Projection quantum mechanics and neutrino mixing
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ELEMENTARY PARTICLES AND FIELDS Theory
Projection Quantum Mechanics and Neutrino Mixing∗ ´ z´ ** and M. Go´ zd ´ z´ *** A. Go´ zd Faculty of Mathematics, Physics and Computer Science University of Maria Curie–Skłodowska, Lublin, Poland Received May 18, 2016
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
1. MOTIVATION Mixing among elementary particles is a common phenomenon. It occurs for the gauge bosons, quarks, neutral leptons, and very weakly for the charged leptons [1]. Its origin is usually linked to the spontaneous breaking of some higher symmetry, which in the low-energy regime results in the redefinition of the quantum numbers. The details of this mechanism, however, are still not established. Mixing of particles inevitably leads to the possibility of their oscillation between different quantum states. In what follows we take neutrinos as an example. Neutrinos show up in three flavors which correspond to the three families of the charged leptons: e, μ, and τ , and it has been experimentally confirmed that all three types of neutrinos, as well as their antiparticles, are different (see [2] and references therein). It has also been observed, that neutrinos produced in a weak reaction in a definite flavor eigenstate change into an other flavor eigenstate during free propagation to the detector [3–5]. To explain the neutrino oscillations the standard ¨ Schrodinger time evolution of a superposition of states is usually evoked [6]. In a more detailed analysis wave packets or quantum field theory is used, leading ultimately to the same final formula. Namely, ∗
The text was submitted by the authors in English. E-mail: [email protected] *** E-mail: [email protected] **
if one assumes that the L-chiral neutrino field is given by Uαi νiL , (1) ναL = i=1,2,3
where U is the mixing matrix, one finishes with two bases: the flavor (interaction) basis labeled α = e, μ, τ , and the second basis labeled by i = 1, 2, 3. In order to explain oscillations between flavor states, this second basis has to correspond to neutrino propagation, thus it has to represent physical particles wi
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