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Geometry of quantum dynamics in infinite dimension

Janusz Grabowski, Marek Kus, Giuseppe Marmo, Tatiana Shulman

Abstract

We develop a geometric approach to quantum mechanics based on the concept of the Tulczyjew triple. Our approach is genuinely infinite-dimensional and including a Lagrangian formalism in which self-adjoint (Schroedinger) operators are obtained as Lagrangian submanifolds associated with the Lagrangian. As a byproduct we obtain also results concerning coadjoint orbits of the unitary group in infinite dimension, embedding of the Hilbert projective space of pure states in the unitary group, and an approach to self-adjoint extensions of symmetric relations.

Geometry of quantum dynamics in infinite dimension

Abstract

We develop a geometric approach to quantum mechanics based on the concept of the Tulczyjew triple. Our approach is genuinely infinite-dimensional and including a Lagrangian formalism in which self-adjoint (Schroedinger) operators are obtained as Lagrangian submanifolds associated with the Lagrangian. As a byproduct we obtain also results concerning coadjoint orbits of the unitary group in infinite dimension, embedding of the Hilbert projective space of pure states in the unitary group, and an approach to self-adjoint extensions of symmetric relations.

Paper Structure

This paper contains 26 sections, 19 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Implicit dynamics in Classical Mechanics
  3. Implicit differential equations
  4. Lagrangian submanifolds
  5. Tangent lifts of forms and Hamiltonian vector fields
  6. The Tulczyjew triple
  7. The Tulczyjew triple - Lagrangian formalism
  8. Euler-Lagrange equations
  9. The Tulczyjew triple - Hamiltonian formalism
  10. Constrained Lagrangians and Hamiltonians
  11. The unitary group and topology
  12. Coadjoint orbits in $\operatorname{u}_*({\cal H})$
  13. Closedness problem
  14. Geometry and topology of quantum states
  15. Local embedding of pure states into the unitary group
  16. ...and 11 more sections

Key Result

Proposition 2.1

The range ${\cal L}=焋(Q)$ of a one-form $焋$ on $Q$ viewed as a section $焋:Q\to{{\fam\ssfam T}}^*Q$ of the cotangent bundle, is a Lagrangian submanifold in ${{\fam\ssfam T}}^*Q$ if and only if $焋$ is a closed form. Moreover, the above Lagrangian submanifolds can be characterized as those for which th

Theorems & Definitions (48)

  • Definition 2.1
  • Example 2.1
  • Example 2.2
  • Remark 2.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Example 2.3
  • Example 3.1
  • Remark 4.1
  • ...and 38 more