Hamilton--Jacobi meet M\"obius

TL;DR

This paper explores how the Hamilton--Jacobi formalism, when adapted to quantum mechanics, reveals a M"obius symmetry that underpins energy quantization and the nature of quantum trajectories, suggesting a new foundational perspective.

Contribution

It provides a pedagogical proof of the M"obius symmetry in the quantum Hamilton--Jacobi formalism and discusses its implications for quantum foundations and spacetime topology.

Findings

M"obius symmetry leads to energy quantization.

Quantum trajectories are inherently undefined under this formalism.

Space must be compact for the symmetry to hold, implying a finite ultraviolet scale.

Abstract

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.