1. The group SU2 defines the spacial part of the space-time invarience group and is independent of whether Galilean or Lorentz invarience is invoked. The generators of rotations in three dimensional space are the angular momentum operators; ergo, rotation invarience is equivalent to angular momentum conservation. SU2 is homomorphic to the group O3 in that the former admits of 1/2 integer intrinsic quantized angular momentum (e.g protons have intrinsic spin 1/2, the N* resonance has intrinsic spin 3/2, etc.)

The time part of the space-time invarience group depends on whether Lorentz or Galilean invarience is invoked. In either case the bottom line is that the laws of physics are invarient under transformations from one inertial frame to another. The difference is that Galilean invarience admits of a frame of absolute rest, Lorentz invarience does not.

2. The Lorentz group which includes all Lorentz transformations (e.g. rotations in space-time) is the group SL2(C), again because it admits of 1/2 integer intrinsic angular momentum (spin).

3. The Poincare group combines the Lorentz group and the 4 dimentional translation group as the semi-direct product of the two. Again, invarience under static space translations is equivalent to conservation of linear momentum (the generatior of space translations) while invarience under static time translations is equivalent to conservation of energy (the generator of time translations).

]]>You really need to read some Christopher Moore books, you’ll be laughing so hard that you’ll want to read the book again, right after finishing. (Which I in fact did, with 3 of his books. …finished one, then immediately read it again.)

You also need to add “Sex Drugs and Cocoa Puffs” to the list that you’re planning on reading. …and some dr. seuss…

]]>Don’t forget the space-time groups which are the mathematical basis of the conservation laws of physics (I discussed these in a comment on Carrolls’ blog about 2 months ago). For instance, invarience under coordinate rotations (the group SU(2) space time as contrasted with the group SU(2) isotopic spin) is equivalent to conservation of angular momentum.

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