Generalised Complex Geometry and the Planck Cone

TL;DR

This paper explores how generalised complex geometry provides a unified framework for understanding dualities in classical and quantum mechanics, relating phase space structures to the Planck cone concept.

Contribution

It introduces the role of generalised complex geometry in describing dualities in mechanics and interprets the Planck cone as a boundary connecting different geometric regions.

Findings

Dualities connect interior and exterior points of the Planck cone.

Generalised complex geometry unifies symplectic and complex structures in mechanics.

The Planck cone analogy with the light-cone offers new insights into phase space geometry.

Abstract

Complex geometry and symplectic geometry are mirrors in string theory. The recently developed generalised complex geometry interpolates between the two of them. On the other hand, the classical and quantum mechanics of a finite number of degrees of freedom are respectively described by a symplectic structure and a complex structure on classical phase space. In this letter we analyse the role played by generalised complex geometry in the classical and quantum mechanics of a finite number of degrees of freedom. We identify generalised complex geometry as an appropriate geometrical setup for dualities. The latter are interpreted as transformations connecting points in the interior of the Planck cone with points in the exterior, and viceversa. The Planck cone bears some resemblance with the relativistic light-cone. However the latter cannot be traversed by physical particles, while dualities do connect the region outside the Planck cone with the region inside, and viceversa.