Prescribed external electromagnetic coupling P5a: the minimal-coupling model and its gauge transformations

Adopt as a physical assumption of the framework, additional to P1-P5 and valid only for the model of this page, the following prescribed-coupling statement for a spinless particle. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice), the choice hypotheses of the cited magnetic-operator suppliers.

(i) Frame and particle data. Fix a named Newtonian inertial frame in the class adopted by Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators (P5) and a declared spinless particle of mass parameter $m>0$ and charge parameter $q\in \mathbb{R}$; the model state space is the scalar component of the massive model, $L^2({\mathbb{R}}^3;\mathbb{C})$, with the position and momentum operators of the massive model. The frame, the mass and the charge are part of the model specification; the frame is not derived from the potentials, and $m$ and $q$ are empirical parameters of a species when one is identified.

(ii) Admissible prescribed potentials. The external data are an admissible pair $({\mathbf{A}}_{\mathrm{e}\mathrm{x}\mathrm{t}},{\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}})$ in the sense of Bounded-coefficient magnetic Schrodinger operator: real classes ${\mathbf{A}}_{\mathrm{e}\mathrm{x}\mathrm{t}}=(A_1,A_2,A_3)$ with $A_j\in L^{\mathrm{\infty }}({\mathbb{R}}^3;\mathbb{R})$ whose weak first derivatives $D_k{A}_j$ are bounded, and ${\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}}\in L^{\mathrm{\infty }}({\mathbb{R}}^3;\mathbb{R})$ (Weak derivative of a locally integrable function, Integer-order Sobolev spaces and their norms). The potentials are prescribed: no field equation, source model or back-reaction is assumed.

(iii) Energy operator. The model's energy operator is the minimal-coupling magnetic Schrodinger operator $H_{\mathrm{e}\mathrm{x}\mathrm{t}}:=K_{{\mathbf{A}}_{\mathrm{e}\mathrm{x}\mathrm{t}},{\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}}},D(H_{\mathrm{e}\mathrm{x}\mathrm{t}})=W^{2,2}({\mathbb{R}}^3;\mathbb{C}),$ of Bounded-coefficient magnetic Schrodinger operator, that is $H_{\mathrm{e}\mathrm{x}\mathrm{t}}=\frac{1}{2m}{\sum }_{j=1}^3(P_j-q{A}_{\mathrm{e}\mathrm{x}\mathrm{t},j}(Q){)}^2+q{\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}}(Q)$ on the displayed domain. This operator is self-adjoint with domain $W^{2,2}({\mathbb{R}}^3;\mathbb{C})$ by Self-adjointness of the bounded-coefficient magnetic Schrodinger operator, so it satisfies the self-adjointness requirement of the dynamics postulate P4 in the reading of Assigned energy observable, unitary flow and the two pictures of autonomous evolution; the adoption here is the identification of this operator as the energy of the declared particle, which is a model assumption and not a consequence of P1-P5.

(iv) Units. Use the units of Dimensions, SI unit conventions and quantity types of the framework; the additional assignments are: ${\mathbf{A}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$ in $\mathrm{T}\mathrm{m}=\mathrm{V}\mathrm{s}{\mathrm{m}}^{-1}$, ${\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}}$ in V, $q$ in C, so $q{\mathbf{A}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$ has momentum units and $q{\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}}$ energy units. The exponential of clause (vi) uses The complex exponential by its power series and $\chi$ is measured in $\mathrm{J}\mathrm{s}{\mathrm{C}}^{-1}$, so the phase $q\chi \mathrm{/}\mathrm{\hslash }$ is dimensionless.

(v) Field definitions. The coefficient classes in (ii) are static spatial data on $\mathrm{\Omega }:={\mathbb{R}}^3$. For any open time interval $I\subseteq \mathbb{R}$ lift them to ${\tilde{\mathbf{A}}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(x,t):={\mathbf{A}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(x)$ and ${\tilde{\mathrm{\Phi }}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(x,t):={\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}}(x)$ on $\mathrm{\Omega }\times I$. Define $\mathbf{B}:=(D_2{A}_{\mathrm{e}\mathrm{x}\mathrm{t},3}-D_3{A}_{\mathrm{e}\mathrm{x}\mathrm{t},2},D_3{A}_{\mathrm{e}\mathrm{x}\mathrm{t},1}-D_1{A}_{\mathrm{e}\mathrm{x}\mathrm{t},3},D_1{A}_{\mathrm{e}\mathrm{x}\mathrm{t},2}-D_2{A}_{\mathrm{e}\mathrm{x}\mathrm{t},1})\in L^{\mathrm{\infty }}(\mathrm{\Omega }\times I;{\mathbb{R}}^3)\text{in T}\mathrm{.}$ $\mathbf{E}:=-{\mathrm{\nabla }}_x{\tilde{\mathrm{\Phi }}}_{\mathrm{e}\mathrm{x}\mathrm{t}}-{\mathrm{\partial }}_t{\tilde{\mathbf{A}}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\in {\mathcal{D}}^{\mathrm{\prime }}(\mathrm{\Omega }\times I;{\mathbb{R}}^3)\text{in }\mathrm{V}{\mathrm{m}}^{-1}\mathrm{.}$ The curl uses the weak spatial derivatives already required in (ii); ${\mathrm{\partial }}_t{\tilde{\mathbf{A}}}_{\mathrm{e}\mathrm{x}\mathrm{t}}=0$. Here ${\mathcal{D}}^{\mathrm{\prime }}$ denotes vector-valued distributions: each component is a continuous linear functional on $C_c^{\mathrm{\infty }}(\mathrm{\Omega }\times I)$, with continuity meaning that for every compact $K\subset \mathrm{\Omega }\times I$ it is bounded by a constant times the suprema on $K$ of finitely many derivatives of each test function supported in $K$. The displayed derivatives are distributional derivatives. Explicitly, for every real test function $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega }\times I)$ and component $j$, $⟨E_j,\phi ⟩={\int }_{\mathrm{\Omega }\times I}{\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}}(x){\mathrm{\partial }}_{x_j}\phi (x,t)dxdt+{\int }_{\mathrm{\Omega }\times I}{A}_{\mathrm{e}\mathrm{x}\mathrm{t},j}(x){\mathrm{\partial }}_t\phi (x,t)dxdt\mathrm{.}$ The second integral vanishes for the static lift; the first is well defined because ${\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}}\in L^{\mathrm{\infty }}$ and $\phi$ has compact support. The same distributional convention applies to the locally integrable time-dependent coefficients in clause (vi). These are definitions of the model's field symbols; they assert no Maxwell equation, no relation to measured fields, and no coupling of the particle back on the sources.

(vi) Gauge transformations. Let $I\subseteq \mathbb{R}$ be an open interval. A real function $\chi :\mathrm{\Omega }\times I\to \mathbb{R}$ is admissible for this clause when (1) for each $t\in I$, $\chi (\cdot ,t)$ is $C^2$ and its first and second spatial derivatives are bounded uniformly in $t$, and (2) for each $x\in \mathrm{\Omega }$, $t\mapsto \chi (x,t)$ is differentiable on $I$ with ${\mathrm{\partial }}_t\chi$ bounded on $\mathrm{\Omega }\times I$. These bounds make $\chi$ jointly Lipschitz and continuous; its time derivative is measurable as a pointwise limit of time difference quotients, and its spatial derivatives are jointly measurable as limits of spatial difference quotients and agree with their weak derivatives. For every admissible $\chi$, use the unitary $G_{\chi }(t)$ defined in Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables clause (c) on $L^2(\mathrm{\Omega };\mathbb{C})$. The model relates the pair $({\mathbf{A}}_{\mathrm{e}\mathrm{x}\mathrm{t}},{\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}})$ to ${\mathbf{A}}_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{\prime }}={\mathbf{A}}_{\mathrm{e}\mathrm{x}\mathrm{t}}+\mathrm{\nabla }\chi ,{\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{\prime }}={\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}}-{\mathrm{\partial }}_t\chi ,$ and the state by ${\psi }^{\mathrm{\prime }}=G_{\chi }(t)\psi =e^{iq\chi \mathrm{/}\mathrm{\hslash }}\psi$. The operator consequences, the criterion under which the multiplier is a symmetry of a fixed model, and the propagator transported to the transformed pair are proved on this page for the stated regularity; the reading that the two coefficient pairs describe the same physical situation is part of this assumption, not a theorem, and the theorem statements of this page are independent of that reading.

(vii) Status and scope. Clauses (i)-(vi) are adopted model assumptions for the declared potential class only. They assert nothing about a laboratory realization, about field sources, about singular or unbounded potentials, about time-dependent data beyond the hypotheses of the gauge items, about statistical or measurement models, or about the accuracy of the spinless minimal-coupling description of any species; the identification of a physical arrangement with the model is a tested identification in the sense of P7.

• What is assumed and what is proved. The minimal-coupling form of the energy operator, the field definitions and the gauge transformation formulas are adopted; self-adjointness on $W^{2,2}$ is a mathematical theorem cited from the magnetic-operator development, and the covariance, symmetry and propagator consequences are proved on this page.
• Bounded coefficients only. The assumption is restricted to admissible pairs with bounded potentials and bounded weak first derivatives; Coulomb, singular, unbounded and general time-dependent potentials are separate model realizations requiring their own self-adjointness and evolution statements.
• No Maxwell theory. The field symbols of clause (v) are definitions; no Maxwell equation, constitutive relation or measured field value appears.