Off-shell input and normalization

The time-ordered inputs retain their off-shell local-functional labels until the product is formed. Representation on the free on-shell Fock space is performed afterward. In particular, $\varphi (P\rho )=0$ as a represented free field does not require the time-ordered insertion with that label to vanish.

Our first retarded coefficient is

This specifies the ordering and sign convention.

For this pair only, the normalization requires:

1. the fixed vacuum Wick prescription at one insertion;

2. chronological operator ordering on ordered separated supports and symmetry of the time-ordered product;

3. the independent-field Wick expansion displayed below;

4. extension of each numerical coefficient without increasing its actual fixed-mass relative scaling degree;

5. distributional differentiation of normalized contractions, including their field-equation contacts.

These conditions do not define all higher causal products.