  Our conventional system of physical units is based on local or microscopic
{\it dimensional} quantities which are {\it defined}, for convenience or
otherwise aesthetic reasons, to be spacetime-independent. A more general choice
of units may entail variation of fundamental physical quantities (`constants')
in spacetime. The theory of gravitation generally does not satisfy conformal
symmetry, i.e. it is not invariant to local changes of the unit of length.
Consequently, the {\it dimensionless} action associated with the
Einstein-Hilbert action ($S_{EH}$) of gravitation, $\phi_{EH}=S_{EH}/\hbar$, is
not invariant to local changes of the length unit; clearly an unsatisfactory
feature for a dimensionless quantity. Here we amend the phase by adding extra
terms that account for spacetime variation of the physical `constants' in
arbitrary unit systems. In such a unit system, all dimensional quantities are
implicitly spacetime-dependent; this is achieved by a conformal transformation
of the metric augmented by appropriate metric-dependent rescalings of the
dimensional quantities. The resulting modified dimensionless action is
scale-invariant, i.e. independent of the unit system, as desired. The deep
connection between gravitation, dimensionless physical quantities, and quantum
mechanics, is elucidated and the implicit ambiguity in interpretations of
dimensional quantities is underlined.
