# Covariant Hamiltonian field theory

In mathematical physics, Hamiltonian field theory usually means the symplectic Hamiltonian formalism when applied to classical field theory, that takes the form of the instantaneous Hamiltonian formalism on an infinite-dimensional phase space, and 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 field theory where canonical momenta pμi 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, i.e. the real line ℝ.

## References

1. ^ 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).
2. ^ Giachetta, G., Mangiarotti, L., Sardanashvily, G., "Advanced Classical Field Theory", World Scientific, 2009, ISBN 978-981-283-895-7.
3. ^ Krupkova, O., Hamiltonian field theory, J. Geom. Phys. 43 (2002) 93.
4. ^ Giachetta, G., Mangiarotti, L., Sardanashvily, G., Covariant Hamiltonian equations for field theory, J. Phys. A32 (1999) 6629; arXiv: hep-th/9904062.
5. ^ Echeverria-Enriquez, A., Munos-Lecanda, M., Roman-Roy, N., Geometry of multisymplectic Hamiltonian first-order field theories, J. Math. Phys. 41 (2002) 7402.
6. ^ 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.