Hamilton--Jacobi meet M\"obius

Adaptation of the Hamilton--Jacobi formalism to quantum mechanics leads to a cocycle condition, which is invariant under $D$--dimensional M\"obius transformations with Euclidean or Minkowski metrics. In this paper we aim to provide a pedagogical presentation of the proof of the M\"obius symmetry underlying the cocycle condition. The M\"obius symmetry implies energy quantization and undefinability of quantum trajectories, without assigning any prior interpretation to the wave function. As such, the Hamilton--Jacobi formalism, augmented with the global M\"obius symmetry, provides an alternative starting point, to the axiomatic probability interpretation of the wave function, for the formulation of quantum mechanics and the quantum spacetime. The M\"obius symmetry can only be implemented consistently if spatial space is compact, and correspondingly if there exist a finite ultraviolet length scale. Evidence for non--trivial space topology may exist in the cosmic microwave background radiation.