Explicit charged Dirac/Maxwell local coupling model

Adopt a smooth local classical Maxwell–Dirac model on an open region of the flat spacetime specified by Quantum field quantities and compatible SI normalizations. Let $A_{\mathrm{S}\mathrm{I},a}$ be a supplied real $C^{\mathrm{\infty }}$ electromagnetic covector potential and let $\psi :\text{\hspace{0.17em}⁣}U\to {\mathbb{C}}^4$ be a $C^{\mathrm{\infty }}$ commuting spinor field. The charge $q_{\mathrm{S}\mathrm{I}}$ and rest mass $m_{\mathrm{S}\mathrm{I}}>0$ are model inputs; for the electron take $q_{\mathrm{S}\mathrm{I}}=-e_{\mathrm{S}\mathrm{I}}$. Use the component, metric, Clifford, charge, field-normalization, and unit conventions of Charged gauge data and SI-natural conventions, Charge, canonical photon field and observer units, Quantum field quantities and compatible SI normalizations, and the electromagnetic suppliers of Positive transverse free photon model. In particular, $D_a={\mathrm{\partial }}_a-i{q}_{\mathrm{S}\mathrm{I}}{A}_{\mathrm{S}\mathrm{I},a}\mathrm{/}\mathrm{\hslash }={\mathrm{\partial }}_a-i{q}_{\mathrm{n}}{a}_a$ and $\mu =m_{\mathrm{S}\mathrm{I}}c\mathrm{/}\mathrm{\hslash }$.

Adopt the local action density in natural normalization

${\mathcal{L}}_{\mathrm{n}\mathrm{a}\mathrm{t}}=-\frac{1}{4}{f}_{ab}{f}^{ab}+\frac{i}{2}(\bar{\psi }{\gamma }^a{D}_a\psi -(D_a\bar{\psi }){\gamma }^a\psi )-\mu \bar{\psi }\psi ,$

where $f_{ab}={\mathrm{\partial }}_a{a}_b-{\mathrm{\partial }}_b{a}_a$ on the region where the potential is supplied, $\bar{\psi }={\psi }^{†}\beta$, and $D_a\bar{\psi }={\mathrm{\partial }}_a\bar{\psi }+i{q}_{\mathrm{n}}{a}_a\bar{\psi }$. The physical action is $S_{\mathrm{c}\mathrm{l}}=\mathrm{\hslash }{\int }_U{\mathcal{L}}_{\mathrm{n}\mathrm{a}\mathrm{t}}{d}^{4}x$ with $x^0=ct$. For compactly supported local variations this is the adopted classical Maxwell–Dirac formulation; its Euler–Lagrange consequences are not postulated as a proof in this item.

For a smooth gauge function $\chi$ in webers, the adopted local transformation is $A_{\mathrm{S}\mathrm{I},a}\mapsto A_{\mathrm{S}\mathrm{I},a}+{\mathrm{\partial }}_a\chi$ and $\psi \mapsto e^{i{q}_{\mathrm{S}\mathrm{I}}\chi \mathrm{/}\mathrm{\hslash }}\psi$. It leaves $F=dA$ invariant and makes $D_a\psi$ transform by the same phase. This is the covector form of the SI potential rule in Potential representations, gauge transformations and the Lorenz and Coulomb conditions. Gauge functions in the local variation argument are compactly supported; no assertion is made about large gauge transformations, nontrivial bundles, boundary charges, or asymptotic sectors.

The model adopts the separately specified free Dirac representation of Specified bosonic scalar and CAR Dirac model adoption and the positive free photon representation of Positive transverse free photon model as the free-field correspondence used to describe the selected species. This extra correspondence does not quantize the interacting classical action. In particular, this postulate supplies no interacting continuum four-dimensional QED Hilbert space, no globally well-posed coupled PDE, no pointwise operator-valued composite current, no scattering or LSZ construction, and no regulator-removal or nonperturbative-existence result.