Geometry of quantum dynamics in infinite dimension

TL;DR

This paper introduces a geometric framework for infinite-dimensional quantum mechanics using the Tulczyjew triple, providing new insights into operator theory, state space embedding, and self-adjoint extensions.

Contribution

It develops a novel geometric approach to quantum dynamics in infinite dimensions, including a Lagrangian formalism and analysis of operator and state space structures.

Findings

Lagrangian formalism for infinite-dimensional quantum systems

Embedding of pure states into the unitary group

Results on self-adjoint extensions of symmetric relations

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.