Appendix D
Global and Local Gauge Symmetries
Gauge symmetries are at the basis of the formulation of very successful and important physical theories, like the standard model of elementary particles and general
relativity and, apparently, they seem to play a crucial role in our understanding of
the physical world. This raises the somewhat philosophical/conceptual question of
the physical/operational meaning of gauge symmetries.
To this purpose, as it will be clear below, it is convenient to separately consider
the global and the local gauge symmetries.
The fact that (global) gauge symmetries are not seen by the observables has been
regarded as a puzzling obstacle for their physical meaning,
28 and has led to the
widespread conclusion that gauge symmetries do not have an empirical/operational
meaning. It becomes then difficult to understand how physical effects may be
explained and traced back to something which is devoid of physical/empirical meaning.
29
Actually, such a hasty conclusion does not take into account the (often overlooked)
fact that a complete description of a physical system involves both its algebra of
observables and its possible states, as stressed in Chap. 11.
30
The physical meaning of a symmetry is more direct if it is operationally defined in
terms of a transformation of the observables, like space translations and rotations, but
it may also emerge in the operational classification of the states, which, for global
28 For a comprehensive account of the philosophical discussions on the empirical meaning of gauge
symmetries, in view of their successful physical consequences, see K. Brading and E. Castellani
eds., Symmetries in Physics: Philosophical Reflections, Cambridge Univ. Press 2003; L. Felline, A.
Ledda, F. Paoli and E. Rossanese, New directions in logic and the philosophy of science, College
Publications 2016.
29 See the contributions by C.A. Martin and J. Earman in Symmetries in Physics: Philosophical
Reflections, quoted above.
30 For a detailed discussion of the operational/experimental description of a physical (not necessarily
quantum) system and of the corresponding mathematical formulation, see F. Strocchi, The physical
Principles of Quantum Mechanics, A Critical Review, Eur. Phys. J. Plus, 127: 12 (2012).
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0
249
Global and Local Gauge Symmetries
Gauge symmetries are at the basis of the formulation of very successful and important physical theories, like the standard model of elementary particles and general
relativity and, apparently, they seem to play a crucial role in our understanding of
the physical world. This raises the somewhat philosophical/conceptual question of
the physical/operational meaning of gauge symmetries.
To this purpose, as it will be clear below, it is convenient to separately consider
the global and the local gauge symmetries.
The fact that (global) gauge symmetries are not seen by the observables has been
regarded as a puzzling obstacle for their physical meaning,
28 and has led to the
widespread conclusion that gauge symmetries do not have an empirical/operational
meaning. It becomes then difficult to understand how physical effects may be
explained and traced back to something which is devoid of physical/empirical meaning.
29
Actually, such a hasty conclusion does not take into account the (often overlooked)
fact that a complete description of a physical system involves both its algebra of
observables and its possible states, as stressed in Chap. 11.
30
The physical meaning of a symmetry is more direct if it is operationally defined in
terms of a transformation of the observables, like space translations and rotations, but
it may also emerge in the operational classification of the states, which, for global
28 For a comprehensive account of the philosophical discussions on the empirical meaning of gauge
symmetries, in view of their successful physical consequences, see K. Brading and E. Castellani
eds., Symmetries in Physics: Philosophical Reflections, Cambridge Univ. Press 2003; L. Felline, A.
Ledda, F. Paoli and E. Rossanese, New directions in logic and the philosophy of science, College
Publications 2016.
29 See the contributions by C.A. Martin and J. Earman in Symmetries in Physics: Philosophical
Reflections, quoted above.
30 For a detailed discussion of the operational/experimental description of a physical (not necessarily
quantum) system and of the corresponding mathematical formulation, see F. Strocchi, The physical
Principles of Quantum Mechanics, A Critical Review, Eur. Phys. J. Plus, 127: 12 (2012).
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0
249