Gauge symmetries for the classification of the physical states

This note focuses the problem of motivating the use of gauge symmetries (being the identity on the observables) from general principles, beyond their practical success, starting from global gauge symmetries and then by emphasizing the substantially different role of local gauge symmetries. In the latter case, a deterministic time evolution of the local field algebra, necessary for field quantization, requires a reduction of the full local gauge group G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}$$\end{document} to a residual local subgroup Gr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}_r$$\end{document} satisfying suitable conditions. A non-trivial residual local gauge group allows for the use of a local field algebra, otherwise precluded if G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}$$\end{document} is reduced to the identity. Moreover, in the non-abelian case a residual Gr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}_r$$\end{document} allows to exploit its topology, which provides the (gauge invariant) topological invariants which classify the vacuum structure with important physical effects. Furthermore, it provides a general mechanism of spontaneous symmetry breaking without Goldstone bosons.