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Dirac matricies

  1. May 26, 2010 #1
    I wonder if I can chose any 4x4 matrices [tex]\gamma^\mu[/tex] which fullfil anticommutationn relations
    [tex]\{\gamma^\mu,\gamma^\nu \}=2g^{\mu\nu} [/tex] as a matricies
    in Dirac equation:
    [tex]
    i \gamma^\mu \partial_\mu \psi= m \psi
    [/tex].
    What changes in the theory if I chose different matricies?
    (of course I have to consistently use this different matricies)
    What if this matricies has explicit time dependence and I'm
    looking for solutions evolving in time as [tex] \exp (-i\omega t) [/tex].
     
  2. jcsd
  3. May 26, 2010 #2

    dextercioby

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    In the derivation by Dirac of his famous equation the "gamma's" are constant matrices. If you want to, you can pass the time dependency of the spinor wave functions onto the "gamma's" and keep the spinor functions depending on p/x only and not on time anymore.
     
  4. May 26, 2010 #3
    Thanks for answer. I'm interested in slightly more complicated tansformation.
    For example instead of traditional [tex]\gamma[/tex] matrices, let's chose the following:
    [tex] \tilde{\gamma}^0=\cosh at \gamma^0 + \sinh at \gamma^1,
    \tilde{\gamma}^1=\sinh at \gamma^0 + \cosh at \gamma^1 ,
    \tilde{\gamma}^2=\gamma^2,\tilde{\gamma}^3=\gamma^3
    [/tex].
    Is it true that the Dirac equation is still
    [tex] i\tilde{\gamma}^\mu \partial_\mu \psi = m\psi[/tex]
    but I have to use this different matrices everywhere
    (i.e. the coupling with electromagnetic filed would be [tex] A_\mu \psi^\dagger \tilde{\gamma}^\mu \psi[/tex])
     
  5. May 26, 2010 #4

    Ben Niehoff

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    If you want to make the gammas coordinate-dependent, I suspect you may have to change Dirac's equation to

    [itex]i\partial_\mu (\gamma^\mu \psi) = m\psi[/itex]

    However, this is a guess, not based on any rigorous derivation. Try deriving the equation from scratch to be sure.
     
  6. May 26, 2010 #5

    tom.stoer

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    Gamma matrices become coordinate dependent in GR.
     
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