The interaction picture of the declared driven model: readings, the interaction propagator and picture equivalence of outcome laws

(a) Setting. Adopt the framework data and the declared driven model of The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models in the bounded-control class of its clause (b)(ii): $\mathfrak{H}$ is the model's state space (Hilbert space), $H_0$ is self-adjoint on its exact dense domain $D(H_0)$ (Symmetric, self-adjoint and essentially self-adjoint operators), $V:I\to \mathcal{B}(\mathfrak{H})$ is strongly continuous with self-adjoint values and $v={\sup }_{t\in I}\mathrm{\parallel }V(t)\mathrm{\parallel }<\mathrm{\infty }$ (A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum), $H(t)=H_0+V(t)$ on $D(H_0)$, and $U(t,s)$ is the mild propagator of clause (c)(ii) constructed in Strong bounded interaction picture. The readings of P1 and P2 are those of State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, The state space: density operators, pure and mixed states, and purity and Assigned sharp observables and the outcome law of the framework; all times lie in $I$ and are read in seconds as in Dimensions, SI unit conventions and quantity types of the framework. Assume the Axiom of Countable Choice and the Axiom of Choice as the choice hypotheses of the cited suppliers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

(b) The free-flow factor. The free-flow factor of the model is the autonomous flow of the unperturbed part, $E(t):=e^{-it{H}_0\mathrm{/}\mathrm{\hslash }}(t\in \mathbb{R}),$ the reading of clause (b) of Assigned energy observable, unitary flow and the two pictures of autonomous evolution applied to the assigned energy observable $(H_0,\mathrm{J})$; it is a strongly continuous one-parameter unitary group on all real arguments. This auxiliary group supplies $E(-t)$ and $E(t-s)$ even when those arguments lie outside $I$; the physical driven evaluation times $s,t$ remain in $I$. The interaction-picture control is the family $V_I:I\to \mathcal{B}(\mathfrak{H})$, $V_I(t):=E(-t)V(t)E(t),$ which is strongly continuous with self-adjoint values and $\mathrm{\parallel }{V}_I(t)\mathrm{\parallel }=\mathrm{\parallel }V(t)\mathrm{\parallel }\le v$. The interaction propagator is $U_I(t,s):=E(-t)U(t,s)E(s)(s,t\in I),$ and the model's clause (c)(ii) supplies, as the cited theorem proves, the decomposition $U(t,s)=E(t)U_I(t,s)E(-s)\mathrm{.}$ The interaction propagator is unitary, jointly strongly continuous, satisfies $U_I(s,s)=I$ and the cocycle $U_I(t,r)=U_I(t,s)U_I(s,r)$, and its first-endpoint map is strongly differentiable with $i\mathrm{\hslash }{\mathrm{\partial }}_t{U}_I(t,s)\psi =V_I(t)U_I(t,s)\psi (\psi \in \mathfrak{H}),$ all as supplied by Strong bounded interaction picture; these are readings of that theorem in the model's vocabulary, not new assumptions.

(c) Interaction states and preparations. For a Schroedinger-picture state vector $\psi (t)=U(t,s)\psi (s)$ the interaction-picture state is ${\psi }_I(t):=E(-t)\psi (t),\text{equivalently}{\psi }_I(t)=U_I(t,s){\psi }_I(s),$ and for a preparation $\rho$ with $\rho (t)=U(t,s)\rho (s)U(t,s{)}^{\ast }$ the interaction-picture preparation is ${\rho }_I(t):=E(-t)\rho (t)E(t),$ which is again a preparation of the model by Density operators and scalar spectral probability; for a pure preparation $[\psi (s)]$, $[{\psi }_I(t)]$ is pure. The map $\psi (t)\mapsto {\psi }_I(t)$ is unitary at every $t$, so $\mathrm{\parallel }{\psi }_I(t)\mathrm{\parallel }=\mathrm{\parallel }\psi (t)\mathrm{\parallel }$.

(d) The two pictures assign the same outcome laws. Let $(A,u_A)$ be an assigned sharp observable with $A\in \mathcal{B}(\mathfrak{H})$ bounded and let $A_I(t):=E(-t)AE(t)$ be its interaction-picture representative, which is again a bounded self-adjoint observable with $E_{A_I(t)}(B)=E(-t)E_A(B)E(t)$ for every Borel set $B$ by clause (d) of Heisenberg flow of a bounded operator: conjugates, covariance and the equation of motion applied to the flow $E$ of the self-adjoint $T=-H_0\mathrm{/}\mathrm{\hslash }$. Then for every preparation $\rho$ specified at $s$ and every $t\in I$, ${\mu }_{\rho (t)}^A(B)={\mu }_{{\rho }_I(t)}^{A_I(t)}(B)(B\subseteq \mathbb{R}\text{ Borel}),$ because $\rho (t)=E(t){\rho }_I(t)E(-t)$ and $\mathrm{tr}(E(t){\rho }_{I}E(-t)E_A(B))=\mathrm{tr}({\rho }_{I}E(-t)E_A(B)E(t))$ by cyclicity of the trace; equivalently, for a pure preparation, $\mathrm{\parallel }{E}_A(B)\psi (t)\mathrm{\parallel }=\mathrm{\parallel }{E}_{A_I(t)}(B){\psi }_I(t)\mathrm{\parallel }$. This identity is the precise sense in which the two pictures are two bookkeeping devices for the same model; it is a reading of P2 together with the cited mathematics, not a claim about apparatuses.

(e) Energy-basis transition amplitudes. Let ${\phi }_a,{\phi }_b\in D(H_0)$ be unit eigenvectors, $H_0{\phi }_a=E_a{\phi }_a$, $H_0{\phi }_b=E_b{\phi }_b$, and let $A_{ba}(t,s)$ and $p_{ba}(t,s)$ be the transition amplitude and probability of clause (e) of The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models. Then $A_{ba}(t,s)=e^{is(E_a-E_b)\mathrm{/}\mathrm{\hslash }}⟨U_I(t,s){\phi }_a,{\phi }_b⟩,\text{hence}{p}_{ba}(t,s)=\mid ⟨U_I(t,s){\phi }_a,{\phi }_b⟩{\mid }^2,$ by the decomposition of clause (b) and $E(-s){\phi }_a=e^{is{E}_a\mathrm{/}\mathrm{\hslash }}{\phi }_a$, $⟨E(t)\chi ,{\phi }_b⟩=e^{-it{E}_b\mathrm{/}\mathrm{\hslash }}⟨\chi ,{\phi }_b⟩$; the phases are of modulus one (Real and complex inner product spaces, with the inner product linear in the first argument). The transition probability is therefore computable in either picture, and the definition records this as the reading used by the page's amplitude theorem; no measurement frequency or rate is defined.

(f) Status and scope. Clauses (b)-(e) are readings of the declared driven model, of clauses (b)-(e) of The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models, and of P1 and P2, in the vocabulary of the cited mathematics. The interaction picture is a change of description of the same model: it adds no assumption, no physical process, no measurement and no empirical claim, and it asserts nothing about detectors, records or the laboratory realization of the model.