Time is not an observable of the framework: the conditional timescale bound

Time is a parameter, not an observable. The framework's evolution postulate P4 assigns to an isolated system a self-adjoint energy operator and the flow $U(t)=\exp (-itH\mathrm{/}\mathrm{\hslash })$ with $t$ a real parameter of the curve (Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Assigned energy observable, unitary flow and the two pictures of autonomous evolution). The framework's assigned sharp observables are self-adjoint operators with spectral projection-valued measures and outcome laws (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes); time is not among them: there is no assigned self-adjoint time operator whose outcome law would enter the Robertson relation of The Robertson relation for preparation spreads of assigned sharp observables, and this page introduces none. Consequently the expression $\mathrm{\Delta }H\mathrm{\Delta }t$ is not an instance of the Robertson relation, and the symbol $\mathrm{\Delta }t$ has no definition until a timescale constructed from observables is specified.

What the framework does support. The graph-domain Ehrenfest theorem Graph-domain Ehrenfest identities and the conditional timescale bound is a conditional mathematical statement about a curve $\psi :I\to H$ that is $C^1$ into the graph norm of a self-adjoint $A$, stays in $D(AK)$ (and in $D(KA)$ where stated), has unit norm and satisfies ${\psi }^{\mathrm{\prime }}=-\frac{i}{\mathrm{\hslash }}K\psi$. Under those hypotheses, with $f(t):=⟨A\psi (t),\psi (t)⟩$ and $f^{\mathrm{\prime }}(t_0)\mathrm{\ne }0$, the timescale ${\tau }_A(t_0):=\mathrm{\Delta }A(t_0)\mathrm{/}\mathrm{\mid }{f}^{\mathrm{\prime }}(t_0)\mathrm{\mid }$ satisfies $\mathrm{\Delta }K(t_0){\tau }_A(t_0)\ge \frac{\mathrm{\hslash }}{2}\mathrm{.}$ With $K=H$ the energy, the Heisenberg-picture theorem identifies $f^{\mathrm{\prime }}(t_0)$ with the derivative of the mean of the representative observable when $A$ is bounded and preserves the energy domain, $AD(H_{\mathrm{o}\mathrm{p}})\subseteq D(H_{\mathrm{o}\mathrm{p}})$ (The Heisenberg picture: equivalent predictions and the domain-qualified equation of motion clause (c)). Boundedness alone does not imply this domain-invariance condition. For a domain-qualified, possibly unbounded $A$, the graph-domain theorem still gives the Schroedinger-picture derivative stated above, but no Heisenberg-picture identification is asserted here. The spreads $\mathrm{\Delta }A(t_0)$ and $\mathrm{\Delta }K(t_0)$ are preparation spreads in the sense of Preparation variance and spread of an assigned sharp observable, and the theorem's clause (d) records that when $f^{\mathrm{\prime }}(t_0)=0$ the quotient supplies no timescale at all.

What is not claimed. The timescale ${\tau }_A$ depends on the chosen observable $A$ and on the state: different choices give different numbers, and no universal "lifetime" or "duration" of the system is defined. No time operator, no energy-time Robertson inequality, no claim that an energy uncertainty is a linewidth and no relation such as ${\gamma }_E\approx \mathrm{\hslash }\mathrm{\Gamma }$ is asserted by P1-P6; such statements require additional model assumptions (an apparatus or clock model, a decay model or an analytic continuation) that are outside this page. The heuristic accounts of measurement disturbance in the literature are not identities of the framework and are not used as premises here.

• Relation to the preparation-spread results. The timescale bound is a Robertson-type estimate applied to the pair (energy, chosen observable) along a curve; it does not weaken or strengthen the preparation-spread relation of The Robertson relation for preparation spreads of assigned sharp observables, which concerns a single preparation and two observables.
• No empirical content. No measured lifetime, linewidth or energy uncertainty is recorded here, and no comparison with data is attempted.