Species of spaces

TL;DR

This paper explores how quantum systems' classical limits lead to novel conceptions of space, emphasizing persistent noncommutative features and their implications for quantum geometry, chaos, and classical-quantum relations.

Contribution

It introduces a framework analyzing noncommutative geometries emerging from quantum limits, connecting quantum indeterminism with classical unpredictability.

Findings

Noncommutative structures persist as Planck constant vanishes.

Classical chaos and quantum indeterminism are compared and fused.

Noncommutative geometry provides new insights into quantum-classical transition.

Abstract

Classical limits of quantum systems are shown to lead to different conceptions of spaces different from the classical one underlying the process of quantization of such systems. The accent is put in situations where traces of noncommutativity, witness of an emblematic feature of quantum mechanise remains when the Planck constant vanishes, in the framework of noncommutative geometry. Complex canonical transformations, spin-statistics, topological quantum fields theory, long time semiclassical approximation and underlying chaotic dynamics are considered, together with a comparison/fusion of classical unpredictability with quantum indeterminism.