Group Transformations of Semiclassical Gauge Systems

TL;DR

This paper explores the symmetry transformations of semiclassical gauge systems by modeling states as a bundle structure, analyzing automorphisms, and focusing on gauge-invariant sections to understand their properties.

Contribution

It introduces a bundle framework for semiclassical gauge systems and studies automorphisms and gauge invariance within this geometric approach.

Findings

Automorphisms correspond to transformations of bundle sections.

Infinitesimal transformations relate to Lie algebra structures.

Gauge groups act on the semiclassical bundle, affecting gauge-invariant sections.

Abstract

Semiclassical systems being symmetric under Lie group are studied. A state of a semiclassical system may be viewed as a set (X,f) of a classical state X and a quantum state f in the external classical background X. Therefore, the set of all semiclassical states may be considered as a bundle ("semiclassical bundle"). Its base {X} is the set of all classical states, while a fibre is a Hilbert space of quantum states in the external background X. Symmetry transformation of a semiclassical system may be viewed as an automorphism of the semiclassical bundle. Automorphism groups can be investigated with the help of sections of the bundle: to any automorphism of the bundle one assigns a transformation of section of the bundle. Infinitesimal properties of transformations of sections are investigated; correspondence between Lie groups and Lie algebras is discussed. For gauge theories, some points of the semiclassical bundle are identified: a gauge group acts on the bundle. For this case, only gauge-invariant sections of the semiclassical bundle are taken into account.