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Simplifying gamma matrices

  1. Dec 5, 2011 #1
    1. The problem statement, all variables and given/known data

    I'm trying to find [itex]P_L \displaystyle{\not}p P_L[/itex] for a left-handed particle.

    (I think the answer is zero...)

    2. Relevant equations

    [itex]P_L = \frac{1}{2} (1-\gamma_5)[/itex] (the left-handed projection operator)
    [itex]\displaystyle{\not} p = \gamma_\mu p^\mu[/itex] (pμ is the 4-momentum)

    μ, γ5 are the gamma matrices)

    3. The attempt at a solution

    So, [itex]P_L \displaystyle{\not} p P_L =
    \frac{1}{4} (1-\gamma_5) \gamma_\mu p^\mu (1-\gamma_5) =
    \frac{1}{4} [ \gamma_\mu p^\mu - \gamma_5 \gamma_\mu p^\mu - \gamma_\mu p^\mu \gamma_5 + \gamma_5 \gamma_\mu p^\mu \gamma_5 ] =
    \frac{1}{4} \gamma_\mu [ p^\mu + \gamma_5 p^\mu - p^\mu \gamma_5 - \gamma_5 p^\mu \gamma_5] [/itex] (*)
    (using the anticommutation relation {γ5μ} = 0).

    Can this be simplified further? Since pμ is a 4x1 matrix, I'm guessing you can't just swap the order of [itex]\gamma_5[/itex] and [itex]p^\mu[/itex].

    However, you can rewrite (*) as [itex]\gamma_\mu P_R p^\mu P_L[/itex], and then because it's a left-handed particle, P_R acting on the momentum gives zero?

    Any help would be appreciated,

    Thanks.
     
    Last edited: Dec 5, 2011
  2. jcsd
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