Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Dirac equation and friends

  1. Feb 28, 2008 #1
    Dirac equation and friends :)

    I was playing with Dirac equations and deriving some usefull details,
    Note sure for a calculation, is all the math right?

    we require for a pure Lorentz trasf that the spinor field trasform linearly as:

    [tex]\psi'(x')=S(\Lambda)\psi(x)[/tex] (1)

    where [tex]x'=\Lambda x[/tex] and [tex]S(\Lambda)[/tex] is a linear operator that we can write follow:

    [tex]S(\Lambda)=exp(-\frac{i}{a}\sigma_{ab}\omega^{ab})[/tex] (2)

    If we take the [tex]\dagger[/tex] and use Dirac gammas on (1) we obtain the transformation law for the dagger spinor:


    and using gamma zero we have:


    Now what i want to show is that:


    correct me in the following equalities if i make something wrong:


    Now using the properties of gamma zero on the matrices:

    [tex]\gamma_{0}^2=Id[/tex]; [tex]\gamma_{0}\sigma_{ab}\gamma_{0}=(\sigma_{ab})^{\dagger}[/tex]

    we get to:

    [tex]\gamma^{0}exp(\frac{i}{a}(\omega^{ab})^{\dagger}(\sigma_{ab})^{\dagger})\gamma^{0}=exp(\gamma^{0}\frac{i}{a}(\omega^{ab})^{\dagger}\gamma^{0}\gamma^{0}(\sigma_{ab})^{\dagger}\gamma^{0})=exp(\frac{i}{a}\sigma_{ab}\omega^{ab})\equiv S^{-1}(\Lambda)[/tex]

    The last follow from (2).

    Am i correct??

    thanks in advance.

  2. jcsd
  3. Feb 28, 2008 #2
    Sorry i wrote a instead of 4 in the S operator underneath the i...
  4. Feb 28, 2008 #3


    User Avatar
    Science Advisor
    Homework Helper

    It looks right to me. And you can edit your posts here for 24 hours so you can fix things like the a versus 4 detail.
  5. Mar 1, 2008 #4
    thanx because my prof....Im sure will be very miticoulous on this "Not so important" calculus....
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?