Fisher Information and Electromagnetic Interacting Dirac Spinors

In earlier works, it was demonstrated that Schr\"{o}dinger's equation, which includes interactions with electromagnetic fields, can be derived from a fluid dynamic Lagrangian framework. This approach treats the system as a charged potential flow interacting with an electromagnetic field. The emergence of quantum behavior was attributed to the inclusion of Fisher information terms in the classical Lagrangian. This insight suggests that quantum mechanical systems are influenced not just by electromagnetic fields but also by information, which plays a fundamental role in driving quantum dynamics. This methodology was extended to Pauli’s equations by relaxing the constraint of potential flow and employing the Clebsch formalism. While this approach yielded significant insights, certain terms remained unexplained. Some of these unresolved terms appear to be directly related to aspects of relativistic Dirac theory. In a recent work, the analysis was revisited within the context of relativistic flows, introducing a novel perspective for deriving relativistic quantum theory but neglecting the interaction with electromagnetic fields for simplicity. This is rectified in the current work showing the implications of the field in the current context.