Finite-precision quantum scattering: from quantum mechanics to algebraic quantum field theory
File(s)
Preprint version
Author(s)
Edalat, Abbas
Type
Working Paper
Abstract
Standard quantum scattering theory idealises experimental preparation by assigning an exact incoming quantum state. In practice, preparation procedures provide only finite operational information and therefore justify only a finite-precision description of the incoming system. Building on interval quantum mechanics, we develop a finite-precision formulation of quantum scattering in which operational specifications are represented by weakly open convex subsets of state space, called parcels, while the underlying quantum dynamics and observables remain exact.
For reversible scattering, the unitary scattering operator induces an affine weak homeomorphism between incoming and outgoing state spaces. Using the scattering graph, we establish a Transfer Principle for graph parcels, prove exact preservation of finite operational specifications, and show that singleton reduction is preserved under unitary scattering. The framework is illustrated by a one-dimensional rectangular barrier and by three-dimensional scattering in the first Born approximation, where finite beam preparation and detector resolution lead naturally to interval-valued angular predictions. For effective irreversible channels, interval-valued predictions are transported exactly by the adjoint channel, while forward evolution is controlled by trace-norm contraction; information loss is characterized by channel fibres and by the observables retained through the adjoint channel.
We then extend the framework to the simplest relativistic setting of algebraic quantum field theory with an isolated one-particle mass shell. Haag--Ruelle wave operators, spectral projections, and detector observables remain part of the exact theoretical scaffolding, whereas states used as operational inputs are specified only to finite precision. This distinction leads to a finite-precision replacement of the exact-vacuum preparation $A\Omega$: a vacuum parcel and its physical preparation capability are propagated through a normalized local one-particle preparation map. We distinguish operational parcels from preparation families, introduce a regularity condition sufficient for uniform predictions, and recover the exact-vacuum construction as an ideal reference case; Haag--Swieca compactness proves regularity of the corresponding reference families.
For reversible scattering, the unitary scattering operator induces an affine weak homeomorphism between incoming and outgoing state spaces. Using the scattering graph, we establish a Transfer Principle for graph parcels, prove exact preservation of finite operational specifications, and show that singleton reduction is preserved under unitary scattering. The framework is illustrated by a one-dimensional rectangular barrier and by three-dimensional scattering in the first Born approximation, where finite beam preparation and detector resolution lead naturally to interval-valued angular predictions. For effective irreversible channels, interval-valued predictions are transported exactly by the adjoint channel, while forward evolution is controlled by trace-norm contraction; information loss is characterized by channel fibres and by the observables retained through the adjoint channel.
We then extend the framework to the simplest relativistic setting of algebraic quantum field theory with an isolated one-particle mass shell. Haag--Ruelle wave operators, spectral projections, and detector observables remain part of the exact theoretical scaffolding, whereas states used as operational inputs are specified only to finite precision. This distinction leads to a finite-precision replacement of the exact-vacuum preparation $A\Omega$: a vacuum parcel and its physical preparation capability are propagated through a normalized local one-particle preparation map. We distinguish operational parcels from preparation families, introduce a regularity condition sufficient for uniform predictions, and recover the exact-vacuum construction as an ideal reference case; Haag--Swieca compactness proves regularity of the corresponding reference families.
Date Issued
2026-08-29
Citation
2026
Publisher
arXiv
Copyright Statement
© 2026 The Author(s).