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Orbifolds as configuration spaces of systems with gauge symmetries

Abstract

In systems like Yang-Mills or gravity theory, which have a symmetry of gauge type, neither phase space nor configuration space is a manifold but rather an orbifold with singular points corresponding to classical states of non-generically higher symmetry. The consequences of these symmetries for quantum theory are investigated. First, a certain orbifold configuration space is identified. Then, the Schrödinger equation on this orbifold is considered. As a typical case, the Schrödinger equation on (double) cones over Riemannian manifolds is discussed in detail as a problem of selfadjoint extensions. A marked tendency towards concentration of the wave function around the singular points in configuration space is observed, which generically even reflects itself in the existence of additional bound states and can be interpreted as a quantum mechanism of symmetry enhancement.

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Communicated by A. Jaffe

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Emmrich, C., Römer, H. Orbifolds as configuration spaces of systems with gauge symmetries. Commun.Math. Phys. 129, 69–94 (1990). https://doi.org/10.1007/BF02096779

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Keywords

  • Neural Network
  • Manifold
  • Wave Function
  • Phase Space
  • Quantum Mechanism