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On covariant and canonical Hamiltonian formalisms for gauge theories

Alejandro Corichi, Juan D. Reyes, Tatjana Vukasinac

Abstract

The Hamiltonian description of classical gauge theories is a very well studied subject. The two best known approaches, namely the covariant and canonical Hamiltonian formalisms have received a lot of attention in the literature. However, a full understanding of the relation between them is not available, specially when the gauge theories are defined over regions with boundaries. Here we consider this issue, by first making precise what we mean by equivalence between the two formalisms. Then we explore several first order gauge theories, and assess whether their corresponding descriptions satisfy the notion of equivalence. We shall show that, even when in several cases the two formalisms are indeed equivalent, there are counterexamples that signal that this is not always the case. Thus, non-equivalence is a generic feature for gauge field theories. These results call for a deeper understanding of the subject.

On covariant and canonical Hamiltonian formalisms for gauge theories

Abstract

The Hamiltonian description of classical gauge theories is a very well studied subject. The two best known approaches, namely the covariant and canonical Hamiltonian formalisms have received a lot of attention in the literature. However, a full understanding of the relation between them is not available, specially when the gauge theories are defined over regions with boundaries. Here we consider this issue, by first making precise what we mean by equivalence between the two formalisms. Then we explore several first order gauge theories, and assess whether their corresponding descriptions satisfy the notion of equivalence. We shall show that, even when in several cases the two formalisms are indeed equivalent, there are counterexamples that signal that this is not always the case. Thus, non-equivalence is a generic feature for gauge field theories. These results call for a deeper understanding of the subject.
Paper Structure (21 sections, 76 equations, 2 figures)

This paper contains 21 sections, 76 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on covariant and canonical Hamiltonian analysis
  3. Covariant Hamiltonian analysis
  4. Canonical Hamiltonian analysis
  5. Comparison and statement of the problem
  6. Maxwell theory
  7. Covariant approach
  8. Canonical approach
  9. Pontryagin theory
  10. Covariant approach
  11. Canonical approach
  12. Chern-Simons theory on the boundary
  13. Covariant approach
  14. Canonical approach
  15. Maxwell-Pontryagin theory
  16. ...and 6 more sections

Figures (2)

  • Figure 1: Gauge 'plane' on $\Gamma_\text{cov}$: Directions $\delta\bold{A}$ along the fiber $\bar{\Pi}^{-1}(d)$ necessarily represent infinitesimal gauge transformations on $\mathcal{M}$ with support outside $\Sigma_0$. Transverse gauge directions on $\Gamma_\text{cov}$ (infinitesimal gauge transformations on $\mathcal{M}$ which are nontrivial on $\Sigma_0$) should connect fibers over the gauge orbit of $d$ in the constraint surface $\bar{\Gamma}_{\text{can}}$. Note that there are several transverse vectors at $s$ that project to the same vector in $d$.
  • Figure 2: $\tilde{\Omega}$, the pullback under the canonical projection map of the pre-symplectic structure $\bar{\Omega}$ on the constraint surface, defines a pre-symplectic structure on the covariant phase space. Directions $\delta\bold{A}$ along the fiber $\bar{\Pi}^{-1}(d)$ are degenerate directions of $\tilde{\Omega}(s)$. Since $\tilde{\Omega}$ is closed, by Cartan's formula then $\mathcal{L}_{\delta\bold{A}}\tilde{\Omega}=0$. So $\tilde{\Omega}$ also has a well defined projection.