What is true Hamiltonian field theory?

Classical field theory is Lagrangian theory where field equations are Euler – Lagrange equations derived from a Lagrangian (see my post Classical field theory is complete: the strict geometric formulation).  This is the case of all observable classical fields, which are electromagnetic, gravitational and Dirac fermion fields. One also considers classical non-Abelian gauge fields described by the Yang – Mills Lagrangian. A problem is Hamiltonian field theory.

Applied to field theory, the familiar symplectic Hamiltonian technique takes the form of instantaneous Hamiltonian formalism on an infinite-dimensional phase space, where canonical variables are field functions at some instant of time. The corresponding Hamilton equations are ill defined, and they are not differential equations. Therefore, this Hamiltonian formalism fails to be a counterpart of the Lagrangian one, and it is applied only to quantization of fields, e.g., in quantum gauge theory.

In mechanics, Hamilton equations are first order dynamic equations, and Hamilton formalism is associated to the first order Lagrangian one. Therefore, let us restrict our consideration to first order Lagrangian field theory when a Lagrangian depends on derivatives of fields of not more than first order. Note that Lagrangians of all realistic field models are of this type. These are Lagrangians of gauge and Dirac fermion fields. The second order Hilbert – Einstein Lagrangian of General Relativity is equivalent to the first order one, and Lagrangians of metric-affine gravitation theory, where dynamic variables are both a pseudo-Riemannian metric and a linear connection, also are of first order.

Let us follow the conventional geometric formulation of classical field theory where fields are represented by sections of some fibre bundle Y->X. Their configuration space of first order Lagrangian theory on Y->X is the first order jet manifold JY of Y->X.  A first order Lagrangian L is defined as a density on JY, and it provides the Legendre map L of the configuration space JY to the phase space P which is a finite-dimensional manifold endowed a polysymplectic form. The true Hamiltonian counterpart of classical first order Lagrangian field theory is covariant Hamiltonian formalism on P where canonical momenta correspond to derivatives of fields with respect to all world coordinates, not only the time. Covariant Hamilton equations are equivalent to the Euler – Lagrange equations in the case of hyperregular Lagrangians L when the Legendre map L is a diffeomorphism. If a Lagrangian L is degenerate, we come to a multi-Hamiltonian theory on the constraint subspace L(JY) of the phase space P in general. This is the case of all the above mentioned field theories.

References:

Giachetta, G., Mangiarotti, L., Sardanashvily, G., Advanced Classical Field Theory (World Scientific, 2009)

Giachetta, G., Mangiarotti, L., Sardanashvily, G., Covariant Hamilton equations for field theory, Journal of Physics  A: Theoretical and Mathematical, 32 (1999) 6629-6642

WikipediA: Covariant Hamiltonian field theory