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 $c^2\Delta m\Delta \tau \ge \hslash /2$ 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 $g_{\mu \nu }\frac{d{x}^{\mu }}{d\tau }\frac{d{x}^{\nu }}{d\tau }$.

9 pages, uses subeqn.sty