General $U(N)$ Gauge Transformations in the Realm of Covariant Hamiltonian Field Theory

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 $\fancyscript{H}$ that is form-invariant under local $U(N)$ gauge transformations.