Bargmann Invariants, Geometric Phases and Recursive Parametrization with Majorana Fermions

TL;DR

This paper extends the connection between Bargmann invariants and geometric phases from Dirac to Majorana fermions, introducing a recursive parametrization for unitary matrices involving Majorana particles, with applications to neutrino physics.

Contribution

It generalizes the formalism linking Bargmann invariants and geometric phases to Majorana fermions and develops a recursive parametrization for unitary matrices including Majorana fermions.

Findings

Relates Majorana Bargmann invariants to CP violation measures

Defines proper quantum spaces for Majorana fermions

Provides a recursive parametrization useful for neutrino mixing studies

Abstract

A generalized connection between the quantum mechanical Bargmann invariants and the geometric phases was established for the Dirac fermions. We extend that formalism for the Majorana fermions by defining proper quantum mechanical ray and Hilbert spaces. We then relate both the Dirac and Majorana type Bargmann invariants to the rephasing invariant measures of CP violation with the Majorana neutrinos, assuming that the neutrinos have lepton number violating Majorana masses. We then generalize the recursive parametrization for studying any unitary matrices to include the Majorana fermions, which could be useful for studying the neutrino mixing matrix.