# .

Applied to classical field theory, the familiar symplectic Hamiltonian formalism takes the form of instantaneous Hamiltonian formalism on an infinite-dimensional phase space, where canonical coordinates are field functions at some instant of time.[1] This Hamiltonian formalism is applied to quantization of fields, e.g., in quantum gauge theory.

The true Hamiltonian counterpart of classical first order Lagrangian field theory is covariant Hamiltonian formalism where canonical momenta p^\mu_i correspond to derivatives of fields with respect to all world coordinates xμ.[2] Covariant Hamilton equations are equivalent to the Euler-Lagrange equations in the case of hyperregular Lagrangians. Covariant Hamiltonian field theory is developed in the Hamilton-De Donder[3], polysymplectic[4], multisymplectic[5] and k-symplectic[6] variants. A phase space of covariant Hamiltonian field theory is a finite-dimensional polysymplectic or multisymplectic manifold.

Hamiltonian non-autonomous mechanics is formulated as covariant Hamiltonian field theory on fiber bundles over the time axis \mathbb R.

References

^ Gotay, M., A multisymplectic framework for classical field theory and the calculus of variations. II. Space + time decomposition, in "Mechanics, Analysis and Geometry: 200 Years after Lagrange" (North Holland, 1991).
^ Giachetta, G., Mangiarotti, L., Sardanashvily, G., "Advanced Classical Field Theory", World Scientific, 2009, ISBN 9789812838957.
^ Krupkova, O., Hamiltonian field theory, J. Geom. Phys. 43 (2002) 93.
^ Giachetta, G., Mangiarotti, L., Sardanashvily, G., Covariant Hamiltonian equations for field theory, J. Phys. A32 (1999) 6629; arXiv: hep-th/9904062.
^ Echeverria-Enriquez, A., Munos-Lecanda, M., Roman-Roy, N., Geometry of multisymplectic Hamiltonian first-order field theories, J. Math. Phys. 41 (2002) 7402.
^ Rey, A., Roman-Roy, N. Saldago, M., Gunther's formalism (k-symplectic formalism) in classical field theory: Skinner-Rusk approach and the evolution operator, J. Math. Phys. 46 (2005) 052901.