NDLI: Nonlinear Reformulation of Heisenberg's Dynamics

Content Provider	Springer Nature Link
Author	Ziegler, Martin Fuchssteiner, Ben
Copyright Year	2005
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 the corresponding operators: $$\displaylines{ \frac{d}{dt}Q=\{Q,H\} \quad \frac{d}{dt}P=\{P,H\} \quad\hbox{versus}\ \quad \frac{d}{dt}{\mathbb Q}=\frac{\rm i}{\hbar}[{\mathbb Q},{\mathbb H}] \quad \frac{d}{dt}{\mathbb P}=\frac{\rm i}{\hbar}[{\mathbb P},{\mathbb H}]}$$ where ℚℙ; – ℙ;ℚ. 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 nonlinear yet Hamiltonian reformulation of QM evolution.
Starting Page	693
Ending Page	717
Page Count	25
File Format	PDF
ISSN	00207748
Journal	International Journal of Theoretical Physics
Volume Number	44
Issue Number	7
e-ISSN	15729575
Language	English
Publisher	Kluwer Academic Publishers-Plenum Publishers
Publisher Date	2005-01-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	quantum mechanics Heisenberg's picture symplectic manifolds nonlinear symmetries integrability Physics Quantum Physics Mathematical and Computational Physics Elementary Particles, Quantum Field Theory
Content Type	Text
Resource Type	Article
Subject	Physics and Astronomy Mathematics

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