  We explore the problem of time in quantum gravity in a point-particle
analogue model of scale-invariant gravity. If quantized after reduction to true
degrees of freedom, it leads to a time-independent Schr\"odinger equation. As
with the Wheeler--DeWitt equation, time disappears, and a frozen formalism that
gives a static wavefunction on the space of possible shapes of the system is
obtained. However, if one follows the Dirac procedure and quantizes by imposing
constraints, the potential that ensures scale invariance gives rise to a
conformal anomaly, and the scale invariance is broken. A behaviour closely
analogous to renormalization-group (RG) flow results. The wavefunction acquires
a dependence on the scale parameter of the RG flow. We interpret this as time
evolution and obtain a novel solution of the problem of time in quantum
gravity. We apply the general procedure to the three-body problem, showing how
to fix a natural initial value condition, introducing the notion of complexity.
We recover a time-dependent Schr\"odinger equation with a repulsive
cosmological force in the `late-time' physics and we analyse the role of the
scale invariant Planck constant. We suggest that several mechanisms presented
in this model could be exploited in more general contexts.
