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Phrase 'the field is in [certain] representation of a [certain] group'

  1. Oct 25, 2013 #1
    I have a question about quantum field theory.

    What does the phrase 'the field is in [certain, e. g. fundamental] representation of a [certain, e. g. SU(2)] group' mean?

    I know mathematical definitions of groups and their representations, but what does this specific phrase mean?
     
  2. jcsd
  3. Oct 25, 2013 #2
    Let's take the Lorentz group. When we say that "the field ##\phi_a(x)## is in the representation ##R## of the Lorentz group" we mean that if we perform a Lorentz transformation ##\Lambda## of our system, the new version of the field is given by ##\phi_a'(x) = {D(\Lambda)_a}^b \phi_b(\Lambda^{-1} x)##. The ##D(\Lambda)## is a matrix (there is one for each possible Lorentz transformation ##\Lambda##) and the set of ##D(\Lambda)## matrices are a representation ##R## of the Lorentz group. For example, if ##D(\Lambda) = 1## for all ##\Lambda## then we have the trivial representation. If the ##D(\Lambda)## are the familiar 4x4 Lorentz matrices, we call this the "vector" representation. And so on.
     
  4. Oct 26, 2013 #3
    Thank you very much!
     
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