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Stupid question on Dirac alpha and beta matrices

  1. Jan 18, 2012 #1
    1. The problem statement, all variables and given/known data
    Dirac proposed that a relativistic wave equation that is linear in both space and time (unlike the Klein-Gordon equation, which is second order) has the form
    [itex]i\frac{\partial}{\partial t}\Psi = (\mathbf{\alpha} \cdot \mathbf{p)+\beta m)\Psi[/itex]
    After squaring this, we'd like it to satisfy the equation [itex]E^2=p^2+m^2[/itex]. So, after some algebra, you conclude that α and β must be matrices that satisfy:
    [itex][\alpha_{i}, \alpha_{j}]_{+} \equiv \alpha_{i}\alpha_{j}+\alpha_{j}\alpha_{i}=0 [/itex]
    [itex][\alpha_{i}, \beta]_{+}=0[/itex]
    [itex](\alpha_{i})^2=(\beta)^2=1[/itex]
    Show that α and β are traceless, Hermetian, have eigenvalues +1 and -1, are of even dimension greater than 4
    2. Relevant equations
    The commutation relations I gave above.


    3. The attempt at a solution
    I honestly have no idea where to start...
    Maybe write out the equations in terms of the trace?

    Ugh, I'm such an idiot....
     
  2. jcsd
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