Massless neutrino mixing

Abstract

Neutrinos are enigmatic subatomic particles. To date, three distinct generations of neutrinos have been discovered. There is compelling evidence that neutrinos can change generation and existing theory then suggests that neutrinos from the three generations have differing, but extremely small rest-masses. However, the contention of this thesis is that the case for non-zero neutrino rest masses is not yet completely convincing. All analyses of massless neutrinos have started from the assumption that they must be described by either Weyl's 2-component equations or Dirac's 4-component equation with rest masses set equal to zero. However, there is another equation, Tokuoka's equation, that describes massless particles with spin 1/2 in a quite different way. In spite of the fact that every solution of Tokuoka's equation is a massless wave function, the equation contains a parameter k, with dimensions of mass. This suggests the enticing possibility of massless neutrino mixing.In Chapter 1 a brief description of subatomic particles is presented including neutrinos. Following this, the massive/massless neutrino debate is summarized.Chapter 2 introduces wave equations used in relativistic quantum mechanics, including Weyl's equations and Dirac's equation. The successful applications of these wave equations are discussed, with an emphasis on the roles of the Lagrangians and gauge transformations. The equations used in quantum electrodynamics are mentioned.In Chapter 3 Harish-Chandra's [1] formulation of Maxwell's equation for the photon is introduced. This equation involves a 10-component wave function comprised of 6 field components and 4 potential components. Tokuoka's [2] equation, which is a spin 1/2 analogue of Harish-Chandra's photon equation is then discussed. Tokuoka's equation has two serious deficiencies. Tokuoka's proposed Lagrangian densities do not have extremals and Tokuoka's Hamiltonian operator is not hermitian with respect to the usual scalar product.In Chapter 4 we extend Tokuoka's equation into multiple generation tensor spaces. This allows the production of appropriate Lagrangian densities. However, as both Tokuoka's equation and our extended Tokuoka equation have Hamiltonian operators that are not hermitian with respect to the usual scalar product, difficulties arise when obtaining meaningful observables from these equations. We eventually find that these difficulties largely disappear by the use of appropriate scalar products acquired from treatments analogous to those employed on both Harish-Chandra's and Dirac's equations.In the first half of Chapter 5, several unsuccessful attempts to produce a massless mixing model based on Tokuoka's work are presented. We conclude that the parameter K found in Tokuoka's equation cannot be used to achieve a massless mixing mechanism. An attempt to produce a massless mixing mechanism based on the time evolutions of the standard deviations of probability densities is presented. This attempt also proved to be unsuccessful, but helped point the way to a more successful approach.In the second half of Chapter 5 we follow Barut's [3] investigation of de Broglie's [4] stationary wave solutions. We find that an adjusted version [5] of these solutions provides some more promising massless mixing models. It should be noted that massive mixing models have two main problems associated with them. Massive particles do not have a fixed handedness and the probablity densities of particles described in massive mixing models have slightly unorthodox time evolutions, at a single neutrino level. We find that the massless mixing models do not have these problems that are associated with their massive mixing model counterparts.