Uncertainty relations on nilpotent Lie groups.

Michael Ruzhansky,Durvudkhan Suragan

doi:10.1098/rspa.2017.0082

Abstract

We give relations between main operators of quantum mechanics on one of most general classes of nilpotent Lie groups. Namely, we show relations between momentum and position operators as well as Euler and Coulomb potential operators on homogeneous groups. Homogeneous group analogues of some well-known inequalities such as Hardy's inequality, Heisenberg–Kennard type and Heisenberg–Pauli–Weyl type uncertainty inequalities, as well as Caffarelli–Kohn–Nirenberg inequality are derived, with best constants. The obtained relations yield new results already in the setting of both isotropic and anisotropic , and of the Heisenberg group. The proof demonstrates that the method of establishing equalities in sharper versions of such inequalities works well in both isotropic and anisotropic settings.

Highlights

The uncertainty principle of Werner Heisenberg [1] forms a fundamental element of quantum mechanics
The first rigorously proven uncertainty relation for position and momentum operators is due to Earle Kennard [2]
There is a vast literature on uncertainty relations and their applications

Summary

Uncertainty relations on nilpotent Lie groups

We give relations between main operators of quantum mechanics on one of most general classes of nilpotent Lie groups. We show relations between momentum and position operators as well as Euler and Coulomb potential operators on homogeneous groups. Homogeneous group analogues of some well-known inequalities such as Hardy’s inequality, Heisenberg–Kennard type and Heisenberg–Pauli– Weyl type uncertainty inequalities, as well as Caffarelli–Kohn–Nirenberg inequality are derived, with best constants. The obtained relations yield new results already in the setting of both isotropic and anisotropic Rn, and of the Heisenberg group. The proof demonstrates that the method of establishing equalities in sharper versions of such inequalities works well in both isotropic and anisotropic settings

Introduction

We define

The operators P and M clearly satisfy the relation

Combining this with

Euler and Coulomb potential operators can be defined by