Dirac operator partial integration

Question

When you have an action with bosonic $X$ and fermionic $\psi$ (Majorana) fields and perform a SUSY transformation $\epsilon$ (the constant, infinitesimal parameter of transformation, a real, anticommuting spinor) can you do normal partial integration on the Dirac operator just like you're used doing with a derivative? For example: $$ \begin{align} S &= \int \mathrm d^{2}\sigma\; \bar{\epsilon}[\left(\gamma^{\alpha}\partial_{\alpha}\gamma^{\beta}\partial_{\beta}\right)X(\sigma)]\psi(\sigma)\\ &= \int \mathrm d^{2}\sigma\; \bar{\epsilon}[{\not}\partial{\not}\partial X(\sigma)]\psi(\sigma)\\ &= -\int \mathrm d^{2}\sigma\; \bar{\epsilon}[{\not}\partial X(\sigma)][{\not}\partial\psi(\sigma)] + `boundary` \end{align} $$

This would make calculations much easier because I'm lost with all matrix multiplications and properties of the Gamma matrices...I wanted to check because I couldn't find any information about this.

Answer 1

In general you cannot, but in your special case it works out.

You should be aware of what the objects in your expression actually are, and how they relate to each other. $X$ is a bosonic field and as such does not feel the presence of gamma matrices at all. Your first line could be rewritten as $$ S = \int \mathrm d^2 \sigma \, \bar \epsilon \gamma^\alpha \gamma^\beta \psi(\sigma) \, \partial_\alpha \partial_\beta X(\sigma) $$ The expression $\bar \epsilon \gamma^\alpha \gamma^\beta \psi$ does not have any free spinor indices! Now, you can use partial integration to get $$ S = -\int \mathrm d^2 \sigma \partial_\beta \left(\bar \epsilon \gamma^\alpha \gamma^\beta \psi(\sigma) \right) \partial_\alpha X(\sigma)$$ which is $$ S = -\int \mathrm d^2 \sigma \bar \epsilon \gamma^\alpha \not \partial \psi(\sigma) \, \partial_\beta X$$ and this is your last line.