Foundations of Physics

, Volume 44, Issue 12, pp 1317–1335

On the Relation Between Gauge and Phase Symmetries

Article

DOI: 10.1007/s10701-014-9817-x

Cite this article as:
Catren, G. Found Phys (2014) 44: 1317. doi:10.1007/s10701-014-9817-x

Abstract

We propose a group-theoretical interpretation of the fact that the transition from classical to quantum mechanics entails a reduction in the number of observables needed to define a physical state (e.g. from \(q\)and\(p\) to \(q\)or\(p\) in the simplest case). We argue that, in analogy to gauge theories, such a reduction results from the action of a symmetry group. To do so, we propose a conceptual analysis of formal tools coming from symplectic geometry and group representation theory, notably Souriau’s moment map, the Mardsen–Weinstein symplectic reduction, the symplectic “category” introduced by Weinstein, and the conjecture (proposed by Guillemin and Sternberg) according to which “quantization commutes with reduction”. By using the generalization of this conjecture to the non-zero coadjoint orbits of an abelian Hamiltonian action, we argue that phase invariance in quantum mechanics and gauge invariance have a common geometric underpinning, namely the symplectic reduction formalism. This stance points towards a gauge-theoretical interpretation of Heisenberg indeterminacy principle. We revisit (the extreme cases of) this principle in the light of the difference between the set-theoretic points of a phase space and its category-theoretic symplectic points.

Keywords

Quantum mechanics Gauge theories Moment map Symplectic reduction Symplectic “category” 

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Laboratoire SPHERE (UMR 7219)Université Paris Diderot - CNRSParisFrance
  2. 2.Facultad de Filosofía y LetrasUniversidad de Buenos Aires - CONICETBuenos AiresArgentina