Classical O(N) nonlinear sigma model on the half line: a study on consistent Hamiltonian description

TL;DR

This paper investigates the Hamiltonian structure of the O(N) nonlinear sigma model on a half-line, explicitly constructing Poisson brackets for certain boundary conditions and proving incompatibility for others.

Contribution

It provides a detailed analysis of boundary conditions in the sigma model, explicitly constructing Poisson brackets for some and proving inconsistency for mixed types.

Findings

Explicit Poisson brackets for Neumann and Dirichlet boundaries

Incompatibility of mixed boundary conditions with nontrivial subgroups

Framework for current algebra construction with boundary conditions

Abstract

The problem of consistent Hamiltonian structure for O(N) nonlinear sigma model in the presence of five different types of boundary conditions is considered in detail. For the case of Neumann, Dirichlet and the mixture of these two types of boundaries, the consistent Poisson brackets are constructed explicitly, which may be used, e.g. for the construction of current algebras in the presence of boundary. While for the mixed boundary conditions and the mixture of mixed and Dirichlet boundary conditions, we prove that there is no consistent Poisson brackets, showing that the mixed boundary conditions are incompatible with all nontrivial subgroups of $O((N)$.