1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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]
    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted

Similar Discussions: Stupid question on Dirac alpha and beta matrices
  1. Dirac gamma matrices (Replies: 2)