Gaussian free vacuum and time-ordered carrier

The physical field in the selected model is the map in [F1], and its action on the common finite-particle domain is supplied by the mathematical construction [F2]. The vacuum $\mathrm{\Omega }$ is the zero-particle vector in that definition.

For real $f$, [F2] identifies the physical Weyl operator as $W_{\mu }(f)=W_{\mathrm{S}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{l}}(\sqrt{2}{K}_{\mu }f)$. Substituting $h=\sqrt{2}{K}_{\mu }f$ into [F3] gives $⟨\mathrm{\Omega },W_{\mu }(f)\mathrm{\Omega }{⟩}_2=\exp (-\mathrm{\parallel }\sqrt{2}{K}_{\mu }f{\mathrm{\parallel }}_2^2\mathrm{/}4)=\exp (-\mathrm{\parallel }{K}_{\mu }f{\mathrm{\parallel }}_2^2\mathrm{/}2)$.

For an ordered product, write its first field as creation plus annihilation. The creation term has zero pairing against the left vacuum. Commute the annihilation term through the later fields using the CCR in [F2]; each commutator is the associated two-point pairing, and the final annihilator kills the right vacuum. Thus the $n$-point moment is the sum, over the possible partner of the first field, of that two-point pairing times the $(n-2)$-point moment of the remaining ordered fields. Starting with vacuum moment $1$ at $n=0$, induction gives zero for odd $n$ and the sum over pairings for even $n$.

The distribution $H_{\mu }$ is defined by the exact partial spatial Fourier formula and extension in [F4]. That supplier proves its temperedness, the ordered Wightman restrictions off coincidence, and the Green equation with the stated $(-+++)$ convention. Therefore these properties hold for the selected scalar model, with no extra factor $i$ in the definition.