Operational Meaning of the Mandelstam-Tamm Time Scale in a Spin-1/2 Clock

Abstract

Georgiev's analysis of a freely precessing spin correctly emphasizes that the Mandelstam-Tamm quantity ΔT=ΔÂ/|d⟨Â⟩/d t| is not the variance of a time operator. He concludes, however, that its divergence near the extrema of the clock signal is devoid of physical meaning and that ΔT does not disclose how well the system measures the passage of time. For the same projective σ̂_x readout, we show that ΔT equals the inverse square root of the classical Fisher information in one binary outcome at regular operating points. Its divergence at submaximal-coherence extrema therefore identifies a loss of local sensitivity in the selected readout, not an infinite variance of physical time; a complementary σ̂_y measurement remains informative. This interpretation also separates Georgiev's exactly known tick--tock period from the statistical precision with which a tick can be located. It preserves his warning against a time-operator interpretation while qualifying his stronger conclusion that the Mandelstam--Tamm scale lacks physical content.

Versions

10.3390/sym18111762Published version10.3390/sym18111762

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BibTeX RIS

BibTeX

@misc{rog2026operational,
  title = {{Operational Meaning of the Mandelstam-Tamm Time Scale in a Spin-1/2 Clock}},
  author = {Jakub Rog},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24056},
  eprint = {2610.24056},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24056},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Operational Meaning of the Mandelstam-Tamm Time Scale in a Spin-1/2 Clock
AU  - Jakub Rog
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.24056
UR  - https://arxiv.org/abs/2610.24056
AB  - Georgiev's analysis of a freely precessing spin correctly emphasizes that the Mandelstam-Tamm quantity ΔT=ΔÂ/|d⟨Â⟩/d t| is not the variance of a time operator. He concludes, however, that its divergence near the extrema of the clock signal is devoid of physical meaning and that ΔT does not disclose how well the system measures the passage of time. For the same projective σ̂_x readout, we show that ΔT equals the inverse square root of the classical Fisher information in one binary outcome at regular operating points. Its divergence at submaximal-coherence extrema therefore identifies a loss of local sensitivity in the selected readout, not an infinite variance of physical time; a complementary σ̂_y measurement remains informative. This interpretation also separates Georgiev's exactly known tick--tock period from the statistical precision with which a tick can be located. It preserves his warning against a time-operator interpretation while qualifying his stronger conclusion that the Mandelstam--Tamm scale lacks physical content.
ER  -

Abstract

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