Operationally induced preferred basis in unitary quantum mechanics

• URL: http://arxiv.org/abs/2601.18856v1
• Date: Mon, 26 Jan 2026 17:22:03 GMT
• Title: Operationally induced preferred basis in unitary quantum mechanics
• Authors: Vitaly Pronskikh,
• Abstract summary: The preferred-basis problem and the definite-outcome aspect of the measurement problem persist even if the detector is modeled unitarily.<n>A change of mathematical type constitutes the core of the 'cut': a structurally necessary interface from group-based kinematics to set-based counting.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The preferred-basis problem and the definite-outcome aspect of the measurement problem persist even if the detector is modeled unitarily, because experimental data are necessarily represented in a Boolean event algebra of mutually exclusive records whereas the theoretical description is naturally formulated in a noncommutative operator algebra with continuous unitary symmetry. This change of mathematical type constitutes the core of the 'cut': a structurally necessary interface from group-based kinematics to set-based counting. In the presented view the basis relevant for recorded outcomes is not determined by the system Hamiltonian alone; it is induced by the measurement mapping, i.e., by the detector channel together with the coarse-grained readout that defines an instrument. The probabilistic mapping is anchored in symmetry and measure theory: by Gleason-type uniqueness (Gleason for projections in $d>2$ and Busch's extension for Positive Operator-Valued Measures (POVMs) including $d=2$), the trace rule is the unique probability measure consistent with additivity over exclusive events and basis-independence of the unitary sector. A compact qubit--pointer model yields an induced unsharp POVM $E_\pm=\tfrac12(\id\pm η\,σ_z)$ with $η$ fixed by pointer resolution, displaying explicitly how the detector induces the relevant basis. Finally, nested-observer paradoxes are tightened into a non-composability lemma: joint assignment of outcome propositions is obstructed unless a joint instrument exists. This relocates the origin of randomness to the stochasticity of the transition rules.

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