  We formulate quantum mechanics in spacetimes with real-order fractional
geometry and more general factorizable measures. In spacetimes where
coordinates and momenta span the whole real line, Heisenberg's principle is
proven and the wave-functions minimizing the uncertainty are found. In spite of
the fact that ordinary time and spatial translations are broken and the
dynamics is not unitary, the theory is in one-to-one correspondence with a
unitary one, thus allowing us to employ standard tools of analysis. These
features are illustrated in the examples of the free particle and the harmonic
oscillator. While fractional (and the more general anomalous-spacetime) free
models are formally indistinguishable from ordinary ones at the classical
level, at the quantum level they differ both in the Hilbert space and for a
topological term fixing the classical action in the path integral formulation.
Thus, all non-unitarity in fractional quantum dynamics is encoded in a
contribution depending only on the initial and final state.
