Conservation And Autonomy Are Distinct

Abstract

Conservation of one observable is distinct from autonomy of an observable space.

Theorem 1.1 (An autonomous observable space need not be stationary).

$$\begin{gathered} \forall n,\mathrm{finite}\left(n\right),H,A\in \mathrm{Mat}\left(n,C\right),\mathrm{star}\left(H\right)=H,\mathrm{star}\left(A\right)=A, \\ ([H,A]=0\Rightarrow \forall t\in R,U\left(H,-t\right)AU\left(H,t\right)=A)\wedge (\mathrm{star}\left(Z\right)=Z\wedge \mathrm{star}\left(X\right)=X\wedge X\in \ker \left(tr\right)\wedge \\ (\forall B\in \ker \left(tr\right),[Z,B]\in \ker \left(tr\right))\wedge [Z,X]\ne 0). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Dynamics/ConservationAutonomySeparation.conservation_and_autonomy_are_distinct (✓ std3). ∎

Source. Repository-derived.

Commentary.

For finite complex matrices, a zero Hamiltonian commutator makes the Heisenberg conjugation of the observable constant at every real time.

The explicit contrast uses the self-adjoint qubit Z and X matrices. The trace-zero observable space contains X and is preserved by commutation with Z, while the commutator of Z and X is nonzero.

References

• Truth anchor: D5/S3/Quantum/Dynamics/ConservationAutonomySeparation.conservation_and_autonomy_are_distinct
• Dependency: D5/S3/Quantum/Dynamics/ProjectionProbabilityFlow
• Dependency: D5/S3/Quantum/FiniteDimensional