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.

为狄拉克费米子建立了量子力学Bargmann不变量与几何相位之间的广义联系。通过定义适当的量子力学射线和希尔伯特空间，我们将形式主义扩展到马约拉那费米子。然后，假设中微子的轻子数违反马约拉纳质量，我们将狄拉克和马约拉纳类型的Bargmann不变量与CP违约的相变不变测度与马约拉纳中微子联系起来。然后，我们将递归参数化推广到任何单一矩阵的研究中，以包括Majorana费米子，这对于研究中微子混合矩阵可能是有用的。