Conserved Noether Currents, Utiyama's Theory of Invariant Variation, and Velocity Dependence in Local Gauge Invariance

TL;DR

This paper extends previous work on gauge invariance by demonstrating the existence of additional conserved Noether currents when gauge fields depend on velocity coordinates, using a generalized approach inspired by Utiyama and Mills.

Contribution

It introduces a more general framework for gauge fields that depend on velocity coordinates, revealing new conserved currents beyond those identified in earlier studies.

Findings

Additional conserved Noether currents are identified.

Handling gauge fields as functions of velocity coordinates extends existing theories.

The approach preserves information while revealing new invariants.

Abstract

The paper discusses the mathematical consequences of the application of derived variables in gauge fields. Physics is aware of several phenomena, which depend first of all on velocities (like e.g., the force caused by charges moving in a magnetic field, or the Lorentz transformation). Applying the property of the second Noether theorem, that allowed generalised variables, this paper extends the article by Al-Kuwari and Taha (1991) with a new conclusion. They concluded that there are no extra conserved currents associated with local gauge invariance. We show, that in a more general case, there are further conserved Noether currents. In its method the paper reconstructs the clue introduced by Utiyama (1956, 1959) and followed by Al-Kuwari and Taha (1991) in the presence of a gauge field that depends on the co-ordinates of the velocity space. In this course we apply certain (but not full) analogies with Mills (1989). We show, that handling the space-time coordinates as implicit variables in the gauge field, reproduces the same results that have been derived in the configuration space (i.e., we do not lose information), while the proposed new treatment gives additional information extending those. The result is an extra conserved Noether current.