Hamilton-Jacobi meet Möbius

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.