  The basic tenet of the present work is the assumption of the lack of external
and fixed time in the Universe. This assumption is best embodied by general
relativity, which replaces the fixed space-time structure with the
gravitational field, which is subject to dynamics. The lack of time does not
imply the lack of evolution but rather brings to the forefront the role of
internal clocks which are some largely arbitrary internal degrees of freedom
with respect to which the evolution of timeless systems can be described. We
take this idea seriously and try to understand what it implies for quantum
mechanics when the fixed external time is replaced by an arbitrary internal
clock. We put the issue in a solid, mathematically rigorous framework. We find
that the dynamical interpretation of a quantum state of a timeless system
depends on the employed internal clock. In particular, we find that the
continuous spectra of well-known dynamical observables like the position of a
free particle on the real line may turn discrete if measured in unusual clocks.
We discuss the meaning of our result for attempts at quantization of global
gravitational degrees of freedom.
