Adjoints and propagation

The gauge parameter is taken as a vector, the defect as a covector:

If instead both are written with opposite index placement, the component equations must be raised or lowered, not silently copied.

For classification,

so its massless kernel is $+S_{\mathrm{w}}(t)$. For the scalar commutator,

whose spatial Fourier kernel is $-\sin (t{\Omega }_m)/{\Omega }_m$. Thus the two uses of a causal propagator have opposite orientation by explicit notation, not by an implicit change within a proof.

The positive current construction is

Both index raising operations are included. Changing to a retarded-minus-advanced $q$ would require a corresponding minus sign in $J_C$.