This defines SM (ϕ) as an element of D (M 2) and by Schwartz's kernel theorem we can associate to it an operator from D(M ) to D (M ). Actually, since LM (f ) is local, the second derivative has support on the diagonal, so SM (ϕ) can be evaluated on smooth functions...
Recall that for P we have the retarded and advanced solutions and we can construct the causal propagator ∆ = ∆R − ∆A. Before I give the explicit formula for the WF set of ∆, I need a one more definition. 3We do not require linearity since in quantum field theory the ...