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Scale Invariant Gravitation and Unambiguous Interpretation of Physical Theories

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, $ϕ_{EH}=S_{EH}/\hbar$, 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.