A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of non-stationary Poisson brackets, i.e. corresponding Poisson tensor is allowed to explicitly depend on time. Starting from this pseudo-Hamiltonian formulation we develop a consistent deformation quantization procedure involving a non-stationary star-product $*_t$ and an ``extended'' operator of time derivative $D_t=\\partial_t+...$ , differentiating the $\\ast_t$ -product. As in the usual case, the $\\ast_t$ -algebra of physical observables is shown to admit an essentially unique (time dependent) trace functional $\\mathrm{Tr}_t$ . Using these ingredients we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of non-stationary Poisson brackets, i.e. corresponding Poisson tensor is allowed to explicitly depend on time. Starting from this pseudo-Hamiltonian formulation we develop a consistent deformation quantization procedure involving a non-stationary star-product $*_t$ and an ``extended'' operator of time derivative $D_t=\\partial_t+...$, differentiating the $\\ast_t$-product. As in the usual case, the $\\ast_t$-algebra of physical observables is shown to admit an essentially unique (time dependent) trace functional $\\mathrm{Tr}_t$. Using these ingredients we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.

Jan 1, 20057 citations

Deformation quantization of linear dissipative systems

V G Kupriyanov, S L Lyakhovich, A A Sharapov

A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of a non-stationary Poisson bracket, i.e. the corresponding Poisson tensor is allowed to explicitly depend on time. Starting from this pseudo-Hamiltonian formulation, we develop a consistent deformation quantization procedure involving a non-stationary star-product ∗ t and an 'extended' operator of time derivative D t = ∂ t + ⋅⋅⋅, differentiating the ∗ t -product. As in the usual case, the ∗ t -algebra of physical observables is shown to admit an essentially unique (time-dependent) trace functional Tr t . Using these ingredients, we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of a non-stationary Poisson bracket, i.e. the corresponding Poisson tensor is allowed to explicitly depend on time. Starting from this pseudo-Hamiltonian formulation, we develop a consistent deformation quantization procedure involving a non-stationary star-product ∗ t and an 'extended' operator of time derivative D t = ∂ t + ⋅⋅⋅, differentiating the ∗ t -product. As in the usual case, the ∗ t -algebra of physical observables is shown to admit an essentially unique (time-dependent) trace functional Tr t . Using these ingredients, we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.

arXiv:quant-ph/0505023Journal of Physics A Mathematical and GeneralAug 31, 200519 citations

*_t

and an ``extended'' operator of time derivative

D_t=\\partial_t+...

, differentiating the

\\ast_t

-product. As in the usual case, the

\\ast_t

-algebra of physical observables is shown to admit an essentially unique (time dependent) trace functional

\\mathrm{Tr}_t

. Using these ingredients we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.