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 φ t of transformations on a Hilbert space, 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 φ t . The second one makes use of the arXiv:1908.03699v1 [quant-ph]

Lagrangian description of Heisenberg and Landau–von Neumann equations of motion

“…In particular, we have shown how the Lagrangian description of the Landau-von Neumann equation may be obtained from the Lagrangian description of the Heisenberg equation by means of the pullback procedure introduced in Ref. [8]. This is in line with the recent developments of classical and quantum information geometry in which the focus is on parametrized submanifold of classical and quantum states rather than on the whole space of states.…”

“…The Lagrangian formulation introduced for the Heisenberg equation can be used to induce a dynamics on the set S in analogy with what is done in Ref. [8]. Indeed, the pull-back of the Lagrangian function in Eq.…”

Nonlinear Description of Quantum Dynamics. Generalized Coherent States

“…Now, considering the Hilbert space of quadratically integrable functions L 2 (R, dq), the immersed submanifold i(C) corresponds to the Gaussian wave packets, i.e. one has the well-known immersion [20,21] i…”

Covariant reduction of classical Hamiltonian Field Theories: From D'Alembert to Klein-Gordon and Schrödinger

“…This idea has been used, [7] to show how it is possible to obtain nonlinear equations by reducing linear ones, or to obtain approximating solutions of the initial equations. When the selected submanifold is an invariant submanifold for the original equations of motion, we may perform also the reduction by restricting directly the equations of motion to the submanifold.…”