Quantum Field Theory of Space-like Neutrino

Abstract

We performed a Lorentz covariant quantization of the spin-1/2 fermion field assuming the space-like energy-momentum dispersion relation. We a\-chieved the task in the following steps: ($i$) determining the unitary realizations of the inhomogenous Lorentz group in the preferred frame scenario by means of the Wigner-Mackey induction procedure and constructing the Fock space; ($ii$) formulating the theory in a manifestly covariant way by constructing the field amplitudes according to the Weinberg method; ($iii$) obtaining the final constraints on the amplitudes by postulating a Dirac-like free field equation. Our theory allows to predict all chiral properties of the neutrinos, preserving the Standard Model dynamics. We discussed the form of the fundamental observables, energy and helicity, and show that non-observation of the $+\tfrac{1}{2}$ helicity state of the neutrino and the $-\tfrac{1}{2}$ helicity state of the antineutrino could be a direct consequence of the "tachyoneity" of neutrinos at the free level. We found that the free field theory of the space-like neutrino is not invariant under the C and P transformations separately but is CP-invariant. We calculated and analyzed the electron energy spectrum in tritium decay within the framework of our theory and found an excellent agreement with the recent measurement of KATRIN. In our formalism the questions of negative/imaginary energies and the causality problem does not appear.