# Hermitian conjugate of Dirac field bilinear

by phypar
Tags: bilinear, conjugate, dirac, field, hermitian
 P: 11 In the standard QFT textbook, the Hermitian conjugate of a Dirac field bilinear $\bar\psi_1\gamma^\mu \psi_2$ is $\bar\psi_2\gamma^\mu \psi_1$. Here is the question, why there is not an extra minus sign coming from the anti-symmetry of fermion fields?
 P: 321 You don't need to anti-commute anything. The transposition automatically change the position of the fields: $$(\bar \psi_1\gamma_\mu\psi_2)^\dagger=\psi_2^\dagger\gamma_\mu^\dagger \gamma_0\psi_1=\psi_2^\dagger\gamma_0\gamma_\mu\psi_1=\bar \psi_2\gamma_\mu\psi_1$$
P: 11
This is exactly what i don't understand, so in the transposition there is a change of the postion of the fermions fields, but according to the anti-commutation rule of them, shouldn't there be a minus sign? I know there is something wrong in my understanding, but just cannot figure it out.

 Quote by Einj You don't need to anti-commute anything. The transposition automatically change the position of the fields: $$(\bar \psi_1\gamma_\mu\psi_2)^\dagger=\psi_2^\dagger\gamma_\mu^\dagger \gamma_0\psi_1=\psi_2^\dagger\gamma_0\gamma_\mu\psi_1=\bar \psi_2\gamma_\mu\psi_1$$

 Quote by phypar In the standard QFT textbook, the Hermitian conjugate of a Dirac field bilinear $\bar\psi_1\gamma^\mu \psi_2$ is $\bar\psi_2\gamma^\mu \psi_1$. Here is the question, why there is not an extra minus sign coming from the anti-symmetry of fermion fields?
There IS an extra minus sign. $\gamma^\mu$ is Hermitian, but $\bar\psi\gamma^\mu \psi$ is anti-Hermitian. The Hermitian quantity is $i \bar\psi\gamma^\mu \psi$.
 P: 321 I'm not anti-commuting the fields. It's just the definition of transpose. Anti-commuting means, for example, to take $\bar \psi_1\gamma_\mu\psi_2$ and move $\psi_2$ on the other side, i.e. you are writing the same operator in a different way. When you take the transpose you are not rearranging the same operator, it's a new one (the transpose) and it is defined with the inverse order of operators.
 P: 872 Is $\gamma^{\mu}$ hermitian? I am not so sure... $\gamma^{0}$ is but $\gamma^{i}$ is antihermitian. The conjugate of the gamma matrices, defined by Clifford Algebra $\left\{ \gamma^{\mu},\gamma^{\nu}\right\} = 2 n^{\mu \nu} I_{4}$ is given by: $(\gamma^{\mu})^{\dagger} = \gamma^{0}\gamma^{\mu}\gamma^{0}$ In fact you have: $(\bar{\psi_{1}} \gamma^{\mu} \psi_{2} )^{\dagger}=\psi_{2}^{\dagger} (\gamma^{\mu})^{\dagger} (\gamma^{0})^{\dagger} \psi_{1} = \psi_{2}^{\dagger} \gamma^{0}\gamma^{\mu}\gamma^{0}\gamma^{0} \psi_{1} = \bar{\psi_{2}} \gamma^{\mu} \psi_{1}$ The first is by definition of any "matrix", whether commuting or not, $(AB)^{\dagger}= B^{\dagger}A^{\dagger}$. If you now want to anticommute the AB: $(AB)^{\dagger}=-(BA)^{\dagger}= -A^{\dagger}B^{\dagger}=B^{\dagger}A^{\dagger}$.
 P: 321 $\gamma_\mu$ is not Hermitian, but the additional $\gamma_0$ that you get from its conjugate goes together to $\psi^\dagger$ to form $\bar \psi$. I honestly don't think there should be any extra minus