Based on what I've learned, I gather the Poincaré group is the group of isometries of Minkowski spacetime and it is a non-abelian Lie group.

Why is it non-abelian?

Or perhaps rather, does the fact that it is non-abelian mean anything significant for the differences we see between pre-SR and SR theory?

  • $\begingroup$ Do you know that already 3D spatial rotations are non-abelian? $\endgroup$ – Qmechanic Jan 30 '15 at 22:50
  • $\begingroup$ The group of rotations of ordinary 3-space is non-abelian as well $\endgroup$ – doetoe Jan 30 '15 at 22:50
  • 2
    $\begingroup$ @doetoe: Touche! $\endgroup$ – Qmechanic Jan 30 '15 at 22:51
  • $\begingroup$ @Qmechanic Oh, is that related to the right-handed rule idea? I haven't really learned much about this subject formally from a geometrical perspective. But I'm planning to. $\endgroup$ – Stan Shunpike Jan 30 '15 at 23:13

There is nothing special about the group of isometries being non-abelian:

Already the simplest isometry you encounter in physics, that of rotations in ordinary Euclidean three space plus (or, rather, semi-direct product with) translations, is non-abelian, since the group of rotations $\mathrm{SO}(3)$ alone is non-abelian - it is not the same to first rotate about one axis and then about another or to do so in reverse.

If you've got more than one dimension, you will always have non-abelian symmetry groups.


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