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

Georgiev’s analysis of a freely precessing spin correctly emphasizes that the Mandelstam-Tamm quantity $\mathrm{\Delta }T=\mathrm{\Delta }\widehat{A}\mathrm{/}\mathrm{\mid }d⟨\widehat{A}⟩\mathrm{/}dt\mathrm{\mid }$ 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 $\mathrm{\Delta }T$ does not disclose how well the system measures the passage of time. For the same projective ${\widehat{\sigma }}_x$ readout, we show that $\mathrm{\Delta }T$ 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 ${\widehat{\sigma }}_y$ 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.