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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?

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2 Answers 2

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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

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  • $\begingroup$ Why are the formulations different? $\endgroup$
    – resgh
    Commented Dec 1, 2012 at 6:37
  • $\begingroup$ @namehere: The single particle case is different from multiparticle case as there is no good definition of eigentimes for multiple particles. The two field versions are related, though one is well-developed in the classical case only, and the other in the quantum case. $\endgroup$ Commented Dec 2, 2012 at 14:05
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The SHP (Stuckelberg, Horwitz, Piron) Hamiltonian formulation is manifestly covariant. The equations of motion are

$$\frac{\mathrm{d}x^\mu}{\mathrm{d} \tau} = \frac{\partial K }{\partial p_\mu}$$

$$\frac{\mathrm{d}p^\mu}{\mathrm{d} \tau} = - \frac{\partial K }{\partial x_\mu}$$

$K=K(x^\mu,p^\mu)$ is the covariant Hamiltonian and $\tau$ the invariant evolution parameter (in general it differs from proper time $s$). The basic monograph is Classical Relativistic Many-Body Dynamics

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  • $\begingroup$ What is this evolution parameter? $\endgroup$
    – resgh
    Commented Dec 1, 2012 at 6:37
  • $\begingroup$ $\tau$ is the relativistic generalization of the Newtonian concept of time. It is the parameter that sincronizes the many-particle correlations and labels each dynamical configuration in the covariant $8N$ phase space. $\tau$ was first introduced by Stuckelberg and Feynman although Hortwitz and Piron extended it to the many particle case. $\endgroup$
    – juanrga
    Commented Dec 2, 2012 at 12:08
  • $\begingroup$ The problem with the Horwitz-Piron approach is that it doesn't reproduce the standard results, and hence is irrelevant for the applications. It has too many observables. $\endgroup$ Commented Dec 2, 2012 at 14:07
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    $\begingroup$ @namehere: The basic fields depend (in momentum space coordinates) on arbitrary 4-vectors $p$ rather than only on time/lightlike vectors on one or several mass shells. $\endgroup$ Commented Dec 2, 2012 at 16:19
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    $\begingroup$ @ChoMedit It is nothing of what he mentioned. See slide 13 on kipdf.com/… $\endgroup$
    – juanrga
    Commented Oct 21, 2022 at 12:10

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