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OPERA Collaboration have observed phase speed of neutrino wave function

Shi-Yuan Li

Published Oct 3, 2011• 2 citations
OpenAlex

Abstract

First we call the attention that velocity defined by ratio between some intervals of space and time respectively is sometimes ambiguous, in the framework of quantum theory. Velocity in general is not possible to be well defined as some generator of certain space-time symmetry operation. Then in section 3 by analyzing the OPERA experiment we show that the OPERA Collaboration may have measured the phase speed of the neutrino wave function. Employing a very (maybe too) simple model which is just a reproduction from Brillouin's classical book, we demonstrate the phase velocity and group velocity. These are just a qualitative illustration rather than aiming to quantitively explain the OPERA data. However, how to rigorously define the phase containing the particle production ordering information as a space-time distribution observable in QFT needs more investigations. Due to the finite life time of the pions (w.r.t. its flying time in the 1 km long tunnel) which decay to the muon neutrinos, the wave form for neutrinos recorded by OPERA is deformed from that of the proton. Such an effect is estimated to lead to an advanced time displacement of the order of 100ns, which coincides with the measured. Such an effect can only be significant when measuring phase speed and the recorded neutrino events per extraction of proton smaller than one, as the OPERA case.

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First we call the attention that velocity defined by ratio between some intervals of space and time respectively is sometimes ambiguous, in the framework of quantum theory. Velocity in general is not possible to be well defined as some generator of certain space-time symmetry operation. Then in section 3 by analyzing the OPERA experiment we show that the OPERA Collaboration may have measured the phase speed of the neutrino wave function. Employing a very (maybe too) simple model which is just a reproduction from Brillouin's classical book, we demonstrate the phase velocity and group velocity. These are just a qualitative illustration rather than aiming to quantitively explain the OPERA data. However, how to rigorously define the phase containing the particle production ordering information as a space-time distribution observable in QFT needs more investigations. Due to the finite life time of the pions (w.r.t. its flying time in the 1 km long tunnel) which decay to the muon neutrinos, the wave form for neutrinos recorded by OPERA is deformed from that of the proton. Such an effect is estimated to lead to an advanced time displacement of the order of 100ns, which coincides with the measured. Such an effect can only be significant when measuring phase speed and the recorded neutrino events per extraction of proton smaller than one, as the OPERA case.
The result of the OPERA experiment revealed that the velocity of muon-neutrinos was larger than the speed of light. We argue that this apparent superluminal velocity can be interpreted as a weak value, which is a new concept recently studied in the context of quantum physics. The OPERA experiment setup forms a scheme that manifests the neutrino velocity as a weak value. The velocity defined in the scheme of weak measurement can exceed the speed of light. The weak velocity is not a concept associated to a single phenomenon but it is a statistical concept defined by accumulating data at separated places and by comparing the data. Neither information nor physical influence is conveyed at the weak velocity. Thus the superluminal velocity in the sense of weak value does not contradict the causality law. We propose also a model for calculating the neutrino velocity with taking neutrino oscillation into account.
We consider a quantum field theory (QFT) model of neutrino oscillations in vacuum that attempts to take into account that the neutrino source particle and the neutrino detection particle both interact with their respective environments by collisions. Our model is minimal in a twofold sense. Firstly we simply assume that neutrino production and detection take place in between collisions in time intervals ΔtΔtΔt and τττ, respectively. Secondly, we only introduce the two wave packets that are absolutely necessary which are those of the neutrino source and the neutrino detection particle. Within this model we find that, for all practical purposes, there are no decoherence effects in the neutrino oscillation amplitude and oscillations occur in space not in time. Moreover, our model leads to the correct time correlation between neutrino production and detection and to a factorization of the event rate of the compound neutrino production--detection process into decay rate, oscillation probability and detection cross section.
Simple rigorous quantum mechanics with no hand waving nor loopholes clarifies the confusion between three contradictory descriptions of the coherence between different neutrino mass eigenstates that can give rise to oscillations: (1) The standard textbook description of oscillations in time produced by coherence between states with different masses and different energies. (2) Stodolsky's proof that interference between states having different energies cannot be observed in realistic experiments. (3) The description of a pion decay at rest into an observed muon and unobserved neutrino as a “missing mass” experiment where coherence between different neutrino mass eigenstates is not observable. The known position in space of all realistic detectors is rigorously shown to provide the quantum-mechanical ignorance of the neutrino momentum needed to produce coherence between amplitudes from neutrino states with the same energy and different masses. Conditions are precisely formulated for the loss of coherence when mass eigenstate wave packets moving with different velocities separate. The example of Bragg scattering shows how quantum-mechanically imposed ignorance produces coherence.
The precise definition and physical interpretation of a tunneling time is a fundamental problem in quantum mechanics. The lack of a well-defined time operator in quantum theory precludes the calculation of time in terms of an expectation value. Consequently, model calculations and simulation studies have been proposed and used to determine the time for a particle to traverse a quantum-mechanical barrier. The results offer both qualitatively and quantitatively, disparate predictions. One of the more commonly used approaches is the phase method, which we have applied to several one-dimensional model potentials to show that the tunneling time is characterized by the ratio of the typical decay length in the barrier to the incident velocity, τ∼(1/κ)/v0. We also show that the phase and the spin precession methods are equivalent when the magnetic field is applied throughout the space containing the particle wave function and the tunneling barrier. In principle, the spin precession method can be regarded as an operational definition of the tunneling time. It also has been shown that a tunneling time can be defined operationally for a time-dependent barrier, for which the model calculations predict a tunneling time given by the mean barrier width divided by the magnitude of the imaginary velocity of the particle in the barrier, τ∼d/‖v‖. This pseudokinematic result has been inferred by magnetic field dependent tunneling through heterostructures and computer simulation of wave packet tunneling through one-dimensional square barriers. Other tunneling time theories predict one or the other of the above mentioned characteristic times; several theories predict both. We interpret these different results for τ to mean that a tunneling time is not uniquely defined as a function of particle energy and the barrier shape unless the process in which the tunneling is involved is also specified. That is, the tunneling time is a meaningful and unique concept only for particular tunneling processes or definite operational procedures.
Measurements of cosmic neutrinos have a reach potential for providing an insight into fundamental neutrino properties. For this a precise knowledge about an astrophysical environment of cosmic neutrinos propagation is needed. However this is not always possible, and the lack of information can bring about theoretical uncertainties in our physical interpretation of the results of experiments on cosmic neutrino fluxes. We formulate an approach that allows one to quantify the uncertainties using the apparatus of quantum measurement theory. We consider high-energy Dirac neutrinos emitted by some distant source and propagating towards Earth in the interstellar space. We take into account the neutrino magnetic moment interaction with stochastic interstellar magnetic fields and describe the neutrino evolution in the formalism of the Lindblad master equation for the neutrino density matrix. It is supposed that neutrinos on their way to Earth pass through highly magnetized dense layers of a cosmic object that ``stops'' left-handed neutrinos and lets only sterile, right-handed neutrinos to propagate farther. Such a filter mimics the strongest effect on the neutrino flux that can be induced by the cosmic object and that can be missed in the theoretical interpretation of the lab measurements due to the insufficient information about the astrophysical environment of the neutrino propagation. Treating the neutrino interaction with the neutrino filter of the cosmic object as the first, neutrino-spin measurement, whose result is not recorded, we study its invasive effect on the second, neutrino-flavor measurement in the lab. We quantify the invasiveness of the first, blind measurement by means of quantum witness that in the discussed case has an advantage over the well-known Leggett-Garg inequality, since the latter explores two-time correlation functions of the same operator. We solve analytically the Lindblad master equation for time evolution of the neutrino density matrix and on this basis we calculate the quantum witness for measuring cosmic electron neutrinos in the lab. We present numerical illustrations of the robust invasive effect showing that the quantum witness as a function of the distance between Earth and a cosmic object can be an asymptotically nonvanishing quantity despite the thermalization of the neutrino spin induced by stochastic interstellar magnetic fields.