The certainty principle

TL;DR

The paper introduces the quantum angle as a metric on quantum states, formulates and proves the certainty principle, and explores its relation to the Heisenberg uncertainty principle, providing rigorous foundations for various quantum inequalities.

Contribution

It introduces the quantum angle metric and the certainty principle, offering a new perspective and rigorous formulations for quantum uncertainty relations.

Findings

Quantum angle is a metric on quantum states.

The certainty principle is formulated and proved.

Connections to and extensions of the Heisenberg uncertainty principle.

Abstract

The notion of the quantum angle is introduced. The quantum angle turns out to be a metric on the set of physical states of a quantum system. Its kinematics and dynamics is studied. The certainty principle for quantum systems is formulated and proved. It turns out that the certainty principle is closely connected with the Heisenberg uncertainty principle (it presents, in some sense, an opposite point of view). But at the same time the certainty principle allows to give rigorous formulations for wider class of problems (it allows to rigorously interpret and ground the analogous inequalities for the pairs of quantities like time - energy, angle - angular momentum etc.)