Covariant description of the canonical formalism

In a gauge theory, one can define the Poisson brackets of gauge-invariant functions ("observables") by three different methods. The first method is based on the constrained Hamiltonian reformulation of the theory. The other two methods, namely, the Peierls method and the covariant symplectic approach, deal directly with the Lagrangian. It is explicitly shown that these three methods are equivalent for an arbitrary gauge theory. The equivalence proof relies on the invariance of the Poisson structure among the observables under the introduction of auxiliary fields.