Generalized Continuity Equations for Schr\"odinger and Dirac Equations

TL;DR

This paper extends the concept of generalized continuity equations from Schr"odinger systems to Dirac systems, revealing new conservation laws and potential applications in controlling fermionic states.

Contribution

It introduces generalized continuity equations for Dirac equations under $SU(N)$ transformations, expanding the theoretical framework and potential control methods for fermionic quantum systems.

Findings

GCE extends to Dirac systems with $SU(N)$ transformations

Conditions for conserved currents are identified

Potential applications in fermionic state control

Abstract

The concept of the generalized continuity equation (GCE) was recently introduced in [J. Phys. A: Math. and Theor. {\bf 52}, 1552034 (2019)], and was derived in the context of $N$ independent Schr\"{o}dinger systems. The GCE is induced by a symmetry transformation which mixes the states of these systems, even though the $N$-system Lagrangian does not. As the $N$-system Schr\"{o}dinger Lagrangian is not invariant under such a transformation, the GCE will involve source terms which, under certain conditions vanish and lead to conserved currents. These conditions may hold globally or locally in a finite domain, leading to globally or locally conserved currents, respectively. In this work, we extend this idea to the case of arbitrary $SU(N)$-transformations and we show that a similar GCE emerges for $N$ systems in the Dirac dynamics framework. The emerging GCEs and the conditions which lead to the attendant conservation laws provide a rich phenomenology and potential use for the preparation and control of fermionic states.