• DOI:10.1088/1751-8113/40/13/016
• Corpus ID: 18752388

Abelian gerbes as a gauge theory of quantum mechanics on phase space

@article{Isidro2006AbelianGA,
  title={Abelian gerbes as a gauge theory of quantum mechanics on phase space},
  author={Jos{\'e} Maria Isidro and Maurice A. de Gosson},
  journal={Journal of Physics A: Mathematical and Theoretical},
  year={2006},
  volume={40},
  pages={3549 - 3567}
}

• J. Isidro, Maurice A. de Gosson
• Published 14 August 2006
• Physics
• Journal of Physics A: Mathematical and Theoretical

We construct a U(1) gerbe with a connection over a finite-dimensional, classical phase space . The connection is given by a triple of forms A, B, H: a potential 1-form A, a Neveu–Schwarz potential 2-form B, and a field-strength 3-form H = dB. All three of them are defined exclusively in terms of elements already present in , the only external input being Planck's constant ℏ. U(1) gauge transformations acting on the triple A, B, H are also defined, parametrized either by a 0-form or by a 1-form. While H remains gauge invariant in all cases, quantumness versus classicality appears as a choice of 0-form gauge for the 1-form A. The fact that [H]/2πi is an integral class in de Rham cohomology is related to the discretization of symplectic area on . This is an equivalent, coordinate-free reexpression of Heisenberg's uncertainty principle. A choice of 1-form gauge for the 2-form B relates our construction to generalized complex structures on classical phase space. Altogether this allows one to interpret the quantum mechanics corresponding to as an Abelian gauge theory.