dorsal/arxiv
View SchemaNonlinear Reformulation of Heisenberg's Dynamics
| Authors | Martin Ziegler, Benno Fuchssteiner |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0210198 |
| URL | https://arxiv.org/abs/quant-ph/0210198 |
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, 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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"abstract": "A structural similarity between Classical Mechanics (CM) and Quantum\nMechanics (QM) was revealed by P.A.M. Dirac in terms of Lie Algebras: while in\nCM the dynamics is determined by the Lie algebra of Poisson brackets on the\nmanifold of scalar fields for classical position/momentum observables q/p, d/dt\nq={q,H}, d/dt p={p,H}, QM evolves (in Heisenberg\u0027s picture) according to the\nformally similar Lie algebra of commutator brackets of the corresponding\noperators Q/P: d/dt Q=i/h [Q,H], d/dt P=i/h [P,H] where QP-PQ=ih. A further\ncommon framework for comparing CM and QM is the category of symplectic\nmanifolds. Other than previous authors, this paper considers phase space of\nHeisenberg\u0027s picture, i.e., the manifold of pairs of operator observables (Q,P)\nsatisfying commutation relation. On a sufficiently high algebraic level of\nabstraction -- which we believe to be of interest on its own -- it turns out\nthat this approach leads to a truly NON-linear yet Hamiltonian reformulation of\nQM evolution.",
"arxiv_id": "quant-ph/0210198",
"authors": [
"Martin Ziegler",
"Benno Fuchssteiner"
],
"categories": [
"quant-ph"
],
"title": "Nonlinear Reformulation of Heisenberg\u0027s Dynamics",
"url": "https://arxiv.org/abs/quant-ph/0210198"
},
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