= Gaussian generating functional for a Dirac field
{c}
{title2=$Z[J,\bar J]=Z[0,0]e^{-i\bar J K_F^{-1}J}$}
For a free <Dirac field> with action $\int\bar\psi K\psi$ and independent odd sources coupled as $\bar\psi J+\bar J\psi$, translation invariance of the <Berezin integral> and completion of the square give $Z[J,\bar J]=Z[0,0]\exp(-i\bar J K_F^{-1}J)$. Here $K_F^{-1}$ has <Feynman i-epsilon prescription> boundary conditions, and each bilinear includes its spacetime integrations. A left <Grassmann derivative> with respect to $\bar J$, followed by a right <Grassmann derivative> with respect to $J$, has normalized value $-iK_F^{-1}$ at zero sources. Multiplying by $-1$ removes the two insertion factors $i^2$ and yields the <Dirac propagator> $S_F=iK_F^{-1}$. Keeping source order fixed prevents a spurious <fermionic sign>.
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