Quantum mechanics in fractional and other anomalous spacetimes

Understand

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.