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arXiv:quant-ph/0311159 · Physics Letters A

Quantization of non-Hamiltonian and dissipative systems

Vasily Evgenevich Tarasov

Published Sep 1, 2001• 70 citations
arXivHTMLPDF
DOI
10.1016/s0375-9601(01)00548-5

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The work of E. Kanai shows that the quantization of dissipative systems by use of a Hamiltonian formalism involving the time explicitly leads to results in disagreement with experience. We discuss the quantization of dissipative systems in a manner such that the time does not enter explicitly. It is shown that in the Heisenberg representation the only kind of dissipative forces consistent with the formalism are those which are functions of the coordinates alone. There exists no Schr\"odinger representation unless the forces are non-dissipative.
In this paper a generalization of Weyl quantization which maps a dynamical operator in a function space to a dynamical superoperator in an operator space is suggested. Quantization of dynamical operator, which cannot be represented as Poisson bracket with some function, is considered. The usual Weyl quantization of observables can be derived as a specific case of suggested quantization if dynamical operator is an operator of multiplication on a function. This approach allows to define consistent Weyl quantization of non-Hamiltonian and dissipative systems. Examples of the harmonic oscillator with friction and a system which evolves by Fokker-Planck-type equation are considered.