Time evolution of the lepton number of Majorana neutrinos in the Schr\"odinger picture versus Heisenberg picture

TL;DR

This paper investigates the time evolution of Majorana neutrino lepton number expectation values in the Schrödinger picture, demonstrating consistency with the Heisenberg picture and exploring the vacuum structure via Bogolyubov transformations.

Contribution

It provides a detailed analysis of Majorana neutrino lepton number evolution in the Schrödinger picture, linking operator transformations and vacuum states with superpositions.

Findings

Lepton number expectation value matches Heisenberg picture results.

Vacuum states are related through Bogolyubov transformations.

Superposition of energy eigenstates describes time evolution.

Abstract

In this paper, we study the time evolution of the expectation value of Majorana neutrino with the Schr\"odinger picture.The operators with the definite lepton number and operators with the definite mass are related to each other by a Bogolyubov transformation. Then the vacuum with the null lepton number is also related to the vacuum for the massive operator and it is written by the superposition of the vacuum for massive field and Majorana pairs condensed states. We choose the state with a definite lepton number $L$ $=1$ and the momentum ${\bf q}\ne 0$ as an initial state. By writing the state in terms of the superposition of energy eigenstates, we are able to study the time evolution of the state in the Schr\"odinger picture.The expectation value of lepton number operator is computed and it reproduces the same result as that obtained in the corresponding Heisenberg operator.