  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.
