The Heisenberg picture: equivalent predictions and the domain-qualified equation of motion

Adopt the state and composition postulate State and composition postulate P1: Hilbert spaces, preparations and composite systems (P1), the observable and probability postulate in the reading of Assigned sharp observables and the outcome law of the framework (P2), and the autonomous-dynamics postulate Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution (P4), and 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 $(H_{\mathrm{o}\mathrm{p}},u_H)$ be the assigned energy observable of a declared isolated system with state space $H$ (The state space of the framework: preparations, pure rays and density operators) and flow $U(t)=\exp (-it{H}_{\mathrm{o}\mathrm{p}}\mathrm{/}\mathrm{\hslash })$ as in Assigned energy observable, unitary flow and the two pictures of autonomous evolution, and let $(A,u_A)$ be a bounded assigned sharp observable: $A\in \mathcal{B}(H)$ is self-adjoint everywhere defined (A bounded linear operator between normed spaces, Self-adjoint, positive, unitary and normal operators) with spectral projection-valued measure $E_A$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure). Put $A_H(t):=U(t{)}^{\ast }AU(t)=U(-t)AU(t)(t\in \mathbb{R})\mathrm{.}$ Then:

(a) The Heisenberg representative is an assigned sharp observable. For every $t$, $A_H(t)$ is bounded and self-adjoint with $\mathrm{\parallel }{A}_H(t)\mathrm{\parallel }=\mathrm{\parallel }A\mathrm{\parallel }$ and spectrum $\sigma (A_H(t))=\sigma (A)$, and its spectral projection-valued measure is $E_{A_H(t)}(B)=U(-t)E_A(B)U(t)(B\subseteq \mathbb{R}\text{ Borel});$ with the declared unit $u_A$ it is again a bounded assigned sharp observable of the framework, and $A_H(0)=A$.

(b) The two pictures assign the same laws. For every preparation $\rho$ (The state space of the framework: preparations, pure rays and density operators), every Borel set $B\subseteq \mathbb{R}$ and every $t$, ${\mu }_{\rho (t)}^A(B)={\mu }_{\rho (0)}^{A_H(t)}(B),$ where ${\mu }_{\cdot }^{\cdot }$ is the outcome law of Assigned sharp observables and the outcome law of the framework and $\rho (t)=U(t)\rho (0)U(t{)}^{\ast }$; for the pure preparation of a unit vector $\psi$ this reads $\mathrm{\parallel }{E}_A(B)U(t)\psi {\mathrm{\parallel }}^2=\mathrm{\parallel }{E}_{A_H(t)}(B)\psi {\mathrm{\parallel }}^2$. In particular the expectation value of the bounded observable agrees, $\mathrm{tr}(\rho (t)A)=\mathrm{tr}(\rho (0)A_H(t))$.

(c) Heisenberg equation. If $AD(H_{\mathrm{o}\mathrm{p}})\subseteq D(H_{\mathrm{o}\mathrm{p}})$, then for every $x\in D(H_{\mathrm{o}\mathrm{p}})$ the curve $t\mapsto A_H(t)x$ is differentiable at every $t$ and $i\mathrm{\hslash }\frac{d}{dt}{A}_H(t)x=[A_H(t),H_{\mathrm{o}\mathrm{p}}]x=(A_H(t)H_{\mathrm{o}\mathrm{p}}-H_{\mathrm{o}\mathrm{p}}{A}_H(t))x\mathrm{.}$

(d) Energy-compatible observables are frozen. If $A$ is compatible with the energy, that is $E_A(B)E_H(C)=E_H(C)E_A(B)$ for all Borel $B,C$ (Assigned energy observable, unitary flow and the two pictures of autonomous evolution clause (f)), then $A_H(t)=A$ for every $t$, and by (b) the outcome law of $(A,u_A)$ is the same in $\rho (t)$ as in $\rho (0)$ for every preparation.

(e) Status. Clauses (a)-(d) are conditional consequences of P1, P2 and P4 and the cited mathematics. They introduce no new physical assumption, no measurement or apparatus model, no collapse, no driven or open dynamics; they make no claim about which laboratory quantity realizes $(A,u_A)$ and record no empirical result.

• The two pictures are not two theories. Step 2.1 is an identity between outcome laws, not a derivation of one picture from the other; the Schroedinger picture and the Heisenberg picture are two ways of computing the same numbers ${\mu }_{\rho (t)}^A(B)$.
• Domain hypothesis in the equation of motion. The hypothesis $AD(H_{\mathrm{o}\mathrm{p}})\subseteq D(H_{\mathrm{o}\mathrm{p}})$ in (c) is exactly what makes the differentiated products defined; the position, momentum and other unbounded quantities of later pages are not bounded observables and are not covered by this theorem.