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

Does charge conjugation affect parity?

  1. Jul 8, 2015 #1
    "Notice that these transformations do not alter the chirality of particles. A left-handed neutrino would be taken by charge conjugation into a left-handed antineutrino, which does not interact in the Standard Model." --https://en.wikipedia.org/wiki/C-symmetry

    The excerpt above seems to unambiguously answer this question. But, then:

    "You can easily convince yourself (exercise II.1.9) that the charge conjugate of a left handed field is right handed and vice versa." --Quantum Field Theory in a Nutshell, A. Zee

    These statements appear to be contradictory. What's going on here?

    Also, it does seem easy to convince myself of Zee's comment (following Zee's convention that [itex]\psi \to \psi_c = \gamma^2 \psi^\ast[/itex]):

    Suppose [itex]\psi[/itex] is left-handed (i.e. [itex]P_L \psi = \psi[/itex] and [itex]P_R \psi = 0[/itex]), then
    [tex]P_L \psi_c = P_L \gamma^2 \psi^\ast = \gamma^2 P_R \psi^\ast = \gamma^2 (P_R \psi)^\ast = 0[/tex]
    and
    [tex]P_R \psi_c = P_R \gamma^2 \psi^\ast = \gamma^2 P_L \psi^\ast = \gamma^2 (P_L \psi)^\ast = \psi_c[/tex]
    Therefore, it appears that Zee's comment is correct. Can anyone help me understand why the two quotes above are or are not in contradiction?
     
  2. jcsd
  3. Jul 10, 2015 #2

    Avodyne

    User Avatar
    Science Advisor

    Both statements are technically true, but I think Zee's is misleading. If we work in basis where ##\gamma_5## is diagonal, then a Dirac field ##\Psi## can be written as a left-handed Weyl field ##\chi## stacked on top of a right-handed Weyl field ##\xi^\dagger##,
    [tex]\Psi=\pmatrix{\chi\cr\xi^\dagger}[/tex]
    The charge conjugate field is then
    [tex]\Psi^c=\pmatrix{\xi\cr\chi^\dagger}[/tex]
    Now if we set ##\xi=0##, then we recover Zee's statement (and your algebra). But I think it is more correct to say that the charge conjugate of the left-handed field ##\chi## is the left-handed field ##\xi##. Then, if we use ##\Psi## as a Dirac field for neutrinos, ##\chi## creates left-handed neutrinos, and ##\xi## creates left-handed antineutrinos, which is consistent with the wikipedia statement.
     
  4. Jul 10, 2015 #3
    Ok, thanks for the reply. I think I'm still a little confused, but you've put me in a particular direction to begin investigating this further.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook