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From page 21 of "Conformal Field Theory" by Di Francesco, Mathieu, and Sénéchal, the free Fermion Lagrangian is given by:

$$L=\frac{i}{2}\psi_i T_{ij}\dot{\psi}_j-V(\psi)$$

Where the $\psi$ and $\dot{\psi}$ are Grassmann numbers. The antisymmetric part of $T_{ij}$ couples to: $$\psi_i\dot{\psi}_j-\psi_j\dot{\psi}_i=\psi_i\dot{\psi}_j+\dot{\psi}_i\psi_j$$

The authors state that only the symmetric part of $T_{ij}$ is relevant because this part is the total derivative. How does this statement show that either the above equation is zero or the results contradict the anticommutation relations of the Grassmann variables?

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The anti-symmetric part $T^{(as)}$ of $T$ gives a contribution $\frac{d}{dt} \left( \frac{i}{2} \psi_i T_{ij}^{(as)} \psi_j \right)$ which is a total derivative. Consequently, this term does not contribute to equations of motion.

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    $\begingroup$ I see now, so its irrelevance manifests through its canceling in the Euler-Lagrange equations. $\endgroup$
    – QPhysl
    Commented Jan 25 at 17:51
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    $\begingroup$ that's right... $\endgroup$
    – Prahar
    Commented Jan 25 at 17:55

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