On the Uncertainty Principle for Proper Time and Mass

Abstract

In [J. Math. Phys. {\bf 40}, 1237 (1999)] Kudaka and Matsumoto derive the uncertainty relation $c^2 \Delta m \Delta \tau \geq \hbar/2$ between the rest mass $m$ and the proper time $\tau$, by considering the Lagrangian $M(\dot \tau - c^{-1} \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu}) + e A_\mu (x) \dot x^\mu$. In this note we give an alternative derivation based on a special case of the time-like geodesic equation obtained using the general relativistic Lagrangian $g_{\mu \nu} \frac{dx^\mu}{d \tau}\frac{dx^\nu}{d \tau}$.