Closed fermion loop sign
= Closed fermion loop sign
{title2=$-1$}
A closed <fermion loop> carries an additional minus sign from the odd <Grassmann parity> of the fields in the <Wick theorem> contraction expansion. Equivalently, integrating out <Grassmann fields> gives a <determinant>, whose logarithmic interaction expansion has the fermionic sign relative to a bosonic inverse <determinant>. This sign must be included in addition to propagator and vertex phases.