Where can I find a covariant approach to Poisson brackets?

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Discussion Overview

The discussion centers on the search for a covariant approach to Poisson brackets, particularly in the context of relativistic physics. Participants explore the differences between relativistic and non-relativistic formulations, as well as resources for further reading on the topic.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant is looking for detailed discussions on relativistic Poisson brackets.
  • Another suggests looking into the Peierls bracket and references a book on quantum field theory by DeWitt.
  • Some participants question why Poisson brackets should differ between relativistic and non-relativistic contexts, with one noting that Poisson brackets require a choice of a special time coordinate, which may affect their covariance.
  • A participant provides a link to a resource that discusses the covariant phase space, suggesting it may yield more relevant information than a direct search for relativistic Poisson brackets.
  • Another participant mentions that the Peierls bracket does not single out a time coordinate, implying it may be more suitable for a covariant approach.
  • References to additional resources, including a paper by Marsden et al., are provided as potential avenues for exploring covariant approaches to Poisson brackets.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of a special time coordinate in Poisson brackets and whether this affects their relativistic covariance. The discussion remains unresolved regarding the implications of these differences.

Contextual Notes

Some statements depend on specific definitions of covariance and the role of time coordinates in the formulation of Poisson brackets. The discussion does not resolve these nuances.

luxxio
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i am searching for a detailed discussion on the relativistic poisson brackets. where i can found it?
 
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See about Pierls bracket. For example, see the book on QFT by DeWitt.
 
Why should the Poisson bracket be different for relativistic and non-relativistic fields/particles?
 
crackjack said:
Why should the Poisson bracket be different for relativistic and non-relativistic fields/particles?
Poisson brackets require a choice of a special time coordinate, so they are not relativistic covariant.
 
A Yahoo search for "relativistic Poisson bracket" (it may be picky with correct spelling) yields a single hit: http://landau.rice.edu/~aac/pubs/Phys-Plasmas.pdf

However, a better search phrase is "covariant phase space" which yields ca 142,000 hits on Google. Even better, searching for "covariant phase space"+Larsson yields six hits on Yahoo, all apparently written by a younger myself.
 

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