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

Charge Conjugate Dirac Field

  1. Jan 22, 2010 #1

    I'm trying to work my way through Halzen and Martin's section 5.4. I'd appreciate if someone could answer the following question:

    How does

    [tex]j^{\mu}_{C} = -e\psi^{T}(\gamma^{\mu})^{T}\overline{\psi}^{T}[/tex]


    [tex]j^{\mu}_{C} = -(-)e\overline{\psi}\gamma^{\mu}\psi[/tex]

    ? Is there some identity I'm missing?

    Thanks in advance.

  2. jcsd
  3. Jan 23, 2010 #2
  4. Jan 24, 2010 #3
    transpose the entire thing, then use the fact that the psi-bar contains a gamma_0 matrix
  5. Jan 24, 2010 #4
    Why? If I transpose the entire thing, I get the next line without a minus sign. But why do I transpose? Not sure I follow you..
  6. Jan 24, 2010 #5
    ok I can get this:


    so performing a complex conjugation one gets

    {j_C^{\mu}}^* -e\overline{\psi}{\gamma^{\mu}}\psi
    Last edited: Jan 24, 2010
  7. Jan 24, 2010 #6
    I don't think you understand my question here. The two expressions are equal. But are you asking me to transform one to the other by performing a transpose followed by a complex conjugation (in other words asking me to take the Hermitian adjoint)? That is, to prove A = B, I should take the Hermitian adjoint of A and find it to be equal to B. Is that what you're saying?

    (Do you intend to utilize the fact that the current density 4 vector is real? If so, we should merely be taking the complex conjugate.)
  8. Jan 24, 2010 #7
    can you just for completeness write down the four current and the C- transformation?

    there are a couple of conventions out there you know..
  9. Jan 24, 2010 #8
    ok if you go to peskin page 70, if you have it then you can work it you I think, with Halzens definitions I have no clue sorry
  10. Jan 25, 2010 #9
    the current is a spinor scalar, it has no spinor indecies, so do a transpose in spinor space and use that the \psi's anticommute.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook