Distance Formula for Grassmann Manifold --Applied to Anandan--Aharonov Type Uncertainty Relation--

TL;DR

This paper generalizes the time-energy uncertainty relation for quantum states by deriving an explicit distance formula on the Grassmann manifold, linking geometric concepts with quantum uncertainty principles.

Contribution

It introduces a new explicit formula for distances in the Grassmann manifold, extending the Anandan-Aharonov uncertainty relation to multiple quantum states.

Findings

Derived an explicit distance formula for Grassmann manifold points

Generalized the time-energy uncertainty relation to multiple states

Bridged geometric and quantum uncertainty concepts

Abstract

The time-energy uncertainty relation of Anandan-Aharonov is generalized to a relation involving a set of quantum state vectors. This is achieved by obtaining an explicit formula for the distance between two finitely separated points in the Grassmann manifold.