symmetry - Relativistic Hamiltonian Formulations

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Hamiltonian mechanics and special relativity?

The Hamiltonian formulation is beautifully symmetric. It's a shame that the explicit time derivatives in Hamilton's equations mean that the Hamiltonian formulation is not manifestly Lorentz-covariant. Is there any variant of the Hamiltonian formulation that is manifestly relativistic?

Answer

The covariant Hamiltonian version of relativistic classical or quantum mechanics of a single particle is just like the nonrelativistic one, with time replace by eigentime; see, e.g., Thirring's mathematical physics course.

A covariant Hamiltonian version of relativistic classical field theory is the multisymplectic formalism; see, e.g.,

http://arxiv.org/pdf/math/9807080
http://lanl.arxiv.org/abs/1010.0337

A covariant Hamiltonian version of relativistic quantum field theory is the Tomonaga-Schwinger formalism; see, e.g.,
http://arxiv.org/pdf/gr-qc/0405006
http://arxiv.org/pdf/0912.0556
http://sargyrop.web.cern.ch/sargyrop/SDEsummary.pdf