Deriving the Kijowski Arrival-Time POVM from the Schrödinger Current: Minimal Positivity and Uniqueness

Abstract

Quantum backflow refers here to the appearance of a negative Schrödinger current for a state whose momentum support is entirely positive. We ask for the smallest modification of the free Schrödinger current that makes it nonnegative for every such state, while preserving the current of each individual momentum component. We show that the required minimal modification changes the free-particle momentum kernel according to K_Sch(p,p')=(p+p')/2m ⟶ K_min(p,p')=√(pp')/m. The resulting current is positive and normalized and therefore defines an arrival-time POVM. Extending the directional no-backflow requirement to states containing both momentum signs forces the cross-sector kernel to vanish, K_min⁺⁻=K_min⁻⁺=0, so that the full current is the sum of two independent directional contributions. The resulting POVM is exactly the Kijowski time-of-arrival POVM, providing a current-based physical motivation for both its directional kernels and their separation. Within the diagonal-preserving pairwise-minimal construction considered here, the result is unique. The construction itself does not impose a first-arrival condition.

Methods for Deriving Nonlinear Kinetic Equations from the Dynamics of Quantum Systems
V. I. Gerasimenko
arXiv20 Oct 2026PreprintCC0Full textMathematical physics Quantum physics & quantum informationarXiv
Deriving the AdS₃×S³× T⁴ Quantum Spectral Curve II: T-system and QSC
Andrea Cavaglià, Davide Polvara, Nicolò Primi +2
arXiv5 Oct 2026PreprintarXiv licenceFull textHigh-energy & particle physics Statistical & nonlinear physicsarXiv
Deriving Schwarzschild and Kerr from the Einstein Equivalence Principle
G. Alencar
arXiv8 Oct 2026PreprintarXiv licenceFull textGeneral physicsarXiv
An empirical model for deriving information on ion dynamics from μSR data via the conventional Kubo-Toyabe function analysis
Ryosuke Kadono
arXiv12 Oct 2026PreprintCC BY-SA 4.0Full textCondensed matterarXiv
Deriving the second law of thermodynamics and exploring its boundaries
Yu Qiao
arXiv26 Oct 2026PreprintCC BY 4.0Full textStatistical & nonlinear physicsarXiv

Cite

BibTeX RIS

BibTeX

@misc{marchewka2026deriving,
  title = {{Deriving the Kijowski Arrival-Time POVM from the Schrödinger Current: Minimal Positivity and Uniqueness}},
  author = {Avi Marchewka},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08458},
  eprint = {2609.08458},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08458},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Deriving the Kijowski Arrival-Time POVM from the Schrödinger Current: Minimal Positivity and Uniqueness
AU  - Avi Marchewka
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08458
UR  - https://arxiv.org/abs/2609.08458
AB  - Quantum backflow refers here to the appearance of a negative Schrödinger current for a state whose momentum support is entirely positive. We ask for the smallest modification of the free Schrödinger current that makes it nonnegative for every such state, while preserving the current of each individual momentum component. We show that the required minimal modification changes the free-particle momentum kernel according to K_Sch(p,p')=(p+p')/2m ⟶ K_min(p,p')=√(pp')/m. The resulting current is positive and normalized and therefore defines an arrival-time POVM. Extending the directional no-backflow requirement to states containing both momentum signs forces the cross-sector kernel to vanish, K_min^+−=K_min^−+=0, so that the full current is the sum of two independent directional contributions. The resulting POVM is exactly the Kijowski time-of-arrival POVM, providing a current-based physical motivation for both its directional kernels and their separation. Within the diagonal-preserving pairwise-minimal construction considered here, the result is unique. The construction itself does not impose a first-arrival condition.
ER  -

Abstract

Related

Cite