General U(N) gauge transformations in the realm of covariant Hamiltonian field theory

TL;DR

This paper develops a covariant Hamiltonian framework for field theories, deriving gauge transformation rules and form-invariant Hamiltonians under local U(N) gauge transformations, extending canonical transformation concepts.

Contribution

It introduces a covariant Hamiltonian formulation for gauge fields and derives the general form of gauge-invariant Hamiltonians using generating functions.

Findings

Derived the covariant canonical field equations equivalent to Euler-Lagrange equations.

Formulated gauge transformation rules from generating functions in the Hamiltonian context.

Identified the most general form of a Hamiltonian density invariant under local U(N) gauge transformations.

Abstract

A consistent, local coordinate formulation of covariant Hamiltonian field theory is presented. While the covariant canonical field equations are equivalent to the Euler-Lagrange field equations, the covariant canonical transformation theory offers more general means for defining mappings that preserve the action functional - and hence the form of the field equations - than the usual Lagrangian description. Similar to the well-known canonical transformation theory of point dynamics, the canonical transformation rules for fields are derived from generating functions. As an interesting example, we work out the generating function of type F_2 of a general local U(N) gauge transformation and thus derive the most general form of a Hamiltonian density that is form-invariant under local U(N) gauge transformations.