Generalized probability and current densities: A Field theory

TL;DR

This paper develops a field theory framework with a generalized Lagrangian density for quantum particles, leading to new forms of probability and current densities that satisfy the continuity equation.

Contribution

It introduces a non-Hermitian kinetic term in the Lagrangian for quantum particles, extending the traditional formalism.

Findings

Derived a generalized Schrödinger equation from the new Lagrangian

Found extended probability and current densities satisfying the continuity equation

Established a field theory approach for quantum particles with non-Hermitian kinetic terms

Abstract

We introduce a generalized Lagrangian density - involving a non-Hermitian kinetic term - for a quantum particle with the generalized momentum operator. Upon variation of the Lagrangian, we obtain the corresponding Schr\"odinger equation. The extended probability and particle current densities are found which satisfy the continuity equation.