Gauge covariance of the prescribed-coupling model: unitary equivalence, the symmetry criterion and the transported propagator

Adopt the postulates P1, P2 and P4 in the reading of Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables clause (a), for a declared isolated system whose energy operator is the prescribed-coupling minimal-coupling operator of Prescribed external electromagnetic coupling P5a: the minimal-coupling model and its gauge transformations: a declared spinless particle of mass parameter $m>0$ and charge parameter $q\in \mathbb{R}$, with admissible prescribed potentials $({\mathbf{A}}_{\mathrm{e}\mathrm{x}\mathrm{t}},{\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}})$ and energy operator $K_{{\mathbf{A}}_{\mathrm{e}\mathrm{x}\mathrm{t}},{\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}}}$ on the domain $W^{2,2}({\mathbb{R}}^3;\mathbb{C})$ (Bounded-coefficient magnetic Schrodinger operator). 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). Let $I\subseteq \mathbb{R}$ be an open interval and let $\chi :{\mathbb{R}}^3\times I\to \mathbb{R}$ be a real gauge function satisfying the two regularity hypotheses of Gauge multiplication preserves H2 and transports the magnetic momentum (for every $t$, $\chi (\cdot ,t)$ is $C^2$ with bounded first and second spatial derivatives, uniformly in $t$; and $t\mapsto \chi (x,t)$ is differentiable with bounded ${\mathrm{\partial }}_t\chi$). Put $\theta (x,t):=\frac{q}{\mathrm{\hslash }}\chi (x,t),A_j^{\mathrm{\prime }}(\cdot ,t):=A_j+D_j\chi (\cdot ,t),{\mathrm{\Phi }}^{\mathrm{\prime }}(\cdot ,t):={\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}}-{\mathrm{\partial }}_t\chi (\cdot ,t),$ let $G(t)$ be multiplication by $e^{i\theta (\cdot ,t)}$, and put $K:=K_{{\mathbf{A}}_{\mathrm{e}\mathrm{x}\mathrm{t}},{\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}}}$, $H^{\mathrm{\prime }}(t):=K_{{\mathbf{A}}^{\mathrm{\prime }}(t),{\mathrm{\Phi }}^{\mathrm{\prime }}(t)}$ and $U(\tau ):=e^{-i\tau K\mathrm{/}\mathrm{\hslash }}$. Then:

(a) Gauge conjugation and unitary equivalence. If $\chi$ does not depend on $t$, then $G:=G(t)$ is independent of $t$, is unitary with $G{W}^{2,2}({\mathbb{R}}^3;\mathbb{C})=W^{2,2}({\mathbb{R}}^3;\mathbb{C})$, and $GK{G}^{-1}=K_{{\mathbf{A}}_{\mathrm{e}\mathrm{x}\mathrm{t}}+\mathrm{\nabla }\chi ,{\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}}},e^{-it{K}_{{\mathbf{A}}_{\mathrm{e}\mathrm{x}\mathrm{t}}+\mathrm{\nabla }\chi ,{\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}}}\mathrm{/}\mathrm{\hslash }}=G{e}^{-itK\mathrm{/}\mathrm{\hslash }}{G}^{-1}(t\in \mathbb{R}):$ the two prescribed-coupling models are related by the fixed unitary $G$.

(b) Symmetry criterion and the global-phase case. For every $t\in I$ the operator $G(t)$ is unitary, preserves $W^{2,2}$, and is a symmetry of the declared dynamics with energy $K$ in the sense of Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables clause (b) if and only if $K_{{\mathbf{A}}^{\mathrm{\prime }}(t),{\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}}}=K_{{\mathbf{A}}_{\mathrm{e}\mathrm{x}\mathrm{t}},{\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}}},$ that is, if and only if transporting the vector potential while keeping the original scalar potential reproduces the original operator. In particular, if $\chi$ is constant, then ${\mathbf{A}}^{\mathrm{\prime }}(t)={\mathbf{A}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$, ${\mathrm{\Phi }}^{\mathrm{\prime }}(t)={\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}}$ and $G(t)=e^{i\theta }I$ for the constant $\theta =q\chi \mathrm{/}\mathrm{\hslash }$, so that every $G(t)$ is a symmetry of the declared dynamics (a global phase). No claim is made in the nonconstant case beyond the displayed criterion.

(c) Transported propagator and the time-dependent model. For the general gauge function of the statement and every $u_0\in W^{2,2}$ and $s\in I$, the curve $u(t):=G(t)U(t-s)G(s{)}^{-1}{u}_0(t\in I)$ lies in $W^{2,2}$, is differentiable into $L^2$ on $I$, satisfies $u(s)=u_0$ and $i\mathrm{\hslash }{u}^{\mathrm{\prime }}(t)=H^{\mathrm{\prime }}(t)u(t)$ for every $t$, and is the unique such curve; the two-parameter family $U^{\mathrm{\prime }}(t,s):=G(t)U(t-s)G(s{)}^{-1}$ consists of unitaries mapping $W^{2,2}$ onto $W^{2,2}$, with $U^{\mathrm{\prime }}(t,t)=I$ and $U^{\mathrm{\prime }}(t,r)U^{\mathrm{\prime }}(r,s)=U^{\mathrm{\prime }}(t,s)$. Moreover $H^{\mathrm{\prime }}(t)=G(t)KG(t{)}^{-1}-q{\mathrm{\partial }}_t\chi (\cdot ,t),$ so the family $G(t)$ relates the autonomous model with energy $K$ to the transformed model whose generator is $G(t)KG(t{)}^{-1}-q{\mathrm{\partial }}_t\chi$; a genuinely time-dependent family $G(t)$ is a family of unitaries indexed by time and is not claimed to be a symmetry of the autonomous dynamics except at those $t$ where the operator identity of (b) holds.

(d) Scope. The statement concerns the prescribed spinless minimal-coupling model with admissible bounded-coefficient potentials and the stated gauge regularity, and is conditional on P1, P2, P4 and P5a and the cited operator analysis. It asserts no Maxwell equation, no source or back-reaction model, no identification of the two coefficient pairs as descriptions of one physical situation (that reading is part of P5a), no statement about singular or unbounded potentials, and no apparatus or empirical result; the gauge multipliers are unitary transformations between coefficient classes and are symmetries of the fixed dynamics exactly under the criterion displayed in (b).

• Gauge transformation is not automatically a symmetry. A gauge transformation replaces the coefficient pair; the multiplier is a symmetry of the fixed dynamics only when the transported magnetic operator with the original scalar potential coincides with the original operator. Constant gauges give the global phase, which is a symmetry; the nonconstant case is left to the criterion, and no general classification of the pairs for which the criterion holds is attempted here.
• Two readings distinguished. The theorem proves the operator identities; the further reading that the gauge-related coefficient pairs describe the same physical situation is part of the adopted postulate P5a and is not a mathematical consequence of the identities proved here.