Theoretical and Mathematical Physics, 192(1): 1000–1015 (2017)
OSCILLATIONS OF PARTICLES IN THE STANDARD MODEL
A. E. Lobanov
We construct Hilbert spaces of particle states such that all neutrinos and also charged leptons and up and
down quarks are united into multiplets and their components can be treated as diﬀerent quantum states
of a single particle. The phenomenon of neutrino oscillations arises in a theory based on the Lagrangian
of the fermionic sector of the Standard Model modiﬁed according to the proposed approach.
Keywords: Standard Model, neutrino oscillation, indeﬁnite-mass state, Poincar´e group, SU(3) group,
The Standard Model of electroweak interactions based on the assumption about the generation of
particle mass by spontaneous symmetry breaking  and on the non-Abelian gauge symmetry of interac-
tions  and the ideology of particle generation mixing ,  is universally accepted. The predictions of
this theory derived in terms of perturbation theory agree well with the experimental data, and there are
no solid reasons for revising the fundamental assumptions of the model, at least for presently attainable
Nevertheless, the Standard Model of electroweak interactions fails to describe neutrino oscillations in
the vacuum because they imply a possibility of a transformation of diﬀerent types of free fundamental
fermions into one another in this case. This important and experimentally well-justiﬁed phenomenon
(see ) can be described by a phenomenological theory based on the pioneering works by Pontecorvo 
and Maki et al. . This theory is well developed and does not conﬂict with the experimental data. The
description of its fundamental statements is given in excellent reviews and monographs (see, e.g., ).
In addition to the obvious assumption that there are three types of neutrinos with diﬀerent (nonzero)
masses, the neutrinos produced in reactions are postulated to exist in states that are superpositions of
the mass states. These states are called ﬂavor states, and the relation between the basis in the mass
state space and the basis in the ﬂavor state space can be described by a unitary mixing matrix. At
the initial instant, the mass basis elements are assumed to be described by plane waves with the same
(three-dimensional) momentum. The time evolution of ﬂavor states is described by the solution of the
corresponding Cauchy problem. Based on these assumptions, the basic principles of quantum mechanics
allow deriving the equations for the ﬂavor-state transition probability uniquely. The corresponding linear
dependence of the oscillation length on the neutrino energy was experimentally demonstrated.
The conclusions of this theory are applicable when the neutrino masses are small compared with the
neutrino energy m
1, and this holds for all real experiments for neutrino oscillations. But this
Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia, e-mail: email@example.com.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 192, No. 1, pp. 70–88, July, 2017. Original
article submitted May 16, 2016; revised July 4, 2016.
2017 Pleiades Publishing, Ltd.