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Related Experiment Videos

Reduction, the trace formula, and semiclassical asymptotics.

V Guillemin1, A Uribe

  • 1Mathematics Department, Massachusetts Institute of Technology, Cambridge, MA 02139.

Proceedings of the National Academy of Sciences of the United States of America
|November 1, 1987
PubMed
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We connect dimensional reduction in Hamiltonian mechanics to spectral properties of operators. This reveals geometric information about classical particle trajectories from quantum Schrödinger operator spectra as Planck

Area of Science:

  • Mathematical Physics
  • Quantum Mechanics
  • Differential Geometry

Background:

  • Dimensional reduction is a key concept in theoretical physics.
  • Spectral properties of operators encode information about physical systems.
  • Hamiltonian mechanics describes classical particle dynamics.

Purpose of the Study:

  • To establish a theorem linking dimensional reduction in Hamiltonian mechanics with spectral properties of elliptic operators.
  • To demonstrate how the spectrum of the Schrödinger operator provides geometric insights into classical particle trajectories.
  • To generalize these findings to particles with internal degrees of freedom and Yang-Mills fields.

Main Methods:

  • Formulation of a new theorem relating dimensional reduction and spectral theory.

Related Experiment Videos

  • Analysis of the Schrödinger operator's spectrum in the semi-classical limit (h --> 0).
  • Application to systems with internal degrees of freedom and Yang-Mills fields.

    • Main Results:

    • A direct relationship is established between dimensional reduction and spectral properties of symmetric elliptic operators on compact manifolds.
    • The semi-classical limit of the Schrödinger operator's spectrum reveals geometric information about closed classical trajectories.
    • This connection is generalized to complex systems including those with Yang-Mills fields, linking to the Wong-Sternberg-Weinstein system.

    Conclusions:

    • The study provides a novel bridge between quantum spectral theory and classical mechanics.
    • Geometric properties of classical systems can be extracted from quantum spectral data.
    • The findings have implications for understanding complex physical systems in the classical limit.