  When we quantize a system consisting of a single particle, the proper time
$\tau $ and the rest mass $m$ are usually dealt with as parameters. In the
present article, however, we introduce a new quantization rule by which these
quantities are regarded as operators in addition to the position and the
momentum. Applying this new rule to a scalar particle and to a particle with
spin $ 1/2 $, we analyze the time evolution of the operator $\tau $. In the
former case, the evolution of the proper time perfectly matches several
well-established classical formulae. In the case of the particle with spin 1/2,
our new rule implies that an oscillation appears in the time evolution of the
operator $\tau $. This oscillation is similar to Zitterbewegung which is
well-known in the ordinary Dirac theory. We formulate one physical effect of
this oscillation by considering the interaction with a gravitational field, and
estimate how small it is.
