!dα, (6)
which to the end is referred to the probability distribu-
tion described by the wave packet, is desired to invoke where the weight factor in the integral is Gaus-
a detection event for the particle, the measured velocity sian. It can be shown that such an initial state
statistics for neutrinos could find a reasonable explana- is normalizable, as can be proved that t = 0t =
h |
tion without breaking relativity. 0 = 4π e4γ2−γ1α20, in deriving this we have used
Foramasslessfermion,theHamiltonianisgiveninthe thie enc√lo4sγu−r1e relation for the coherent state Φ =
form eφb† 0 , i.e., d(ReΦ)d(ImΦ)e (ReΦ2+ImΦ2) Φ Φ| i =
| i π − | ih |
H =c−→σ ·−→P. (1) 1, and the intRegrals ∞ e−βx2sin(bx)dx = 0 and
Where, c, the velocity of light, is a parameter that con- ∞ e−βx2cos(bx)dx= R−πβ∞e4bβ2.
nectsspaceandtime. Thisdoesn’tnecessarilyimplythat −∞
theparticlehasadefinitevelocityasthatoflight,bearing R Thetemporalevolutioqnofthestaterepresentedbythe
wavepacketinEq.(6)canbederivedinastraightforward
in mind the Dirac equation for the massive electron.
way. At any moment t, it has
For simplicity, without loss of generality, we con-
sdiidreercttiohne,stihtueantitohnetHhaamt itthoenipaanrt(i1c)leisprroewpargitatteensaaslong x- |ti=Z e−γ(α−α0)2e−iE(α)t ee−−1221bb††bb††++aabb††||00ii!dα. (7)
H =cσ pˆ . (2)
x x
Thus the expectation value for the position operator
Now,let’srepresentthe positionoperatorxˆ andmomen-
at any moment t is
tumpˆintermsofthecreationandannihilationoperators
xˆ= i∆(b b†),pˆ= 1 (b†+b). (3) x(t)= ht|xˆ|ti = ht|√i∆2(b−b†)|ti, (8)
√2 − √2∆ t=0t=0 t=0t=0
h | i h | i
With ∆ being a parameter in the dimension of length,
which leads to the result x(t)=√2∆ At, implying that
then the Hamitonian in Eq.(2) turns into the velocity of the packet wave is a·constant of c, as
H =Aσx(b+b†), (4) v = x(t) =√2∆ A=c.
t ·
Where A= c . Now let’s calculate the width of the wave packet at
√2∆ any moment t, Γ = ( xˆ(t)2 xˆ(t) 2)1/2. Note that
proItjeicstioobnvoiofutshethsaptinthofethheelipcaitryt,icηle=on−→tσo·th−→Pe/dpir,eic.tei.o,nthoef ht|xˆ2|ti = −∆22ht|(b −hb†)2|iti−, iht thius has hxˆ(t)2i =
momentum, is a conserved quantity since it commutates th=t|0x2t=|ti0 =2∆2(A2t2+ γ1)− ∆22, consequently
h | i
with the Hamiltonian in Eq.(2). The eigenstates for the
helicity operator are given as follows Γ(t)=[2∆2(A2t2+1) ∆2 (√2∆At)2]1/2 =∆(2 1)1/2.
γ − 2 − γ−2
+ = (e−12b†b†+b†)|0i , (9)
| i (e−21b†b†+b†)0 ! Based on the arguments above one concludes that the
| i
wave packet given in Eq.(6) is non-dispersive, and prop-
= (e−21b†b†−b†)|0i . (5) agatesatavelocityofc,thereforeitcanbeusedtorepre-
|−i (e−21b†b†−b†)0 ! sentamasslessfermionorabeamofmasslessfermions.It
| i
3
is noteworthy to mention that 4 γ 1/4 as required ory. Ifthevelocityofsuchawavepacketisdeterminedby
≥ ≥
by the normalization condition and the positiveness of measuringthetimeofflight,thensolongastimetagging
the wave packet width. is performed with respect to precisely the same position
From Eq.(9) we see that the extension of this non- onthewavepacket,saythecenterofthewavepacketbut
dispersivewavepacketisproportionaltothe freeparam- not necessarily, it will result in the same value, c, even
eters ∆, consequently it can in principle be made suffi- though it has a finite extension. For the muon neutrinos
cientlyextensive. Thisisanalogoustothecaseoflightfor in Ref.[2], it requires a time tagging referring to exactly
which the wave packet can be of an enormously variable the same position of the wave packet at the source and
lengths, since the Compton wavelength of photon is in- atthedetector,aquitechallengingtaskbecausethetime
finitely long and the photons are almost decoupled from taggingwasperformedviadifferentprocesses. The finite
each other. For massless particles such as photons and extension of the wave packet may cause a peculiar sys-
neutrinos, their behaviors are expected to display more tematicdeviationofmeasuredneutrinovelocityfromthe
undulatorycharacteristics. Incontrast,massiveparticles velocityoflight,followingtheCopenhageninterpretation
such as electrons and neutrons have a limited Compton of the wave function.
wavelength, their wave packet can have only a limited The event that a particle initiates as being registered,
extension-in general cases they are just treated as a dis- whether on the surface of a solid detector directly or via
tinct particle instead of a train of wave. an intermediate nuclear reaction, is essentially quantum
Now it is probably appropriate to discuss the statis- mechanical, i.e., in a statistical sense. Suppose that it
tics of velocity measurement for the massless fermions, needs a threshold particle flux, or energy transfer, to in-
e.g., for neutrino. When the neutrinos are represented voke a detection event, e.g., to generate taus to be mea-
bythewavepacketformulatedinEq.(6),doesthatimply suredtoindicatethearrivaloftheneutrinos[2],thenthis
the possibility of measuring a superluminal velocity un- implies that a threshold portion of the wave packet has
der certain circumstances? Or, what kind of an energy- to enter the detection regionor the device [9, 10]. As we
dependent velocity measurement can be anticipated for mentionedabove,thewavepacketforneutrinoscanhave
neutrinos when taken as a non-dispersive wave packet a macroscopic extension as a pulse of light, the super-
with a constant propagation velocity of c? luminal velocity statistics for neutrino can find a natu-
Thedifferenceofthemeasuredvelocitiesformuonneu- ral and straightforward explanation under the threshold
trinos relative to that of light has been given in a statis- probability assumption. In case a larger threshold por-
ticalnotion,andtheoriginaldataofvelocityscatterran- tionofthewavepacket,implyingalargerparticlefluxor
domly aroundthe velocityoflight. Forthe muonneutri- energy transfer, is required to invoke a detection event
noswithanaverageenergyof17GeV,Adamandcowork- that the time tagging at the detector might be retarded
ers, based on the measurement of time-of-flight through relativetothetimetaggingatthesource,thevelocitydis-
a base line of 730 km long, came to the conclusion that tribution will be biased by < c results; and if a smaller
(v c)/c=(2.48 0.28(stat.) 0.30(sys.)) 10 5[2]. A threshold probability of the wave packetis required that
−
− ± ± ×
measurement performed for an average neutrino energy the time-of flight measurement may lead to an early ar-
of 28.1 GeV shows only negligible difference from the rival time, then the velocity distribution may be biased
result above, but the statistics taking events of energy by > c results (Figure 1). Clearly, for the beam of neu-
higher than 20 GeV shows a significantly larger velocity trinos of greater energy, the velocity distribution may
than that for energy below 20 GeV, the early neutrino contain more superluminal data, farther from the value
arrival time δt is 67.1 ns in the former case while only of c.
∼
53.1ns in the latter case [2]. Furthermore, it is worth In summary, non-dispersive wave packet with a con-
∼
noting that the Minos experiment with neutrino energy stant propagation velocity of c has been formulated for
peaking at 3.0 GeV with a tail extending to above 100 masslessfermionsstartingfromtheDiracequation. Such
∼
GeV results in (v c)/c = 5.1 2.9 10−5 [6], but a a wave packet is constructed from the superposition of
− ± ×
stringent limit of (v c)/c < 2 10 9 (given in abso- common eigenstates of the Hamiltonian and the helic-
−
− ×
lute value!) was set by the observation of (anti) neutri- ity operator, which are represented on the basis of the
nos emitted by the SN1987A supernova in the 100 MeV squeezed-coherent states. Based on the assumption that
range [7, 8]. Roughly speaking, the measured neutrino athresholdportionofthewavepacketisdemandedtoin-
velocityseemstosuggestthatfollowingalargerneutrino vokea detectionevent,the wavepacket,macroscopically
energy there may be an increasedprobability of measur- extensive as it may be in principle as a beam of light,
ing a superluminal velocity. can provide a quite reasonableexplanation to the super-
The wave packet formulated in Eq.(6) for massless luminalvelocitystatistics recentlyreportedforneutrino,
fermionsisstrictlynon-dispersive,anditpropagatesata whichisanticipatedtooccurmorefrequentlyandfarther
constant velocity of light. This concludes from the solu- awayfromthevelocityoflightathigherparticleenergies.
tion of the equation of motion for the particles, which is Theresultcanserveasabasisforthediscussionofother
consistentwith both special relativity and quantumthe- problemsconcerningmasslessfermionswhere the formu-
4
Chinese academy of sciences.
[1] L. D. Landau, and E. M. Lifshitz, Quantum Mechanics,
Butterworth Heinemann (1991).
[2] T. Adam et al., http://arxiv.org/abs/1109.4897.
[3] Ch. Weinheimeret al., Phys.Lett. B 460, 219 (1999).
[4] Zh. Wang, X. G. Meng, and H. Y. Fan, Journal of the
Optical Society of America B, (2012), in press.
[5] A.S.Goldhaber,M.Goldhaber,PhysicsTodayMay,40
(2011).
[6] P. Adamson at al., Phys.Rev.D 76,072005 (2007).
[7] K. Hirata et al., Phys.Rev. Lett.58, 1490 (1987).
[8] R. M. Bionta et al., Phys. Rev.Lett. 58, 1494 (1987).
[9] Y.Aharonov,andD.Rohelich,QuantumParadox,Wiley
(2005).
FIG.1: Velocitymeasurementofanon-dispersivewavepacket [10] J. Baggott, Beyond Measure: Modern Physics, Philoso-
formasslessfermionwithaconstantpropagationvelocityofc. phy, and the Meaning of Quantum Theory, Oxford Uni-
Incasethattheportionoftheincidentwavepacketdemanded versity Press (2004).
toinvokeadetectioneventis(a)aheadof,(b)coincidentwith,
or (c) lagging behind the point registered as t=0 , then the
velocitycalculatedfromthetimeofflightwillbev>c,v=c APPENDIX
or v<c, respectively.
1. The module square of the initial state t=0
| i
For
lation of the wave packet may be of formal significance.
Acknowledgment. Oneofthe author(Cao)thanksthe
fidnatainocniaolfsCuphpinoartgsrfarnomt ntoh.e1n0a9t7io4n2a2l7n,athtuermalisnciisetnryceoffosucni-- |t = 0i=Z e−γ(α−α0)2 ee−−1122bb††bb††++aabb††||00ii!dα,
ence and technology grant no. 2009CB930800 , and the
t=0t=0 = e−γ(α−α0)2−γ(α′2−α0)2dαdα′ 2 0e−12bb+αbe−12b†b†+α′b† 0
h | i · h | | i
Z
= e−γ(α−α0)2−γ(α′2−α0)2dαdα′ 2 0e−12bb+αb d(ReΦ)d(ImΦ)e−(ReΦ2+ImΦ2) Φ Φe−21b†b†+α′b† 0
· h | π | ih | | i
Z Z
= 2 e−γ(α−α0)2−γ(α′2−α0)2dαdα′ d(ReΦ)d(ImΦ)e−2ReΦ2+(α+α′)ReΦ+i(α−α′)ImΦ.
π
Z Z
For brevity, replacing ReΦ with ζ and ImΦ with η,
t=0t=0 = 2 e−γ(α−α0)2−γ(α′2−α0)2dαdα′ dζdηe−2ζ2+(α+α′)ζ+i(α−α′)η
h | i π
Z Z
= 2 e−γ(α−α0)2−γ(α′2−α0)2dαdα′ dζdηe−2ζ2+(α+α′)ζ[cos(α α′)η+isin(α α′)η].
π − −
Z Z
Discarding the imaginary part of the integral as it must vanish, one obtains
t=0t=0 = 2 e−γ(α−α0)2−γ(α′2−α0)2dαdα′ dζdηe−2ζ2+(α+α′)ζcos(α α′)η
h | i π −
Z Z
= 2 e−γ(α−α0)2−γ(α′2−α0)2dαdα′ dζdηe−2(ζ−α+4α′)2+(α+8α′)2 cos(α α′)η
π −
Z Z
= 2 e−γ(α−α0)2−γ(α′2−α0)2dαdα′ dζ′dηe−2ζ′2+(α+8α′)2 cos(α α′)η
π −
Z Z
= 2 π e−γ(α−α0)2−γ(α′2−α0)2dαdα′ dηe(α+8α′)2 cos(α α′)η
π 2 −
r Z Z
5
= 2 e−γ(α2−2αα0+α20)+α82+α4α′e−γ(α′−α0)2+α8′2 cos(α α′)ηdαdα′dη
π −
r Z
= 2e−γα20 e−(γ−81)α2+8γα04+α′αe−γ(α′−α0)2+α8′2 cos(α α′)ηdαdα′dη
π −
r Z
= π2e−γα20 e−(γ−81)α′′2cos(α′′ + 88γ(αγ0+1α)′ −α′)ηdα′′e(8γ8(α80γ+−α1′))2−γ(α′−α0)2+α8′2dα′dη
r Z − 8
′
= π2e−8γ8γ2−−21γα20 e−(γ−18)α′′2[cosα′′ηcos8γα0−8γ(8γ1−2)α η
r Z −
′
sinα′′ηsin8γα0−(8γ−2)α η]e8γ1−1[−(8γ2−2γ)α′2+16γ2α0α′]dα′′dα′dη
− 8γ 1
−
′
= π2e−8γ8γ2−−21γα20 8γ8π 1e−8γ2−1η2cos8γα0−8γ(8γ1−2)α η
r Z r − −
8γ2−2γ(α′ 8γ2α0 )2 64γ4α20 ) ′
e− 8γ−1 −8γ2−2γ e(8γ−1)(8γ2−2γ dα dη
×
= 16 e8γ42γ−22γα20 e−8γ2−1η2e−8γ8γ2−−21γα′′′2cos[ 8γα0 8γ−2(α′′′ + 8γ2α0 )]ηdα′′′dη
8γ 1 8γ 1 − 8γ 1 8γ2 2γ
r − Z − − −
= 16 e8γ42γ−22γα20 e−8γ2−1η2e−8γ8γ2−−21γα′′′2cosηα′′′8γ−2dα′′′dη
8γ 1 8γ 1
r − Z −
= 16π e8γ42γ−22γα20 dηe−21γη2
8γ2 2γ
r − Z
= 4π e4γ2−γ1α2. (10)
√4γ 1 0
−
2. Motion of the wave packet The width of the wave packet is given by Γ =
For state t=0 , its temporal evolution is given by [ t(xˆ x¯)2 t / t t ]1/2 which can be simplified to Γ =
| i h | − | i h || i
[ ht|xˆ2|ti x¯2]1/2. Since
|ti=Z e−γ(α−α0)2e−iAαt ee−−1221bb††bb††++ααbb††||00ii!dα. (11) ht=0t|t=xˆ02it−= t(i∆(b b ))2 t
†
h | | i h | √2 − | i
Thus it has x(t) t = 0t = 0 = t i∆(b b )t
· h | i h |√2 − † | i ∆2 ∆2
from which, following the same tricks in deriving (10), = t(b2+b†2 2b†b)t tt .(14)
− 2 h | − | i− 2 h | i
one obtains Thus we just need calculate txˆ2 t + ∆2 tt which,
h | | i 2 h | i
again following the similar techniques in deriving (23),
x t=0t=0 =√2∆At 4π e4γ2−γ1α20. (12) can be found to result in
·h | i √4γ 1
−
Thisimpliesthatx(t)=√2∆At,consequentlytheveloc- txˆ2 t +∆2 tt = 8∆2π e8γ42γ−22γα20(A2t2+1), (15)
h | | i 2 h | i √4γ 1 γ
ity of the wave packet is
−
v = x(t) =√2∆A=c. (13) or ht|xˆ2|ti = 2∆2(A2t2+ 1) ∆2 which further leads to
t Γ(t)ht=|ti∆(2 1)1/2, i.e.γ, t−he 2width remains constant.
γ − 2
3. Dispersion of the wave packet The wave packet is non-dispersive.