Gamma matrices

  1. I'm doing a course which assumes knowledge of Group Theory - unfortunately I don't have very much.

    Can someone please explain this statement to me (particularly the bits in bold):

    "there is only one non-trivial irreducible representation of the Cliford algebra, up to conjugacy"

    FYI The Clifford algebra is just the the relationship between gamma matrices:

    [tex][\gamma_\mu,\gamma_\nu]=2\eta_{\nu\mu}[/tex]

    where [..] is the anticommutator rather than the commutator
     
  2. jcsd
  3. Bill_K

    Bill_K 4,160
    Science Advisor

    The γμ's are initially understood to be abstract objects satisfying the equation you've written. Representation means we assign a matrix to each γμ and interpret the equation as a matrix equation, where the RHS contains the identity matrix I. Conjugacy refers to the fact that for any matrix M, if γμ is a solution then the conjugate set γμ' = M γμ M-1 is also a solution. Irreducible means that the γμ's are not simultaneously block diagonal, nor conjugate to a set that is block diagonal.

    Putting that all together, it means that you can write down any set of γ matrices you can think of that solve the equation, and be assured that any other set γ' you might have chosen instead is related to your set by a conjugation.
     
  4. thanks for the info Bill:)
     
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