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Uncertainty relations on nilpotent Lie groups
Michael Ruzhansky1, Durvudkhan Suragan2,1
1Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK.
Proceedings. Mathematical, Physical, and Engineering Sciences
|June 8, 2017
Summary
This study establishes quantum mechanics operator relations on nilpotent Lie groups, deriving novel inequalities with best constants for momentum, position, and potentials. These findings advance understanding in quantum mechanics, particularly for anisotropic and Heisenberg group settings.
Area of Science:
- Quantum Mechanics
- Lie Groups
- Mathematical Physics
Background:
- Quantum mechanics operators govern physical systems.
- Nilpotent Lie groups provide a general framework for studying symmetries.
- Homogeneous groups are a key class within Lie groups.
Purpose of the Study:
- To establish relations between fundamental quantum mechanical operators.
- To derive analogues of established inequalities on homogeneous groups.
- To obtain best constants for these derived inequalities.
Main Methods:
- Utilizing operator relations on general classes of nilpotent Lie groups.
- Applying methods to homogeneous groups, including isotropic and anisotropic settings.
- Deriving Hardy's, Heisenberg-Kennard, Heisenberg-Pauli-Weyl, and Caffarelli-Kohn-Nirenberg inequalities.
Main Results:
- New relations between momentum, position, Euler, and Coulomb potential operators are presented.
- Best constants for derived inequalities are determined.
- Novel results are achieved for isotropic, anisotropic, and Heisenberg group settings.
Conclusions:
- The established operator relations provide a foundation for further quantum mechanical studies.
- The derived inequalities and their best constants offer significant advancements.
- The proof methodology is robust for both isotropic and anisotropic scenarios.
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