Geometry and symmetry of quantum and classical-quantum variational principles

TL;DR

This paper develops a geometric framework for quantum variational principles, extending it to classical-quantum interactions, and introduces new variational principles for mixed states and hybrid dynamics using symmetry and group theory.

Contribution

It introduces a geometric approach to quantum variational principles, incorporating classical-quantum interactions and deriving new variational principles for mixed states and hybrid systems.

Findings

New variational principles for mixed quantum states.

Geometric formulation of classical-quantum interactions.

Group-theoretic structures underpinning hybrid dynamics.

Abstract

This paper presents the geometric setting of quantum variational principles and extends it to comprise the interaction between classical and quantum degrees of freedom. Euler-Poincar\'e reduction theory is applied to the Schr\"odinger, Heisenberg and Wigner-Moyal dynamics of pure states. This construction leads to new variational principles for the description of mixed quantum states. The corresponding momentum map properties are presented as they arise from the underlying unitary symmetries. Finally, certain semidirect-product group structures are shown to produce new variational principles for Dirac's interaction picture and the equations of hybrid classical-quantum dynamics.