Nonlinear Dynamics from Linear Quantum Evolutions

Abstract

Linear dynamics restricted to invariant submanifolds generally gives rise to nonlinear dynamics. Submanifolds in the quantum framework may emerge for several reasons: one could be interested in specific properties possessed by a given family of states, either as a consequence of experimental constraints or inside an approximation scheme. In this work we investigate such issues in connection with a one parameter group ${\varphi }_t$ of transformations on a Hilbert space, $\mathcal{H}$, defining the unitary evolutions of a chosen quantum system. Two procedures will be presented: the first one consists in the restriction of the vector field associated with the Schrödinger equation to a submanifold invariant under the flow ${\varphi }_t$. The second one makes use of the Lagrangian formalism and can be extended also to non-invariant submanifolds, even if in such a case the resulting dynamics is only an approximation of the flow ${\varphi }_t$. Such a result, therefore, should be conceived as a generalization of the variational method already employed for stationary problems.

arXiv comment38 pages

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• Covariant reduction of classical Hamiltonian Field Theories: From D'Alembert to Klein-Gordon and Schrödinger
• Lagrangian description of Heisenberg and Landau-von Neumann equations of motion
• A symplectic approach to Schrödinger equations in the infinite-dimensional unbounded setting
• Nonlinear Description of Quantum Dynamics. Generalized Coherent States