Local Gauss law and local gauge symmetries in QFT

TL;DR

This paper explores the fundamental relationship between local gauge symmetries and local Gauss laws in quantum field theory, showing how gauge symmetries naturally emerge from the structure of local field algebras and physical states.

Contribution

It demonstrates that local gauge symmetries can be derived from the properties of local field algebras and the implementation of Local Gauss Laws in QFT, providing a new perspective on their foundational role.

Findings

Local gauge symmetries arise from local field algebra representations.

Local Gauss Laws are valid on physical states and linked to gauge symmetries.

The Gauss operator characterizes realizations of gauge QFT in local algebras.

Abstract

Local gauge symmetries reduce to the identity on the observables, as well as on the physical states (apart from reflexes of the local gauge group topology) and therefore their use in Quantum Field Theory (QFT) asks for a justification of their strategic role. They play an intermediate role in deriving the validity of Local Gauss Laws on the physical states (for the currents which generate the related global gauge group); conversely, we show that local gauge symmetries arise whenever a vacuum representation of a local field algebra ${\cal{F}}$ is used for the description/construction of physical states satisfying Local Gauss Laws, just as global compact gauge groups arise for the description of localizable states labeled by superselected quantum numbers. The above relation suggests that the Gauss operator, which by locality cannot vanish in ${\cal{F}}$, provides an intrinsic characterization of the realizations of a gauge QFT in terms of a local field algebra and of the related local gauge symmetries.