Nonlinear reformulation of Heisenberg's dynamics

Abstract

A structural similarity between Classical Mechanics (CM) and Quantum Mechanics (QM) was revealed by P.A.M. Dirac in terms of Lie Algebras: while in CM the dynamics is determined by the Lie algebra of Poisson brackets on the manifold of scalar fields for classical position/momentum observables Q/P, QM evolves (in Heisenberg's picture) according to the formally similar Lie algebra of commutator brackets of th corresponding operators:

d/dtQ = {Q,H} d/dtP = {P,H} versus d/tdQ = i/h [Q,H] d/dtP = 1/h [P,H]

where QP - PQ = ih. A further common frameworks for comparing CM and QM is the category of symplectic manifolds. Other than previous authors, this paper considers phase space of Heisenberg's picture, i.e., the manifold of pairs of operator observables (Q,P) satisfying commutation relation. On a sufficiently high algebraic level of abstractation-which we believe to be of interest on its own-it turns out that this approach leads to a truly nonlinear yet Hamiltonian reformulation of QM evolution.