Two-field scalar Wick identity ($T\{\varphi (x)\varphi (y)\}=:\varphi (x)\varphi (y):+{\Delta }_F(x-y)1$)

For a free real scalar field, split $\varphi ={\varphi }^{(+)}+{\varphi }^{(-)}$ into annihilation and creation parts. Moving the creation operators left gives $\varphi (x)\varphi (y)=:\varphi (x)\varphi (y):+[{\varphi }^{(+)}(x),{\varphi }^{(-)}(y)]$. The commutator is the Wightman function. Choosing either time order gives the displayed identity with the Feynman propagator. It is an identity of operator-valued distributions; coincident-point products need a suitable prescription.