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Physics of Atomic Nuclei

Projection quantum mechanics and neutrino mixing

ANDRZEJ GÓŹDŹ, Marek Góźdź

Published Mar 1, 2017• 1 citation
Publisher

Abstract

The theory of neutrino oscillations rests on the assumption, that the interaction basis and the physical (mass) basis of neutrino states are different. Therefore neutrino is produced in a certain welldefined superposition of three mass eigenstates, which propagate separately and may be detected as a different superposition. This is called flavor oscillations. It is, however, not clear why neutrinos behave this way, i.e., what is the underlying mechanism which leads to the production of a superposition of physical states in a single reaction. In this paper we argue, that one of the reasons may be connected with the temporal structure of the process. In order to discuss the role of time in processes on the quantum level, we use a special formulation of the quantum mechanics, which is based on the projection time evolution. We arrive at the conclusion, that for short reaction times the formation of a superposition of states of similar masses is natural.

DOI
10.1134/s1063778817020181

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The theory of neutrino oscillations rests on the assumption, that the interaction basis and the physical (mass) basis of neutrino states are different. Therefore neutrino is produced in a certain welldefined superposition of three mass eigenstates, which propagate separately and may be detected as a different superposition. This is called flavor oscillations. It is, however, not clear why neutrinos behave this way, i.e., what is the underlying mechanism which leads to the production of a superposition of physical states in a single reaction. In this paper we argue, that one of the reasons may be connected with the temporal structure of the process. In order to discuss the role of time in processes on the quantum level, we use a special formulation of the quantum mechanics, which is based on the projection time evolution. We arrive at the conclusion, that for short reaction times the formation of a superposition of states of similar masses is natural.
It has been firmly established that neutrinos change their flavour during propagation. This feature is attributed to the feature that each flavour eigenstate is a superposition of three mass eigenstates, which propagate with different frequencies. This picture, although widely accepted, is wrong in the simplest approach and requires quite sophisticated treatment based on the wave-packet description within quantum field theory. In this communication we present a novel, much simpler explanation and show that oscillations among massive particles can be obtained in a natural way. We use the framework of quantum mechanics with time being a physical observable, not just a parameter.
Abstract We revisit the theory of neutrino oscillations and describe it through the formalism of weak measurements with postselection. It is well understood that due to the large momentum uncertainty in detection, there is no collapse of the neutrino wavefunction in the momentum or energy basis, and the mass eigenstates are detected coherently. Here we show that postselection, which projects the system to a final flavor state, deforms the system wavefunction in such a way that the momentum detected is not the expectation value of the neutrino mass eigenstates momenta, but the corresponding weak value. We use the weak values to describe the intermediate state in the oscillation process, avoiding problems in defining probability currents for particle states with mass superposition.
We perform such a calculation in quantum field theory, where we include simultaneously the source, detector, and neutrino fields in the Hamiltonian. Within the appropriate limits associated with current neutrino oscillation experiments, we recover the standard oscillation formula. On the other hand, we find that the dominant contributions to the amplitude are associated with different neutrino mass eigenstates being emitted at different times, such that they arrive at the detector at the same time. This is contrary to the neutrino wave packet picture, where they are emitted simultaneously and separate as they travel to the detector. This has direct consequences regarding the mechanisms that lead to a damping of neutrino oscillations for very long baselines. Our analysis also provides a pedagogical example of a measurement process in quantum mechanics.
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.
In charged current weak interaction processes, neutrinos are produced in an entangled state with the charged lepton. This correlated state is disentangled by the measurement of the charged lepton in a detector at the production site. We study the dynamical aspects of disentanglement, propagation, and detection, in particular, the conditions under which the disentangled state is a coherent superposition of mass eigenstates. The appearance and disappearance far-detection processes are described from the time evolution of this disentangled ``collapsed'' state. The familiar quantum mechanical interpretation and factorization of the detection rate emerges when the quantum state is disentangled on time scales much shorter than the inverse oscillation frequency, in which case the final detection rate factorizes in terms of the usual quantum mechanical transition probability provided the final density of states is insensitive to the neutrino energy difference. We suggest possible corrections for short-baseline experiments. If the charged lepton is unobserved, neutrino oscillations and coherence are described in terms of a reduced density matrix obtained by tracing out an unobserved charged lepton. The diagonal elements in the mass basis describe the production of mass eigenstates whereas the off-diagonal ones provide a measure of coherence. It is shown that coherences are of the same order of the diagonal terms on time scales up to the inverse oscillation frequency, beyond which the coherences oscillate as a result of the interference between mass eigenstates.
Coherence, which represents the superposition of orthogonal states, is a fundamental concept in quantum mechanics and can also be precisely defined within quantum resource theory. Thus exploring quantum coherence in neutrino oscillations can not only help in examining the intrinsic quantum nature but can also explore their potential applications in quantum information technologies. Previous studies on quantum coherence have focused on neutrino flavor oscillations (FO). However, FO imply that neutrinos have mass, and this can lead to the generation of a tiny but finite magnetic dipole moment of neutrinos through quantum loop diagrams at higher orders of perturbative expansion of the interaction. This electromagnetic property of neutrinos can induce spin flavor oscillations (SFO) in the presence of an external magnetic field and hence is expected to enrich the study of coherence. In this work, we investigate quantum coherence in neutrino SFO with three flavor mixing within the interstellar as well as intergalactic magnetic fields, quantified by the l 1 norm and the relative entropy of coherence, and express these measures in terms of neutrino SFO probabilities. For FO, coherence measures can sustain higher values (say, within 50% of the maximum) over distances of several kilometers, which are relevant for terrestrial experiments like reactor and accelerator neutrinos. However, for SFO, we find that the coherence scale can extend to astrophysical distances, spanning from kiloparsecs to gigaparsecs.