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.

Adopt as one selected regulated physical model the finite oriented-graph data of Exact compact U(1) rotor plus finite CAR data, with finite nonempty vertex and link sets, compact $U(1)$ rotor variables on links, finite CAR matter at vertices, fixed integer background $b_x\in \{0,\dots ,s\}$, and the displayed Gauss operators. The mathematical definition specifies ${\mathcal{H}}_{\mathrm{k}\mathrm{i}\mathrm{n}}=L^2({\mathbb{T}}^E)\widehat{\otimes }{\mathcal{F}}_a({\mathbb{C}}^{s\mathrm{\mid }V\mathrm{\mid }})$, the integer-spectrum electric operators $E_e$, link unitaries $U_e$, and the Hamiltonian $H=H_0+V_B+V_F$ on its stated maximal domain $D(H_0)$. This physical adoption does not add an embedding of the graph into space or alter the mathematical definition's operator domains.

For the physical-unit assignment, the link angles, $E_e$, $U_e$, and finite CAR operators are dimensionless. Assign units joules to ${\lambda }_E$, every ${\lambda }_p$, and every entry of $M_x$ and $T_e$ in the displayed Hamiltonian; then $H$ has units joules and $tH\mathrm{/}\mathrm{\hslash }$ is dimensionless with $t$ in seconds and $\mathrm{\hslash }$ in joule-seconds, as in Quantum field quantities and compatible SI normalizations. The graph is finite regulator data, not a coordinate lattice with an assigned spacing.

The adopted reading is limited to a model whose mathematical data are actually constructed, consistent with Physical interpretation of a supplied constructed quantum model. The CAR construction inside Exact compact U(1) rotor plus finite CAR data is conditional on Countable Choice and its inspected draft Fock/CAR supplier interfaces; this postulate does not confer proof acceptance on those carriers. It adopts no experimental result, measurement probability, continuum approximation, Lorentz-invariant continuum theory, or claim that the model describes nature.

Assume the fixed finite-graph model adopted by Specified finite spatial compact U(1) gauge regulator: a finite nonempty oriented graph, $s\ge 1$ finite CAR modes per vertex, fixed backgrounds $b_x\in \{0,\dots ,s\}$, link electric operators $E_e$, the joint Gauss operators $G_x$, and $H=H_0+V_B+V_F$ on the exact maximal domain $D(H_0)$ of Exact compact U(1) rotor plus finite CAR data. Assume the Axiom of Choice and a constant $\mathrm{\hslash }>0$ in joule-seconds. Then the physical sector ${\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}={\bigcap }_{x\in V}\ker G_x$ is nonzero. The restriction $H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:=H{\mathrm{\mid }}_{D(H)\cap {\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}$ is self-adjoint, and ${\mathcal{U}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(t)=e^{-it{H}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}\mathrm{\hslash }},t\in \mathbb{R},$ is a strongly continuous unitary group on ${\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ with the exact derivative domain $D(H)\cap {\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$. This is an exact fixed-regulator result conditional on the stated mathematical suppliers; it asserts no continuum dynamics or continuum QED construction.

Assume the finite-graph model of Specified finite spatial compact U(1) gauge regulator and the Axiom of Choice. Write $\mathcal{U}(t)=e^{-itH\mathrm{/}\mathrm{\hslash }}$ for the unitary group of Finite-regulator physical Gauss-sector existence and dynamics, and let ${\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}={\bigcap }_x\ker G_x$ be its nonzero physical sector.

For a bounded operator $O$ on ${\mathcal{H}}_{\mathrm{k}\mathrm{i}\mathrm{n}}$, define its gauge charge $q=(q_x{)}_{x\in V}\in {\mathbb{Z}}^V$ by $U(\alpha )OU(\alpha {)}^{†}=e^{i{\sum }_x{q}_x{\alpha }_x}O(\alpha \in {\mathbb{T}}^V)\mathrm{.}$ For each normalized $\psi \in {\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$, $⟨\psi ,O\psi ⟩=0$ when $q\mathrm{\ne }0$. A finite product of bounded gauge-charge eigenoperators has zero expectation whenever the sum of its charge vectors is nonzero. If $O$ maps $D(G_x)$ into itself and $[G_x,O]$ extends to a bounded operator, then $⟨\psi ,[G_x,O]\psi ⟩=0$. These statements also hold for $O(t)=\mathcal{U}(t{)}^{†}O\mathcal{U}(t)$; the finite-regulator evolution preserves gauge charge and the domains $D(G_x)$.

For a fixed vertex $x$, distinct real insertion times $t_1,\dots ,t_n$ with $n\ge 1$, bounded operators $O_j$ that map $D(G_x)$ into itself, and bounded commutators $[G_x,O_j]$, define ordinary time ordering by placing the factor with larger time to the left. With $O_j(t_j)=\mathcal{U}(t_j{)}^{†}{O}_j\mathcal{U}(t_j)$ and $G_x(t)=\mathcal{U}(t{)}^{†}{G}_x\mathcal{U}(t)$, the following identity holds as a distribution in $t$ between normalized physical/domain vectors: $\frac{d}{dt}⟨\psi ,\mathcal{T}\text{\hspace{0.17em}⁣}[G_x(t){\prod }_{j=1}^n{O}_j(t_j)]\psi ⟩={\sum }_{j=1}^n\delta (t-t_j)⟨\psi ,\mathcal{T}\text{\hspace{0.17em}⁣}([G_x,O_j(t_j)]{\prod }_{i\mathrm{\ne }j}{O}_i(t_i))\psi ⟩\mathrm{.}$ The products are evaluated on the stated common domains and $G_x$ is bosonic, so no grading sign is inserted.

For a region $R\subseteq V$, define its boundary links by $\mathrm{\partial }R=\{e\in E:\text{exactly one endpoint of }e\text{ lies in }R\},$ with $ϵ(e,R)=1$ for a link directed out of $R$ and $-1$ for a link directed into $R$. On ${\mathcal{D}}_R={\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\cap {\bigcap }_{e\in \mathrm{\partial }R}D(E_e)$, put $Q_R^{\mathrm{l}\mathrm{a}\mathrm{t}}={\sum }_{x\in R}(n_x-b_{x}I)$. If bounded $O$ preserves ${\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$, maps each boundary $D(E_e)$ into itself, and $E_{e}O\varphi =O{E}_e\varphi$ for every $\varphi \in D(E_e)$ and every $e\in \mathrm{\partial }R$, then $[Q_R^{\mathrm{l}\mathrm{a}\mathrm{t}},O]\psi =0$ for every $\psi \in {\mathcal{D}}_R$. Consequently, an operator that changes $Q_R^{\mathrm{l}\mathrm{a}\mathrm{t}}$ by a fixed nonzero charge cannot satisfy all these preservation and boundary-commutation hypotheses while acting nontrivially on that domain.

There is also a separate conditional continuum implication; it does not assert that a continuum representation exists. Let ${\mathcal{H}}_c$ be a Hilbert space and $D\subseteq {\mathcal{H}}_c$ a common invariant dense domain. For real $f,g_k\in C_c^{\mathrm{\infty }}({\mathbb{R}}^3)$, let $\rho (f),E_k(g_k):D\to D$ be linear assignments and put $E(\mathbf{g})={\sum }_{k=1}^3{E}_k(g_k)$ for real $\mathbf{g}=(g_1,g_2,g_3)\in C_c^{\mathrm{\infty }}({\mathbb{R}}^3;{\mathbb{R}}^3)$. Fix ${\epsilon }_0>0$ and assume $\rho (f)=-{\epsilon }_0{\sum }_{k=1}^3{E}_k({\mathrm{\partial }}_{k}f)(f\in C_c^{\mathrm{\infty }}({\mathbb{R}}^3))\mathrm{.}$ Let $K\subset {\mathbb{R}}^3$ be compact. Suppose bounded $O$ maps $D$ into itself and $E(\mathbf{g})O\psi =OE(\mathbf{g})\psi$ for every $\psi \in D$ whenever $\mathrm{supp}\mathbf{g}$ is disjoint from $K$. For integers $R\ge R_0$, suppose real ${\chi }_R\in C_c^{\mathrm{\infty }}({\mathbb{R}}^3)$ equals $1$ on a neighborhood of $K$, $\mathrm{supp}\mathrm{\nabla }{\chi }_R$ is disjoint from $K$, and, with $Q_R^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}:=\rho ({\chi }_R)=-{\epsilon }_{0}E(\mathrm{\nabla }{\chi }_R)$, there is $Q:D\to D$ such that $Q_R^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}\psi \to Q\psi$ and $Q_R^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}O\psi \to QO\psi$ for every $\psi \in D$. Then $[Q,O]\psi =0$ on $D$. If also $[Q,O]\psi =qO\psi$ on $D$ for a fixed $q\mathrm{\ne }0$, boundedness of $O$ implies $O=0$ on ${\mathcal{H}}_c$. This clause is conditional on the displayed common-domain, Gauss, locality and limit hypotheses only.

Use the coefficient convention of Explicit formal QED vertex/free-line convention and formal ring of Finite matrix formal kernel and inverse series. Let $\zeta$ denote finite graph link data, $z$ a formal indeterminate, and $K(z;\zeta )=K_0(\zeta )+zV(\zeta )\in M_d(\mathbb{C}\mathrm{⟦}z\mathrm{⟧})$ with $d\ge 1$ and $K_0(\zeta )\in G{L}_d(\mathbb{C})$. For each vertex-angle tuple $\theta$, let ${\zeta }^{\theta }$ and the invertible diagonal matrix $D_{\theta }$ be the finite gauge action and phase matrix of Finite-graph U(1) phases for matrix kernels. Assume the coefficient matrices satisfy $K_0({\zeta }^{\theta })=D_{\theta }{K}_0(\zeta )D_{\theta }^{-1},V({\zeta }^{\theta })=D_{\theta }V(\zeta )D_{\theta }^{-1}\mathrm{.}$ Let $S(z;\zeta )$ be the coefficientwise formal inverse specified by Finite matrix formal kernel and inverse series. Then $S(z;{\zeta }^{\theta })=D_{\theta }S(z;\zeta )D_{\theta }^{-1},\det K(z;{\zeta }^{\theta })=\det K(z;\zeta )\mathrm{.}$ For a real vertex tuple $\theta$, put $\mathrm{\Theta }={\mathrm{diag}}_{x\in X}({\theta }_x{I}_r)$, ${\zeta }^s:={\zeta }^{s\theta }$, and $D_s:=D_{s\theta }=e^{is\mathrm{\Theta }}$. For any coefficient family $A_s=A(z;{\zeta }^s)$, write $\delta A:={\frac{d}{ds}{A}_s\mid }_{s=0}$ coefficientwise. Assume also that $K_0({\zeta }^s)$ and $V({\zeta }^s)$ vary differentiably along the gauge orbit ${\zeta }^s$. Coefficientwise differentiation then gives $\delta S=i[\mathrm{\Theta },S]=-S(\delta K)S\mathrm{.}$ These are finite/formal identities only; the formal variable is not assigned a numerical value and no convergence is asserted.

For the tree-current clause, take $\mu >0$ and four future-shell momenta $p,p^{\mathrm{\prime }},q,q^{\mathrm{\prime }}$ with $h(p,p)=h(p^{\mathrm{\prime }},p^{\mathrm{\prime }})=h(q,q)=h(q^{\mathrm{\prime }},q^{\mathrm{\prime }})={\mu }^2$, in inverse-length units, and put $k=p-p^{\mathrm{\prime }}=q^{\mathrm{\prime }}-q$. Let $J_1^a=\bar{u}(p^{\mathrm{\prime }}){\gamma }^{a}u(p),J_2^b=\bar{u}(q^{\mathrm{\prime }}){\gamma }^{b}u(q),$ where $\bar{u}=u^{†}\beta$ and the spinors satisfy the corresponding positive-energy Dirac equations $(\mathrm{p̸}-\mu )u(p)=0$, $(\mathrm{̸}{p}^{\mathrm{\prime }}-\mu )u(p^{\mathrm{\prime }})=0$, $(\mathrm{q̸}-\mu )u(q)=0$, $(\mathrm{̸}{q}^{\mathrm{\prime }}-\mu )u(q^{\mathrm{\prime }})=0$ and their Dirac-adjoint left equations, in the $h=-\eta$ convention of Explicit formal QED vertex/free-line convention. For a finite scalar $c(k)$, set $\mathrm{\Delta }{D}_{ab}(k)=c(k)k_a{k}_b$. Then $J_1^a\mathrm{\Delta }{D}_{ab}(k)J_2^b=0.$ For either displayed conserved current, the contraction with a covector ${ϵ}_a$ is unchanged by ${ϵ}_a\mapsto {ϵ}_a+\rho k_a$ for finite $\rho \in \mathbb{C}$. If $c(k)$ is a rational coefficient containing $1\mathrm{/}h(k,k)$, require $h(k,k)\mathrm{\ne }0$ at the selected momentum; a distributional $+i0$ boundary value is not a pointwise finite $c(k)$. The adopted off-shell fermion coefficient has denominator $h(p,p)-{\mu }^2$ and is used only when that denominator is nonzero. No internal-line singular product is used here. The tree clause is limited to these explicit finite contractions and does not assert gauge independence for arbitrary diagrams.