We study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n-dimensional quaternionic vector spaces. For quaternionic linear operators the eigenvalue spectrum consists of n complex values. For these operators we give a necessary and sufficient condition for the diagonalization of their quaternionic matrix representations. Our discussion is also extended to complex linear operators, whose spectrum is characterized by 2n complex eigenvalues. We show that a consistent analysis of the eigenvalue problem for complex linear operators requires the choice of a complex geometry in defining inner products. Finally, we introduce some examples of the left eigenvalue equations and highlight the main difficulties in their solution.

By using a weak measurement technique, we investigated the interplay between the angular and the lateral Goos–Hänchen shift of a focused He–Ne laser beam for incidence near the critical angle. We verified that this interplay dramatically affects the composite Goos–Hänchen shift of the propagated beam. The experimental results confirm theoretical predictions that recently appeared in the literature.

We report about recent results on Dirac wave packets in the treatment of neutrino flavor oscillation where the initial localization of a spinor state implies an interference between positive and negative energy components of mass-eigenstate wave packets. A satisfactory description of fermionic particles requires the use of the Dirac equation as evolution equation for the mass eigenstates. In this context, a new flavor conversion formula can be obtained when the effects of chiral oscillation are taken into account. Our study leads to the conclusion that the fermionic nature of the particles, where chiral oscillations and the interference between positive and negative frequency components of mass-eigenstate wave packets are implicitly assumed, modifies the standard oscillation probability. Nevertheless, for ultrarelativistic particles and sharply peaked momentum distributions, we can analytically demonstrate that these modifications introduce correction factors proportional to $m_{1,2}^2/p_0^2$ which are practically undetectable by any experimental analysis.

We discuss the extension of a version of quaternion quantum mechanics to field theory and in particular to the simplest example, the free scalar field. A previous difficulty with the conservation of four-momentum for the "anomalous" bosonic particles is resolved.