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Symplectic realizations and Lie groupoids in Poisson Electrodynamics

Fabio Di Cosmo, Alberto Ibort, Giuseppe Marmo, Patrizia Vitale

Abstract

We define the gauge potentials of Poisson electrodynamics as sections of a symplectic realization of the spacetime manifold and infinitesimal gauge transformations as a representation of the associated Lie algebroid acting on the symplectic realization. Finite gauge transformations are obtained by integrating the sections of the Lie algebroid to bisections of a symplectic groupoid, which form a one-parameter group of transformations, whose action on the fields of the theory is realized in terms of an action groupoid. A covariant electromagnetic two-form is obtained, together with a dual two-form, invariant under gauge transformations. The duality appearing in the picture originates from the existence of a pair of orthogonal foliations of the symplectic realization, which produce dual quotient manifolds, one related with space-time, the other with momenta.

Symplectic realizations and Lie groupoids in Poisson Electrodynamics

Abstract

We define the gauge potentials of Poisson electrodynamics as sections of a symplectic realization of the spacetime manifold and infinitesimal gauge transformations as a representation of the associated Lie algebroid acting on the symplectic realization. Finite gauge transformations are obtained by integrating the sections of the Lie algebroid to bisections of a symplectic groupoid, which form a one-parameter group of transformations, whose action on the fields of the theory is realized in terms of an action groupoid. A covariant electromagnetic two-form is obtained, together with a dual two-form, invariant under gauge transformations. The duality appearing in the picture originates from the existence of a pair of orthogonal foliations of the symplectic realization, which produce dual quotient manifolds, one related with space-time, the other with momenta.
Paper Structure (12 sections, 3 theorems, 28 equations)

This paper contains 12 sections, 3 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Poisson electrodynamics revisited
  3. The electromagnetic field
  4. Gauge transformations
  5. Lie Groupoids and their Lie Algebroids
  6. Bisections of Lie groupoids and the exponential map
  7. Action groupoids and their algebroids
  8. Back to Poisson Electrodynamics
  9. Potentials in the Lie Algebroid setting
  10. Gauge transformations
  11. Covariance and Invariance in the groupoid interpretation of Poisson electrodynamics
  12. Conclusions and Perspectives

Key Result

Theorem 1

Given a Poisson manifold $(M,\Theta)$ and a Poisson spray $V_{\Theta}$, then there exists an open neighborhood $\mathcal{U}\subset T^*M$ of the zero section so that is a symplectic structure on $\mathcal{U}$ and the canonical projection $\pi\,\colon\,\mathcal{U}\,\rightarrow\,M$ is a symplectic realization.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3