  Since future, precise theory of neutrino oscillations should include the
understanding of the neutrino mass generation and a precise, relativistic
description of hadrons, and observing that such a future theory may require
Dirac's FF of Hamiltonian dynamics, we provide a preliminary FF description of
neutrino oscillations using the Feynman--Gell-Mann-Levy version of an effective
theory in which leptons interact directly with whole nucleons and pions,
instead of with quarks via intermediate bosons. The interactions are treated in
the lowest-order perturbative expansion in the coupling constants in the
effective theory, including a perturbative solution of the coupled constraint
equations. Despite missing quarks and their binding mechanism, the effective
Hamiltonian description is sufficiently precise for showing that the standard
oscillation formula results from the interference of amplitudes with different
neutrinos in virtual intermediate states. This holds provided that the inherent
experimental uncertainties of preparing beams of incoming and measuring rates
of production of outgoing particles are large enough for all of the different
neutrino intermediate states to contribute as alternative virtual paths through
which the long-baseline scattering process can manifest itself. The result that
an approximate, effective FF theory reproduces the standard oscillation formula
at the level of transition rates for currently considered long-baseline
experiments--even though the space-time development of scattering is traced
differently and the relevant interaction Hamiltonians are constructed
differently than in the commonly used IF of dynamics--has two implications. It
shows that the common interpretation of experimental results is not the only
one, and it opens the possibility of considering more precise theories taking
advantage of the features of the FF that are not available in the IF.
