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Relational Motivation for Conformal Operator Ordering in Quantum Cosmology

Operator-ordering in quantum cosmology is a major as-yet unsettled ambiguity with not only formal but also physical consequences. We determine the Lagrangian origin of the conformal invariance that underlies the conformal operator-ordering choice in quantum cosmology. It is particularly naturally and simply manifest in relationalist product-type actions (such as the Jacobi action for mechanics or Baierlein-Sharp-Wheeler type actions for general relativity), for which all that is required for the kinetic and potential factors to rescale in compensation to each other. These actions themselves mathematically sharply implementing philosophical principles relevant to whole-universe modelling, the motivation for conformal operator-ordering in quantum cosmology is substantially strengthened. Relationalist product-type actions also give emergent times which amount to recovering Newtonian, proper and cosmic time in the various relevant contexts. The conformal scaling of these actions directly tells us how emergent time scales; if one follows suit with the Newtonian time or the lapse in the more commonly used difference-type Euler--Lagrange or Arnowitt--Deser--Misner type actions, one sees how these too obey a more complicated conformal invariance. Moreover, our discovery of the conformal scaling of the time involved permits relating how it simplifies equations of motion with how affine parametrization simplifies geodesics.