Nonlinear Reformulation of Heisenberg's Dynamics

Details Nonlinear Reformulation of Heisenberg's Dynamics Nonlinear Reformulation of Heisenberg's Dynamics

2002-10-30

arxiv.org/abs/quant-ph/0210198v4

Physics

Quantum Physics

10 pages (+optional appendix which may be omitted for print); partially rewritten upon referee's request

Scientific paper

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, d/dt q={q,H}, d/dt p={p,H}, QM evolves (in Heisenberg's picture) according to the formally similar Lie algebra of commutator brackets of the corresponding operators Q/P: d/dt Q=i/h [Q,H], d/dt P=i/h [P,H] where QP-PQ=ih. A further common framework 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 abstraction -- which we believe to be of interest on its own -- it turns out that this approach leads to a truly NON-linear yet Hamiltonian reformulation of QM evolution.

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