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cedricyu803
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Homework Statement
I am reading Srednicki's QFT up to CPT symmetries of Spinors
In eq. 40.42 of
http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf
I attempted to get the 2nd equation:
[tex]C^{-1}\bar{\Psi}C=\Psi^{T}C[/tex]
from the first one:
[tex]C^{-1}\Psi C=\bar{\Psi}^{T}C[/tex]
Homework Equations
[tex]\bar{\Psi}=\Psi^\dagger \beta[/tex]
where numerically [tex]\beta=\gamma^0[/tex]
[tex]C^\dagger=C^{-1}=C^T=-C[/tex]
The Attempt at a Solution
h.c. of the first equation:
[tex]C^{-1}\Psi^\dagger C=(C^{-1}\Psi C)^\dagger=(C\bar{\Psi}^T)^\dagger
=(C(\Psi^\dagger \beta)^T)^\dagger=(C\beta\Psi^\ast)^\dagger=\Psi^T\beta C^\dagger=\Psi^T C\beta[/tex]
So
[tex]C^{-1}\bar{\Psi}C=C^{-1}\Psi^\dagger \beta C=-C^{-1}\Psi^\dagger C \beta=-(\Psi^T C\beta) \beta=-\Psi^{T}C[/tex]
I got an extra minus sign.
However, if I start from takingg transpose of the first equation I got the equation correctly.
What have I done wrong?
Also, for eq. 40.43
A is some general combination of gamma matrices.
Should it not be
[tex]C^{-1}\bar{\Psi}A\Psi C=\Psi^TA\bar{\Psi}^T[/tex]
?
Why are there C's wedging A??
Thanks a lot