Neutrino Oscillations with Nil Mass

Abstract

An alternative neutrino oscillation process is presented as a counterexample for which the neutrino may have nil mass consistent with the standard model. The process is developed in a quantum trajectories representation of quantum mechanics, which has a Hamilton–Jacobi foundation. This process has no need for mass differences between mass eigenstates. Flavor oscillations and $\nu ,\bar{\nu }$ oscillations are examined.

Appendix: A Outline for a \(\varvec{\psi }\) Representation of Flavor Oscillation

A suggested $\psi$ representation of the quantum trajectories algorithm for neutrino oscillation is now outlined. While the quantum reduced action (a generator of quantum motion) of the quantum trajectories representation accounts for the entanglement between the dichromatic components of the neutrino’s spectrum, a composite $\psi$, Eq. (8), does the same accounting in the $\psi$ representation. In both representations, the two spectral components, $k_{\pm } = \pm k = \pm (E^2 - m^2c^4)^{1/2}/(\hbar c)$, are not manipulated as separate entities but compositely to incorporate their mutual entanglement with each other to describe the dichromatic neutrino’s behavior. As the neutrino is not bound, its $\psi$ is complex (if bound, then $\psi$ would be real) [21, 26]. Consequently, its complex $\psi$ does not have any microstates in the quantum trajectories representation [21]. When encountering an interaction, the $k_{-}$ spectral component of $\psi$ is the proxy for the would-be reflected wave. In this manner, the would-be reflected wave may be considered to be a Majorana entity. This would-be reflected wave acts as a secondary, complementary wave that is entangled with the primary would-be incident wave, which is represented by $k_{+}$ spectral component. In other words, the would-be reflection is already incorporated into the neutrino. The entanglement between the two spectral components produce a dichromatic wave function with compound modulation, Eqs. (8) and (11). In the $\psi$ representation, the neutrino propagates until it encounters a matching flavor-dependent interaction where the complex dichromatic $\psi$ and $\partial _q \psi$ are continuous, $\mathcal {C}^1$, across the interaction [the wave-length and amplitude modulations are not independent of each other, Eq. (19)]. Flavor oscillations are incorporated into the dichromatic $\psi$ for the neutrino by compound modulation, Eqs. (11) and (19). As wavelength modulation, W(q), and amplitude modulation, $[\partial _qW(q)]^{-1/2}$, periodically evolve with q [9], the values for the dichromatic $\psi (q)$, and $\partial _q W(q)$ change with q to produce periodic flavor oscillation. For a deeper development of this algorithm, the interested reader is invited to review Ref. [35], which discusses the non-relativistic tunneling problem from the points of view of both the quantum trajectories and the $\psi$ representations.