Explicit formal QED vertex/free-line convention

Adopt the following coefficient convention for finite formal QED bookkeeping in the Minkowski, units, and Clifford conventions of Quantum field quantities and compatible SI normalizations. In particular, $\eta =\mathrm{diag}(-1,1,1,1)$, $h=-\eta$, $\{{\gamma }^a,{\gamma }^b\}=2h^{ab}{I}_4$, and $\mathrm{p̸}:={\gamma }^a{h}_{ab}{p}^b$. The inverse-length mass is $\mu =m_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t},\mathrm{S}\mathrm{I}}c\mathrm{/}\mathrm{\hslash }$ and a declared physical charge $q_{\mathrm{S}\mathrm{I}}$ has dimensionless value $q_{\mathrm{n}}=q_{\mathrm{S}\mathrm{I}}\sqrt{{\mu }_{0}c\mathrm{/}\mathrm{\hslash }}$.

A formal coefficient graph here is a finite directed graph of interaction vertices and internal lines with a momentum covector assigned to each line and momentum conservation imposed at each vertex. It records a coefficient at a fixed finite or formal order; it is not, by this definition, a convergent amplitude or an operator on a continuum interacting Hilbert space. For a Dirac species of dimensionless charge $g$, use $D_a={\mathrm{\partial }}_a-ig{a}_a$ and adopt the vertex factor $ig{\gamma }^a$. For an electron, $g=-e_{\mathrm{n}}$, so the vertex is $-i{e}_{\mathrm{n}}{\gamma }^a$.

For $\mu >0$ and off-shell momentum $p$ with $h(p,p)-{\mu }^2\mathrm{\ne }0$, set $S_0(p)=\frac{\mathrm{p̸}+\mu }{h(p,p)-{\mu }^2}\mathrm{.}$ The adopted fermion-line coefficient is $i{S}_0(p)$. In Feynman gauge the auxiliary covariant photon-line coefficient is $D_{ab}^F(k)=-\frac{i{h}_{ab}}{h(k,k)+i0}=\frac{i{\eta }_{ab}}{h(k,k)+i0}\mathrm{.}$ The symbol $+i0$ denotes the selected Feynman boundary-value convention when a distributional line is used; it does not define products of singular loop distributions. The coefficient $D_{ab}^F$ is a calculational tensor factor, not a declaration of four positive physical photon polarization states.

For the formal matrix realization, use the coefficientwise inverse specified by the planned interface Finite matrix formal kernel and inverse series: if $K_0$ is an invertible finite matrix and $K(z)=K_0+zV$, the formal inverse is the power series in $z$; no convergence in $z$ is claimed. The planned off-shell/tree identities of Off-shell free Dirac Ward telescoping and tree currents fix the $h=-\eta$ slash and the nonzero-denominator conditions. When a time-ordered free distribution is needed, use the actual carrier $H_{\mu }$ of Tempered time-ordered scalar two-point carrier, for which $(\mathrm{\square }-{\mu }^2)H_{\mu }=i{\delta }_0$ and whose definition has no extra prefactor $i$. The distributional use of that carrier is conditional on The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$); the formal coefficient factors above are adopted independently of that optional distributional identification.

This postulate fixes vertex, free-line, and finite/formal coefficient conventions. It does not assert that the formal series converges, that every diagram is gauge independent, that loop products are defined, or that an exact interacting continuum QED state space or scattering operator exists. The corresponding finite/formal and tree-level deductions are separately stated in Qualified formal/tree gauge-current identities.