From Energy-Time Uncertainty to Symplectic Displacement Energy

Abstract: Heisenberg's Uncertainty Principle is one of the most celebrated features of quantum mechanics, which states that one cannot simultaneously obtain the precise measurements of two conjugated physical quantities such as the pair of position and momentum or the pair of electric potential and charge density. Among the different formulations of this fundamental quantum property, the uncertainty between energy and time has a special place. This is because the time is rather a variable parametrizing the system evolution than a physical quantity waiting for determination. Physicists working on the foundation of quantum theory have understood this energy-time relation by a universal bound of how fast any quantum system with given energy can evolve from one state to another in a distinguishable (orthogonal) way. Recently, there have been many arguing that this bound is not a pure quantum phenomenon but a general dynamical property of Hilbert space. In this talk, in contrast to the usual Hilbert space formalism, we will provide a homological viewpoint of this evolutional speed limit based on a persistence-like distance of the derived category of sheaves : during a time period what is the minimal energy needed for a system to evolve from one sheaf to a status that is distinguishable from a given subcategory? As an application, we will also discuss its geometric incarnation in the dynamics of classical mechanics, namely the notion of symplectic displacement. We will see how this categorical energy manages to characterize the symplectic energy for disjointing a Lagrangian from an open set.

differential geometrydynamical systemsgeometric topologysymplectic geometry

Audience: researchers in the topic

Geometry and Dynamics seminar

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