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In the Supergravity book of Freedman et.al, which uses the signature $(+,-,\dots,-)$, we have defined the charge conjugation matrix for general Clifford Algebra as $(C\Gamma^{(r)})^T = -t_rC \Gamma^{(r)}$, where $\Gamma^{(r)}$ are the basis of the Clifford Algebra with rank $r$ and $t_r = \pm 1$ (Eqn. 3.44). Furthermore, the Majorana Conjugate as $\bar \lambda = \lambda^T C$ and later he states the Majorana Flip Relation (Eqn. 3.51) as

$$ \bar \lambda \gamma_{\mu_1 \dots \mu_r} \chi = t_r \bar \chi \gamma_{\mu_1 \dots \mu_r} \lambda $$

However, when I calculate the expression on the right, I get an additional minus sign:

\begin{align*} \bar \lambda \gamma_{\mu_1 \dots \mu_r} \chi &= \lambda^T C \gamma_{\mu_1 \dots \mu_r} \chi = (\chi^T \gamma^T_{\mu_1 \dots \mu_r} C^T \lambda)^T \\ &= -t_r (\chi^TC \gamma_{\mu_1 \dots \mu_r} \lambda)^T \\ &= - t_r\bar \chi \gamma_{\mu_1 \dots \mu_r} \lambda \end{align*}

where in the last step I use that the expression $\bar \chi\gamma_{\mu_1 \dots \mu_r} \lambda$ is a scalar. He (apparently) explains the lack of the minus sign by saying:

$``$The minus sign obtained by changing the order of Grassmann valued spinor components has been incorporated.$"$

I honestly have no idea what he is talking about and I also don't know where exactly my mistake in the calculation above is wrong.

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  • $\begingroup$ “To negate, or not to negate, that is the freaking question”: $(AB)^T = - B^TA^T$, while $(AB)^\dagger= B^\dagger A^\dagger$, assuming $A$ and $B$ are Grassmann valued. Confused? you should be! See here: physics.stackexchange.com/questions/529496/… $\endgroup$ – MadMax Feb 24 '20 at 17:03
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Recall that in an elementary Linear Algebra course, you proved that for two matrices $A,B$, $(AB)^T=B^T A^T$. Do this in component notation. For this proof you will commute the component of $A$ past the component of $B$. In the case where the components of $A,B$ are anti-commuting (Grassmann) numbers, the same step will give you an extra minus sign. Hence, for this latter case, $(AB)^T=-B^T A^T$.

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Spinor components $\lambda$ and $\chi$ are anti-commuting Grassmann numbers, hence when you change the order of spinors, there will be one minus sign.

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