Hamiltonian study for Chern-Simons and Pontryagin theories

TL;DR

This paper performs a Hamiltonian analysis of Chern-Simons and Pontryagin theories, revealing their topological nature and constraint structure, and extends the Pontryagin theory for a more comprehensive understanding.

Contribution

It provides a detailed Hamiltonian formulation for both theories, including constraint algebra and gauge transformations, and extends the Pontryagin theory for improved analysis.

Findings

Both theories are topological with no local degrees of freedom.

The constraint algebra is closed and consistent with gauge invariance.

Extended analysis of the Pontryagin theory offers a clearer understanding of its structure.

Abstract

The Hamiltonian analysis for the Chern-Simons theory and Pontryagin invariant, which depends of a connection valued in the Lie algebra of SO(3,1), is performed. By applying a pure Dirac's method we find for both theories the extended Hamiltonian, the extended action, the constraint algebra, the gauge transformations and we carry out the counting of degrees of freedom. From the results obtained in the present analysis, we will conclude that the theories under study have a closed relation among its constraints and defines a topological field theory. In addition, we extends the configuration space for the Pontryagin theory and we develop the Hamiltonian analysis for this modified version, finding a best description of the results previously obtained.