Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Classification of the representations of the Lorentz algebra

  1. Oct 6, 2009 #1
    The complexified Lie algebra of the Lorentz group can be written as a direct sum of two commuting complexified Lie algebras of SU(2).

    It is being said, that this enables us to classify the irreducible representations of the Lorentz algebra with two half-integers (m,n). But can someone explain me why this is so? I mean, I know that the irreps. of su(2) are characterized by a half-integer (spin), and since the Lorentz algebra is a direct sum of two su(2)'s, it seems somehow probable that to characterize the irreps. of the Lorentz algebra, we need two such half-integers, but I don't really see how the detailed argument would go like.
     
  2. jcsd
  3. Oct 14, 2009 #2
    The finite-dimensional, irreducible representation of sl(2,C) \oplus sl(2,C) are precisely of the form V \otimes W, where V and W are finite-dimensional, irreducible representations of sl(2,C). The sl(2,C)-irreps V may be classified by a single natural number n, and the irreps V \otimes W of sl(2,C) \oplus sl(2,C) may then be classified by a pair of natural numbers (n,m).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Classification of the representations of the Lorentz algebra
Loading...