What are the fundamental commutation/anti-commutation relations between half integer and integer spin fields? For instance, in QED do we have \begin{equation} [\psi(x),A^{\mu}(y)]=0 \end{equation} or \begin{equation} \lbrace\psi(x),A^{\mu}(y)\rbrace=0 \end{equation} ?
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1$\begingroup$ Related (duplicate?): physics.stackexchange.com/q/235589 In particular, though the question isn't exactly the same, the first answer seems to address yours. $\endgroup$– Chris ♦Commented Jan 30, 2018 at 6:50
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They commute; see e.g. this discussion of the $\mathbb{Z}_2$ grading involved.
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$\begingroup$ Specific references where OP can read about Grassmann grading would make this answer much more helpful. Since googling "Grassmann grading" gives only 226 results and it appears to be mostly research papers and conference proceedings. ;) $\endgroup$– Chris ♦Commented Jan 30, 2018 at 7:48
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$\begingroup$ @Chris Hopefully the latest edit helps more. $\endgroup$– J.G.Commented Jan 30, 2018 at 8:33