OPERA Collaboration have observed phase speed of neutrino wave function, while advanced time displacement is mainly due to finite life time of pion
aa r X i v : . [ h e p - ph ] D ec OPERA Collaboration have observed phasespeed of neutrino wave function, whileadvanced time displacement is mainly due tofinite life time of pion
Shi-Yuan LiSchool of Physics, Shandong University, Jinan, 250100, PRCSeptember 28, 2018
Abstract
We show that the OPERA Collaboration have measured the phasevelocity of the neutrino wave function, based on the analysis of theexperimental method. On the other hand, the significant advancedtime displacement δt reported by OPERA is mainly due to the finitelife time of the pions (w.r.t. its flying time in the 1 km long tunnel)which decay to the muon neutrinos. . The OPERA Collab. [1] announced to have measured the ‘time offlight’ T ( T OF ν in [1]) of the neutrino from CERN to Gran Sasso. Dividingthe distance between these two places L (from BCT to OPERA [1]) by T,one can get a ‘velocity’ whose value turned out to be larger than the speedof light in vacuum, c, i.e., L/T > c. Such a result, at first glance, could be achallenge to Einstein’s Special Relativity, though one is aware that c is one ofthe basic constant in modern physics whose physical meaning is much moredeeper than light’s ‘speed’ whose physical meaning is quite ambiguous. Onlyin the case of strict classical mechanics, when the position (element of somemathematical structure on which differential w.r.t. the parameter ‘time’ canbe defined) is considered as the basic observable of a particle, and as a singlevalue function of time, the velocity is a well-defined quantity. According toN¨other’s theorem, a basic and well defined space-time related observable is1he generator of some kind of space-time symmetry operation. In quantumtheory, the movement of a particle is described by state vector, and generallyposition is not a good quantum number. So in position representation we en-counter a distribution described by wave function of the particle (leaving outthe more complexity of particle creation and annihilation). Hence velocity istoo complex and case-dependent, generally can not be taken as basic observ-able related to some kind of space-time symmetry, but as a quantity definedby others, depending on the concrete cases. E.g. (suppose Lorentz invari-ance kept), for a free particle in energy-momentum eigenstate | E P > , with E = P + m , its velocity can be defined by P /E , but is not easy to relateto some unambiguous distance over time interval as in classical mechanics,even with the knowledge of the wave function. Another intuitive exampleis the electron bound in hydrogen atom, whose 3-momentum is not a ‘goodquantum number’, i.e., not commuted with the Hamiltonian hence not themovement integral. In this case the above definition of velocity fails; andbecause the electron is ‘off-shell’, if we define some ratio between momentumand energy, or space and time, it is very possible to get some ‘velocity’ whichis large than c. One may call the electron in such case as tachyon if one likes,but we know that the basic physical principles like causality of the hydrogensystem, e.g., in process of transition between different energy levels, alwayskeep. Hence we should pay attention that some ‘velocity’ defined by ratiobetween some measured space and time intervals is not always straightfor-ward pointing to the basic principle about the property of space-time. Onaware of the above, we can well understand that even the terminology ‘speedof light in vacuum’ is purely a traditional name. Since we should ask whatkind of ‘light’: a photon? a light pulse? etc. For each case the definition ofvelocity varies. We know that c is further used, e.g., to scale the momentum,mass, energy of a free particle so that E = P + M , etc., etc. However, ifthe wave function of a particle can be deduced from some experiments, oneof course can discuss some ‘displacements’ in space as well as time based onthe wave function. Their ratio could be interesting provided a careful inves-tigation on what such kind of measurement the experiment in fact makes,and what/whether basic physical principles can be further deduced from it.In the following we will recall that the phase speed of the wave functioncan very naturally be larger than c (section 2). With the effort to analyzethe key experiment method, i.e., fitting the wave forms (average time distri-butions in the time interval of the extraction [1]) of the proton as well as theneutrino event to get the time displacement w.r.t. ‘flying in the speed of c’,2e show that the OPERA Collab. have measured the phase of the particlewave function and its speed hence is possibly larger than c (section 3). Sinceneutrino mass is very small, whether phase velocity or group velocity, if prop-erly measured, both should not varying from c significantly, no matter largeror smaller than c. However, due to the finite life time of the pions (w.r.t. itsflying time in the 1 km long tunnel) which decay to the muon neutrinos, thewave form for neutrino is deformed from that of the proton. Such an effectis estimated to lead to an advanced time displacement of the order of 100ns,which coincide with the measured δt reported in [1] (section 4). . The basic things measured by OPERA Collab. are two time distri-butions in two space-time positions. At position x (CERN) and time t ,one measures the proton to deduce the wave function of the neutrino (for theproblems see section 4) and then at t and position x (Gran Sasso) measuresthe wave function by the OPERA detector (of curse we can only deduce thewave function by measuring its mudulus square). To make the discussionssimple, we do not discuss the neutrino oscillation. The good time correspon-dence of the neutrino events and the proton extractions as presented by theCollab. [1] indicates that the neutrino is well propagating and could be mod-eled as a plane wave for the simplest consideration. We will employ wavepacket to describe the neutrino in Section 3 and show that the measuredresult L/T is possibly understood as the phase speed of the propagatingwave function since their key measurement is to calibrate the pace/phasefor the particle in these two space-time positions. L/T larger than c is thestraightforward result from that the physical velocity of neutrino is smallerthan c and well obeys the energy-momentum relation of a massive particle.Suppose a likely experiment is done for a much more massive ‘dark matter’particle, such a phase speed may be significantly larger than c. However, thisdoes not to say such measurement is meaningless. Detailed analysis on suchmeasurement may help to improve the experiments on neutrino oscillationor even absolute value of neutrino mass.The wave function is e − iωt + iκ · x , (1)and we find that ω | κ | = | ∆ x | ∆ t = | x − x | t − t (2)This is the well-known phase velocity of the wave of eq. (1). Here oneencounters the similar problem that Louis de Broglie encountered almost300 years ago [2]. When ω and κ correspond to energy and momentum ofa particle, according to the Einstein relation, we will find that this phasevelocity is never smaller that c but always larger, except for light it equalsto c. And the wise solution of employing more reasonable group velocity byde Broglie is the key corner stone for his matter wave hypothesis which isone of the most important turning points from Bohr’s old quantum theoryto the modern quantum mechanics. . A plane wave as above extends infinitely. If nothing can mark itsphase, our discussions in Section 2 are just for an ‘ideal experiment’. By theabove we only show that for a plane wave, the interval between two space-time positions with the same phase could be space like. For the mechanicalwave like that on a long shaking rope, we can see the oscillation (say, upand down) of the points of the rope with different phases, so that we canobserve the propagation. It turns out that the time structure of the protonPDF (probability density function [1], but in fact number density as to beclarified) and that of the OPERA neutrino events can mark the phase to bemeasurable, which we will make clear in the following based on the analysisof the experiment. It is our key point in this section to make clear whatthe experiment measures with a simple model. Now it is more of reality todescribe the propagation of neutrino by wave packet,Ψ( x, t ) = Z dk Ψ( k ) e − iωt + ikx , (3)where ω = ω ( k ) is defined by the on-shell Einstein relation for four momen-tum (In this section the space related variables like k , x are not speciallydenoted to show they are SO(3) vectors but indicated). Here we do not dis-cuss the details of Ψ( k ), only this wave packet is assumed well localized: Forspace, even the whole neutrino source system of CERN at Prevessin and thedetectors in laboratory at Gran Sasso can be taken as point comparing to thedistance between these two places. So here we need not discuss how well theposition localized which is in fact well investigated by OPERA Collab. Fortime, we have no problem to say that this particle is localized much smallerthan 10 . µs , based on the PDF time structure. Same as [1], we employ thetime when the proton going through the BCT to mark the time of the wavepacket (e.g., the central peak of the wave packet of Eq. (3)). Taking intoaccount the reaction time of the BCT or by carefully analyzing the PDF,one can extract a basic time length l T (say, around 5 ns according to Fig.4OF [1]). Here a proper Ψ( k ) need to lead that Ψ( x, t + ∆ t ) vanishing when4 ∆ t | > l T . Since the wave packet could expand en route of propagation, sothis requirement is only for the time and place at CERN, where the protonsemployed to produce neutrino by its reaction with the target is carefullymeasured, one of the most important instrument for the this measurementis the BCT [1].This is the key point here. We would like to remind the reader that ifhe/she understood that we assign the distribution of the PDF or such kindsas the time distribution of the wave (packet) function of each proton as well asthat of the neutrino, he/she completely misunderstood. Particles producedin CERN in the processes described in Section 2 of [1], of which proton isthe most especially typical, can all be considered as that their wave packetsize is as small as possible (in time and space) so that the inner structure ofthe wave packet function is not possible seen by all detectors. The PDF timestructures exactly is the numbers of particles varying with time. However,this varying value can mark the PHASE of the wave packet of the particle.Since in no way to assume any difference among all the protons, we canemploy the same form of wave packet to describe all the protons as well asneutrinos. Only that, each of the particles (proton/neutrino) correspondingto each ‘point’ ( l T is the unit) t’ at the time axis of PDF distribution has atime displacement w.r.t. the beginning of the extraction: e − iHt ′ Ψ( x, t ) = Z dk Ψ( k ) e − iω ( t + t ′ )+ ikx , (4)with 0 < t ′ < . µs . This phase in other word can be called the paceof the protons/neutrinos, i.e., to mark their order of ‘step’ in time of reac-tionreaction and reaction of each extraction. (Actually this only marks theneutrino’s mother particles, mainly pions and kaons, the problems caused bythe infinite life time during the decay is discussed in details in section 4.) Asthe experiment has put together and makes statistics for all the particles indifferent ‘extractions’, they are to project on a same t’ region by plus/minusn time of 50 ms (the time interval between extractions) for their real timet. Here we emphasize again that the wave packet is so well located thatany ‘interference’ between different wave packets (particles) are neglected, ascan be assumed by the experiment (or effects as beam interaction has beencorrected by experiment). We emphasize this is to point out that those kindof effects are of NO relation with our discussions. And we also point out thatthough in experiment each event and wave form has been stamped by thereal time, but what to be used in analysis is the averaged results obtained by5arge number of samples of extractions and events projected together on theinterval of the extraction. We in fact can not say which proton (neutrino)has which phase but just employ the different numbers of particles (events)in this duration to represent the fact that each wave packet of the particlehas certain definite phase, marked by t’, relative to the, e.g, the beginningtime of the extraction.The wave packet will expand during propagating, but the experimentshow they still keep some peak structure so that can be detected in GranSasso at some certain time. This exact value of time but is only used toextract the t’ to get the event time structure for comparing with the sourcetime structure (PDF), then to calibrate the phase parameter correspondingto t’. The method is, as described by the OPERA Collab., to employ themaximum likehood method to coinside/tune the same pace for these twodistributions. This is just to determine which phase group of proton PDF theparticles/events detected by OPERA belong to. Then it is possible to find thesame phase positions ( t , x ) and ( t , x ) when the phase/pace is calibratedto coincide. The space-time interval ratio hence can be now understood asthe velocity of the phase of the wave packet marked by t’ moves from ( t , x )to ( t , x ) (i.e., from CERN to Gran Sasso). x − x = L is the length of thebaseline presented from [1], while t − t = T is also possible deduced fromthe measured quantities in [1]. In other words, the experiment is comparingthe step order of the proton and reaction events, so that to find two space-time position with the same phase. Here we would like to emphasize againthat the PDF and the coinciding neutrino signal distribution is not the formof wave function but only show the fact that the wave packet’s phase markedby t’ properly propagates from ( x , t ) to ( x , t ). Their interval could bespace like as discussed in the following.The movement of the wave packet from CERN to Gran Sasso, e − iH ( t − t ) Ψ( x , t , t ′ ) (5)in practice is too complex to be employed to calculate the phase velocityand information (or ‘particle’) velocity based on the real form of Ψ( k ), evenwe can get it. Here we only employ the (maybe too) simplified qualitativeexpressions to discuss: Suppose that the effective integral region of k (whereΨ( k ) significantly different from zero), ∆ k , is much smaller than k, i.e.,∆ k << k, which also renders ∆ ω << ω. (6)6his may be valid in high energies. Now we can rewrite the wave functionΨ( x, t, t ′ ) as ∆ k Ψ( k ) e − iω ( t + t ′ )+ ikx e − i ∆ ωt + i ∆ kx . (7)∆ ωt ′ is higher order infinitesimal and neglected from Eq. 7, comparing to along propagation time, since t’ is fixed as a small value between 0 to . µs .So here t’ only appear in the ‘carrier’ wave. From this quite simple expressionwe see the phase propagates with a velocity ω/k , and the same phase space-time position can be marked by t’. While the wave packet propagates at thegroup velocity ∆ ω/ ∆ k . The above simplification is just a reproduction of theclassical discussion from Brillouin’ book, chapter 1 [3]. In the most simplecase the phase velocity is large than c while the group velocity is smaller,just to refer Section 2.The same narrow k distribution assumption (6) for large k approximatelyleads to | Ψ( x, t + t ′ ) | ∼ = δ ( v g ( t + t ′ ) − x ) , (8)with v g = ∆ ω/ ∆ k the group velocity in Eq. (7). This is just the distri-bution function of a classical particle and this interpretes why in most highenergy experiments, one can treat the space-time movement of a single par-ticle as classical with velocity P/E , even with the fact it is almost in energymomentum eigenstate rather than position eigenstate. Such a result also con-firms what we have emphasized above, that in our treatment, we need notto assume anything new from the conventional treatment about the protonbeam, as well as the pions, kaons, and neutrinos in the production tunnel[1]. However, one should be aware that since in quantum mechanics, a singlemicroscopic particle is not able to be marked and measured (e.g, the valuesof the position) many times without dramatically changing its state. For astatic particle flow, this delta function description is of no difference fromthe plane wave description: no signal of movement like classical one particledisplacement in space-time can be observed.Now comes to the description of the time structure of the proton beam,which is also assumed that of the neutrino. This is an inherent property ofthe proton beam system. Here we assumed that many times average (averageparticle number < N ( t ′ ) > ) can cancel the fluctuations from environment,i.e., < N ( t ′ ) > is just the PDF [1]. So the particle system can be described asdirect product of the the single particle wave functions, with the interactions7mong them canceled.Φ( x, t + t ′ ) =
2. Only considering the effect of pion, itleads to the value of the order of 100 ns, which is coincide with the OPERAresult. The result is calculated from the following equations: R dt t e − t/T π R dt e − t/T π . (12)Here for simplicity we take T f as unit. We also emphasize that this effect isnot sensitive to he wave form of the proton, whether the old long time oneand the new short time (pulse) one.However, here we must point out that such an effect is significant butmissed by the Collab. CRUCIALLY depends on the special property of thiskind of measurement and even more CRUCIALLY depends on the under-standing for the physical meaning of this experiment: First, as clarified insection 3, this experiment compares phase/pase (‘wave form’) to measurethe phase velocity. Second, the recorded neutrino events are quite rare, sta-tistically the number of neutrino events recorded by OPERA per protonextraction is much much smaller than one. In such a special case, the decayprocess in the tunnel for each extraction is not possible to be tracked by theneutrino events (requiring number of neutrino events per extraction ≥ − β π ∼ − and pion only fly not more than 1km,which HAS been taken into account in [1].The MINOS [5] experiment is much less affected by the decay effect dis-cussed above since its energy is much lower hence a much smaller γ . All themother particles can be taken as to decay instantly. . After the release of the OPERA data, there have been more than100 papers discussing this result [6]. This has stimulated the study/reviewof the Lorentz invariance violation in various ways, which could be a goodwindow for the future new physics. On the other hand, there are severalpapers devoted to study the systematics of the experiment or suggestingthe velocity could be unphysical one. However, such kind of investigationshould include several key points: first to clarify what the observable ismeasured by analyzing the basic experimental method; second to see whetherthe measured value agree or not with the suggested, if not, where is thesystematic. This paper is trying to do in this way. This is a thing that onecan not escape from for exploring Lorentz invariance violation, not relating toany conservative attitude of believing Einstein, but something learned fromEinstein’s attitude to experiments, no matter sooner or later his theories arefound to be broken through.The author thanks Prof. Dr. WANG Meng for explaining some as-pects of the OPERA experiments. This work is partially supported byNSFC(10935012), SFSD (JQ201101) and SDU (2010JQ006). References [1] T. Adam et al. [ OPERA Collaboration ], [arXiv:1109.4897 [hep-ex]].[2] Louis de Broglie,
Recherches sur la th´eorie des quanta,
PhD Thesis(Paris), 1924. 113] L. Brillouin,
Wave propagation and group velocity,
Academic Press,1960.[4] M. J. Longo, Phys. Rev. D 36 (1987) 3276, Phys. Rev. Lett. 60 (1988)173; L. Stodolsky, Phys. Lett. B (1988) 353. The references forthe observations on the neutrino as well as light signals from Supernova1987A are to be found in these three papers.[5] P. Adamson et al.