Re: Poisson brackets in relativistic mechanics

On Jul 19, 2:39 pm, Gen Zhang <genn...@xxxxxxxxx> wrote:
I've been trying to digest Rovelli's work on what he calls
"relativistic mechanics", essentially a generalised classical
mechanics, treating time on the same footing as any other variable. As
I understand it, the geometric structure works as follows:

The covariant polysymplectic approach to field theory -- otherwise
known as "field theory as field theory" (as opposed to "field theory
as mechanics").

There has been a recent surge of interest in the general approach
since the 1979 Lecture Notes in Physics 107. Rovelli is more of an
observer than a central participant in these endeavors and has not
fully encapsulated all of the insights and subtleties in the general
framework.

This has been developed in far greater detail in Sardanashvily (a
major player in this area), et. al.

Just to give you an idea. From the publication list

http://webcenter.ru/~sardan/lp_ep.html

Multimomentum Hamiltonian formalism in field theory, hep-th/ 9403172
Multimomentum Hamiltonian formalism in quantum field theory, hep-th/
9404001
Multimomentum Hamiltonian formalism in field theory. Geometric
supplementary, hep-th/ 9405040
Five lectures on the jet manifold methods in field theory, hep-th/
9411089
Energy-momentum conservation law in Hamiltonian field theory, gr-qc/
9412041
On the bracket problem in covariant Hamiltonian field theory, hep-th/
9903220
Deformation quantization in covariant Hamiltonian field theory, hep-
th/ 0203044
Ten lectures on jet manifolds in classical and quantum field theory,
math-ph/ 0203040
Geometric quantization of relativistic Hamiltonian mechanics, gr-qc/
0208073
The bracket and the evolution operator in covariant Hamiltonian field
theory, math-ph/ 0209001

a VERY small subset of what's there.

Differential geometry of time dependent mechanics, dg-ga/ 9702020

This actually works out the embedding of the "instantaneous time" (or
"field theory as mechanics") within covariant field theory. The
embedding, however, exploiting subtleties in the polysymplectic
framework actually goes *further*, even within this field-theory-as-
mechanics framework, introducing non-trivial elements not otherwise
seen.

Now, my question is: what is the corresponding Poisson bracket
structure?

As you can begin to see, this is an active area of research.

I've outlined some ideas (while discussing an article in the 2007
Quantum Gravity book from Birkhauser) in

Time in Quantum Theory and General Relativity
http://federation.g3z.com/Physics/index.htm#QG2007_1

One of the BIG subtleties (and opportunities) that arise is discussed
there (the example and the issue of the 3+0 vs. 2+1 split), and
alluded to in the Geometry of Lagrangian Dynamics article below.

Also:

The Generalization of the Poisson Bracket to Field Theory
http://federation.g3z.com/Physics/index.htm#PolyPoisson

some notes taken from Kanatchikov (hep-th/9709229).

A reply to and expansion on a recent s.p.r. article

The Geometry of Lagrangian Dynamics
http://federation.g3z.com/Physics/index.htm#LagrangeGeo

This does a close side-by-side comparison of the jet bundle formalism
vs. the more commonly seen and prosaic formalism used in field theory
in the Physics community.

.

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