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  What is the classification of representations of the d-dimensional Poincare group that can be identified with 'particle representations'?

+ 4 like - 0 dislike

What is the classification of representations of the d-dimensional Poincare group that can be identified with 'particle representations'? I am specifically looking for the classification in terms of SU(2) subgroup labels as Srednicki does in chapter 33 of his textbook. 

Edit: Updated to reflect Arnold's clarification.

asked Aug 26, 2014 in Theoretical Physics by DJBunk (80 points) [ revision history ]
edited Aug 28, 2014 by Arnold Neumaier

1 Answer

+ 5 like - 0 dislike

Srednicki assumes in chapter 33 $d=3$ space dimensions. Otherwise (33.11) doesn't make sense. The Lorentz group is in this case a direct product of two $SO(3)$ groups, hence the irreducible representations are labelled by two generalized spin indices, one for each copy of the $SO(3)$. (You can get the latter in terms of a sub $SO(2)$ and corresponding ladder operations.)

However, it is misleading to think of the irreducible representations of the Lorentz group as ''particle representations''. Particles are associated with unitary positive energy representations of the Poincare group, not with arbitrary representations of the Lorentz group.

answered Aug 26, 2014 by Arnold Neumaier (15,787 points) [ revision history ]
edited Aug 28, 2014 by Arnold Neumaier

+1. @Arnold - Thanks for the help clarifying my question and the info. I agree that eqn 33.11 doesn't hold in general. Do you know if it is possible to write down generalized relations and SU(2) labels for arbitrary d dimensions?

For higher $d$ two labels are not enough; for details you'd have to look into a book on group representation theory (perhaps Gilmore or Cornwell). But only the case $d=3$ has a particle interpretation.

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