Bracket relations for relativity groups

Abstract

Poisson bracket relations for generators of canonical transformations are derived directly from the Galilei and Poincaré groups of space-time coordinate changes. 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 Poincaré 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.