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

Charge conjugation in second quantization

  1. Sep 11, 2012 #1
    We know that under charge conjugation the current operator reverses the sign:

    \hat{C} \hat{\bar{\Psi}} \gamma^{\mu} \hat{\Psi} \hat{C} = - \hat{\bar{\Psi}} \gamma^\mu \hat{\Psi}

    Here [itex] \hat{C} [/itex] is the unitary charge conjugation operator. I was wondering should we consider gamma matrix here as also an entity undergoing transformation (like when we prove form-covariance of Dirac equation under any unitary transformation): [itex] \hat{C} \gamma^{\mu} \hat{C} = \gamma^{\prime \mu} [/itex]? Or gamma matrix is something of a structure ensuring element and should not be changed?
  2. jcsd
  3. Sep 11, 2012 #2


    User Avatar
    Science Advisor

    (Forgive me for writing ψ to mean the adjoint.)

    In second quantization, charge conjugation is represented by a unitary operator ℂ. Associated with it is a 4 x 4 matrix C that acts on the spinor indices. According to Bjorken and Drell vol 2, the action is

    ℂψℂ-1 = C-1ψT
    ψ-1 = - ψTC

    where the matrix C has the property

    μC-1 = - (γμ)T

    From this,

    ℂ(ψγμψ)ℂ-1 = - (ψTC)γμ(C-1ψT) = + ψTμ)TψT = + (ψγμψ)T = - ψγμψ.
  4. Sep 14, 2012 #3
    Thank you, Bill. But I still have some points to think about.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Charge conjugation in second quantization
  1. Second Quantization (Replies: 4)

  2. Second quantization (Replies: 6)

  3. Second Quantization (Replies: 1)

  4. Second Quantization (Replies: 8)

  5. Second quantization (Replies: 9)