Emergent semiclassical time in quantum gravity: I. Mechanical models

The emergent semiclassical time approach to resolving the problem of time in quantum gravity is considered in the arena of relational particle toy models. In situations with 'heavy' and 'light' degrees of freedom, two notions of emergent semiclassical WKB time emerge; these are furthermore equivalent to two notions of emergent classical 'Leibniz-Mach-Barbour' time. I study the semiclassical approach, in a geometric phase formalism, extended to include linear constraints, and with particular care to make explicit the approximations and assumptions used, which are an important part of the semiclassical approach. I propose a new iterative scheme for the semiclassical approach in the cosmologically motivated case with one heavy degree of freedom. I find that the usual semiclassical quantum cosmology emergence of time comes hand in hand with the emergence of other qualitatively significant terms, including back-reactions on the heavy subsystem and second time derivatives. I take my analysis further for relational particle models with linearly coupled harmonic oscillator potentials, which, being exactly soluble by means outside the semiclassical approach to quantum cosmology, are additionally useful for testing the justifiability of some of the approximations and assumptions habitually made therein. Finally, I contrast emergent semiclassical time with its hidden dilational Euler time counterpart.