The M\"obius Symmetry of Quantum Mechanics

TL;DR

This paper presents a geometrical formulation of quantum mechanics based on M"obius symmetry, implying space compactness, energy quantization, and an intrinsic regularization scheme, with potential cosmological evidence.

Contribution

It introduces a M"obius invariance-based approach to quantum mechanics, linking space compactness to fundamental quantum properties without prior wave function interpretation.

Findings

M"obius symmetry leads to space compactness and energy quantization.

The approach provides an intrinsic regularization scheme.

Potential cosmic microwave background evidence supports space compactness.

Abstract

The equivalence postulate approach to quantum mechanics aims to formulate quantum mechanics from a fundamental geometrical principle. Underlying the formulation there exists a basic cocycle condition which is invariant under $D$--dimensional M\"obius transformations with respect to the Euclidean or Minkowski metrics. The invariance under global M\"obius transformations implies that spatial space is compact. Furthermore, it implies energy quantisation and undefinability of quantum trajectories without assuming any prior interpretation of the wave function. The approach may be viewed as conventional quantum mechanics with the caveat that spatial space is compact, as dictated by the M\"obius symmetry, with the classical limit corresponding to the decompactification limit. Correspondingly, there exists a finite length scale in the formalism and consequently an intrinsic regularisation scheme. Evidence for the compactness of space may exist in the cosmic microwave background radiation.