5

Possible Duplicate:
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?

Community
  • 1
resgh
  • 1,154
  • 1
    Possible duplicates: http://physics.stackexchange.com/q/32368/2451 and links therein. Related: http://physics.stackexchange.com/q/38286/2451 – Qmechanic Nov 30 '12 at 18:12

2 Answers2

6

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

Arnold Neumaier
  • 44,880
  • Why are the formulations different? – resgh Dec 01 '12 at 06:37
  • @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 02 '12 at 14:05
2

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

juanrga
  • 5,645
  • What is this evolution parameter? – resgh Dec 01 '12 at 06:37
  • $\tau$ 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 02 '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 02 '12 at 14:07
  • @ArnoldNeumaier I'm not going to buy the book :(. What do you mean it has too many observables? – resgh Dec 02 '12 at 15:52
  • 1
    @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 02 '12 at 16:19
  • @ArnoldNeumaier: http://arxiv.org/abs/hep-th/9407075, http://arxiv.org/abs/quant-ph/0103136, http://arxiv.org/abs/physics/0212036, http://arxiv.org/abs/hep-th/9601021, and so on... – juanrga Dec 03 '12 at 11:41
  • @namehere: The SHP theory is more complex than the simple approaches cited above by Neumaier because the SHP theory was designed to study many-body motion. The field-theoretic approaches cited by Neumaier work for "a single particle" or for trivial many-body problems which can be effectively treated as a collection of single particle problems. – juanrga Dec 03 '12 at 11:48
  • @juanrga Arnold says your approach doesn't reproduce the 'standard' results. What say you? – resgh Dec 03 '12 at 11:50
  • @namehere: I believe that I have answered him above with a collection of links to published references regarding 'standard' results and predictions. Did not notice? – juanrga Dec 03 '12 at 11:53
  • @juanrga: e.g., hep-th/9601021 ''calculate certain elementary processes, including Compton scattering and Moeller scattering. These calculations lead to qualitative deviations from the usual scattering cross-sections'' – Arnold Neumaier Dec 03 '12 at 14:01
  • @ArnoldNeumaier: please continue reading where you stop: "which are, however, small effects, but may be visible at small angles near the forward direction." Can you provide a single reference with experimental data that disproves the predictions on the off-shell sector? Moreover, ordinary QED is recovered in the on-shell limit. – juanrga Dec 03 '12 at 15:27
  • I only claimed that it does not reproduce the standard results. – Arnold Neumaier Dec 03 '12 at 15:44
  • @ArnoldNeumaier: Yes you did, but the four references given above show that the theory reproduces the standard results in well-defined limits. Moreover, the SHP theory was initially developed for applications beyond the scope of field theory. This is all very well explained in the Schieve's monograph cited in my answer. – juanrga Dec 03 '12 at 20:04
  • It seems interesting, because I guess we can do everything same as just usual Hamiltonian dynamics in non-relativistic setting for well-given $s$ and parameters. I haven't read the monograph yet, but maybe there always be way to define $s$ right? Great enough.. – ChoMedit Oct 16 '22 at 05:34
  • @ChoMedit It's been a decade since I wrote that. Yes, we can continue to use Hamiltonian dynamics in relativity regimes, but the modern RHD (Relativistic Hamiltonian Dynamics) approach is much better than the manifestly covariant SHP approach. – juanrga Oct 17 '22 at 17:07
  • @juanrga Oh, indeed. This was written in 2012. Very long time ago. Thanks for comment. But still, i have some question. The modern RHD approach, you mention now, is the thing mentioned by Arnold above? or are there some differences or developements? – ChoMedit Oct 19 '22 at 00:53
  • 1
    @ChoMedit It is nothing of what he mentioned. See slide 13 on https://kipdf.com/relativistic-dynamics-in-nuclear-and-particle-physics_5aee2cdf7f8b9ad4178b4570.html – juanrga Oct 21 '22 at 12:10