Quantum speed limit and categorical energy relative to microlocal projector

TL;DR

This paper introduces a categorical energy concept for sheaves in derived categories, linking microlocal analysis with quantum speed limits to establish bounds on Hofer displacement energy.

Contribution

It defines a new categorical energy for sheaves relative to microlocal projectors and connects it to Hofer displacement energy using algebraic microlocal analysis.

Findings

Categorical energy provides a lower bound for Hofer displacement energy.

Establishes a relative energy-capacity inequality.

Provides a sheaf-theoretic proof of positivity of Hofer displacement energy.

Abstract

Inspired by recent developments of quantum speed limit we introduce a categorical energy of sheaves in the derived category over a manifold relative to a microlocal projector. We utilize the tool of algebraic microlocal analysis to show that with regard to the microsupports of sheaves, our categorical energy gives a lower bound of the Hofer displacement energy. We also prove that on the other hand our categorical energy obeys a relative energy-capacity type inequality. As a by-product this provides a sheaf-theoretic proof of the positivity of the Hofer displacement energy for disjointing the zero section $L$ from an open subset $O$ in $T^*L$ , given that $L \cap O \neq \emptyset$.