# Relativistic Hamiltonian Formulations [duplicate]

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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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## marked as duplicate by Qmechanic♦, Manishearth♦, Emilio PisantyDec 10 '12 at 10:19

Possible duplicates: physics.stackexchange.com/q/32368/2451 and links therein. Related: physics.stackexchange.com/q/38286/2451 –  Qmechanic Nov 30 '12 at 18:12

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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Why are the formulations different? –  namehere Dec 1 '12 at 6:37
@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. –  Arnold Neumaier Dec 2 '12 at 14:05

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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What is this evolution parameter? –  namehere Dec 1 '12 at 6:37
$\tau$ 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. –  juanrga Dec 2 '12 at 12:08
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. –  Arnold Neumaier Dec 2 '12 at 14:07
@ArnoldNeumaier I'm not going to buy the book :(. What do you mean it has too many observables? –  namehere Dec 2 '12 at 15:52
@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. –  Arnold Neumaier Dec 2 '12 at 16:19