Title:

Geometry and symmetry of quantum and classicalquantum dynamics

The symmetry properties of quantum variational principles are considered. EulerPoincaré reduction theory is applied to the DiracFrenkel variational principle for Schrödinger's dynamics producing new variational principles for the different pictures of quantum mechanics (Schrödinger's, Heisenberg's and Dirac's) as well as for the WignerMoyal formulation. In addition, new variational principles for mixed states dynamics have been formulated. The already known geometric characterization of quantum mechanics on the complex projective space is shown to emerge naturally from the EulerPoincaré variational principle. Semidirectproduct structures are seen to produce new variational principles for Dirac's picture. On the other hand, the variational and Hamiltonian approach to Ehrenfest expectation values dynamics is proposed. In the Lagrangian framework, the quantum variational principle for Schrödinger's dynamics is extended to account for both classical and quantum degrees of freedom. First, it is shown that the mean field model of any quantum mechanical system can be derived from a classicalquantum EulerPoincaré Lagrangian on the direct sum Lie algebra of the Heisenberg and unitary groups. Then, the semidirectproduct structure (named Ehrenfest group), is constructed using the displacement operator from the theory of coherent quantum states ( the unitary action of the Heisenberg group on the space of wavefunctions). New classicalquantum equations for Ehrenfest's expectation values dynamics are derived redefining the meanfield model EulerPoincaré Lagrangian on the Lie algebra of the Ehrenfest group. In the Hamiltonian framework, first expectation values of the canonical observables are shown to be equivariant momentum maps for the unitary action of the Heisenberg group on quantum states. Then, the Hamiltonian structure for Ehrenfest's dynamics is shown to be LiePoisson for the Ehrenfest group. The variational formulation is then given a corresponding Hamiltonian structure. The classicalquantum Ehrenfest dynamics equations produce classical and quantum dynamics as special limit cases. In the particular case of Gaussian states, expectation values couple to second order moments, so that GS are completely characterized by first and second moments. When the total energy is computed with respect to a Gaussian state, higher moments can be expressed in terms of the first two, so that the moment hierarchy closes for Gaussian states. Second moments are shown to be equivariant momentum maps for the action of the symplectic group on the space of Wigner functions. Eventually, Gaussian states are shown to possess a LiePoisson structure on the Jacobi group. This structure produces an energyconserving variant of a class of Gaussian moment models that have appeared in the chemical physics literature.
