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Proving a representation of the Lorentz Algebra from Clifford Algebra/Gamma matrices.

  1. Dec 1, 2011 #1
    Paraphrasing Peskin and Schroeder:

    By repeated use of
    [itex]\left\{ \gamma^{\mu} , \gamma^{\nu} \right\}= 2 g^{\mu\nu} \times \textbf{1}_{n \times n} [/itex] (Clifford/Dirac algebra),
    verify that the n-dimensional representation of the Lorentz algebra,
    [itex]S^{\mu \nu}=\frac{i}{4}\left[\gamma^{\mu},\gamma^{\nu}\right][/itex],
    satisfies the commutation relation
    [itex]\left[J^{\mu \nu},J^{\rho \sigma}\right]=i\left(g^{\nu \rho}J^{\mu \sigma}-g^{\mu \rho}J^{\nu \sigma}-g^{\nu \sigma}J^{\mu \rho}+g^{\mu \sigma}J^{\nu \rho}\right)[/itex].

    I've tried many lengthy computations and always seem to be missing something.
    Most obvious thing to try is just
    [itex]\left[S^{\mu \nu},S^{\rho \sigma}\right]=S^{\mu \nu}S^{\rho \sigma}-S^{\rho \sigma}S^{\mu \nu}=\frac{-1}{16}\left(\left[\gamma^{\mu},\gamma^{\nu}\right]\left[\gamma^{\rho},\gamma^{\sigma}\right]-\left[\gamma^{\rho},\gamma^{\sigma}\right]\left[\gamma^{\mu},\gamma^{\nu}\right]\right)[/itex]
    [itex]=\frac{-1}{16}\left(\left(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu}\right)\left(\gamma^{\rho}\gamma^{\sigma}-\gamma^{\sigma}\gamma^{\rho}\right)-\left(\gamma^{\rho}\gamma^{\sigma}-\gamma^{\sigma}\gamma^{\rho}\right)\left(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu}\right)\right)[/itex]
    [itex]=\frac{-1}{16}\left( \gamma^{\mu} \gamma^{\nu} \gamma^{\rho} \gamma^{\sigma} - \gamma^{\mu} \gamma^{\nu} \gamma^{\sigma} \gamma^{\rho} - \gamma^{\nu} \gamma^{\mu} \gamma^{\rho} \gamma^{\sigma} + \gamma^{\nu} \gamma^{\mu} \gamma^{\sigma} \gamma^{\rho} - \gamma^{\rho} \gamma^{\sigma} \gamma^{\mu} \gamma^{\nu} + \gamma^{\rho} \gamma^{\sigma} \gamma^{\mu} \gamma^{\nu} + \gamma^{\sigma} \gamma^{\rho} \gamma^{\mu} \gamma^{\nu} - \gamma^{\sigma} \gamma^{\rho} \gamma^{\nu} \gamma^{\mu} \right) [/itex]

    and then I've tried a few different commutation relations but to no avail.
    Would be very grateful for any help in finishing this off.
     
    Last edited: Dec 1, 2011
  2. jcsd
  3. Dec 1, 2011 #2

    dextercioby

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    Re: Proving a representation of the Lorentz Algebra from Clifford Algebra/Gamma matri

    This is indeed a difficult computation. You must cheat in a way, in the sense that you already know what the final answer looks like. So the the LHS with those 8 terms must lead to the RHS which has also 8 terms (4 times J, but each J has 2 times gammas). The g's will appear when you use the clifford algebra as

    [tex] \gamma^{\mu}\gamma^{\nu} = 2g^{\mu\nu}-\gamma^{\nu}\gamma^{\mu} [/tex]

    So try to group the 8 terms of 4 gammas into the desired form according to how the J's indices occur in the RHS of what you're trying to prove.
     
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