Bracket relations for relativity groups

TL;DR

This paper derives Poisson bracket relations for relativity groups directly from their coordinate transformation groups, clarifying the origin of constants and simplifying the derivation process, with immediate relevance to quantum mechanics.

Contribution

It provides a simple, rigorous method to derive bracket relations from Galilei and Poincaré groups without using differences, clarifying the role of constants and enabling straightforward quantum conversion.

Findings

Constants appear in Galilei but not in Poincaré brackets.

Derivation is simplified using only first-order coordinate changes.

Method allows direct transition to quantum commutation relations.

Abstract

Poisson bracket relations for generators of canonical transformations are derived directly from the Galilei and Poincar\'e groups of changes of space-time coordinates. The method is simple but rigorous. The meaning of each step is clear because it corresponds to an operation in the group of changes of space-time coordinates. Only products and inverses are used; differences are not used. It is made explicitly clear why constants occur in some bracket relations but not in others, and how some constants can be removed, so that in the end there is a constant in the bracket relations for the Galilei group but not for the Poincar\'e group. Each change of coordinates needs to be only to first order, so matrices are not needed for rotations or Lorentz transformations; simple three-vector descriptions are enough. Conversion to quantum mechanics is immediate. One result is a simpler derivation of the commutation relations for angular momentum directly from rotations. Problems are included.