Charge conjugation in second quantization

  • Thread starter LayMuon
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  • #1
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Main Question or Discussion Point

We know that under charge conjugation the current operator reverses the sign:

[tex]
\hat{C} \hat{\bar{\Psi}} \gamma^{\mu} \hat{\Psi} \hat{C} = - \hat{\bar{\Psi}} \gamma^\mu \hat{\Psi}
[/tex]

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?
 

Answers and Replies

  • #2
Bill_K
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(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.
 
  • #3
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Thank you, Bill. But I still have some points to think about.
 

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