| Contents |
| Geometric quantum mechanics has various applications ranging from quantum tomography to quantum computation and control. In finite dimensional quantum systems, deep geometric concepts emerge naturally, such as Stiefel manifolds and Grassmannians, projective spaces and holonomy. While the first part of the talk will illustrate how these structures pop up, the second part will show how they are used in the Lagrangian reduction of the quantum (Dirac-Frenkel) variational principle. For example, the Schr\"odinger and Heisenberg pictures are shown to correspond respectively to the Eulerian and convective frameworks of Euler-Poincar\'e theory. Also, the geometric phase is shown to obey a Kelvin-Noether theorem arising from relabeling symmetry. |
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