Quantization Of Dissipative Systems Research Articles

An alternative approach to canonical quantizationof the radiation damping

Inspired in some works about quantization of dissipative systems, in particular of the damped harmonic oscillator\cite{MB,RB,12}, we consider the dissipative system of a charge interacting with its own radiation, which originates the radiation damping (RD). Using the indirect Lagrangian representation we obtained a Lagrangian formalism with a Chern-Simons-like term. A Hamiltonian analysis is also done, what leads to the quantization of the system.

Quantization of Dissipative Systems with Friction Linear in Velocity

It is shown that a dissipative system with friction linear in velocity can be analytically represented by two different Lagrangians which are harmless at the classical level. But they lead to different quantum mechanical systems. This is explicitly demonstrated by using both canonical quantization procedure as well as path integral method.

Heisenberg’s quantization of dissipative systems

The quantization of the damped one-dimensional harmonic oscillator is carried out. This is implemented by first writing the classical equations of motion that describe the system in a Hamiltonian form, the Hamiltonian being time independent explicitly. Canonical quantization is then implemented by using Heisenberg's picture. When friction is turned off the results agree with those of the one-dimensional harmonic oscillator.

Hamiltonian description and quantization of dissipative systems

Dissipative systems are described by a Hamiltonian, combined with a “dynamical matrix” which generalizes the simplectic form of the equations of motion. Criteria for dissipation are given and the examples of a particle with friction and of the Lotka-Volterra model are presented. Quantization is first introduced by translating generalized Poisson brackets into commutators and anticommutators. Then a generalized Schrodinger equation expressed by a dynamical matrix is constructed and discussed.

The stochastic approach for non-Hamiltonian systems

The stochastic quantization of dissipative systems is discussed. It is shown that in order to stochastically quantize a system with dissipation, one has to restrict the Fourier transform of the space-time variable to the positive half domain in the complex plane. This breaks the time-reversal invariance, which manifests itself in the formulation through the resulting noninvariant forms for the propagators. The relation of the stochastic approach with the Caldeira and Leggett path-integral method is also analyzed.

The Heisenberg picture is not privileged for the canonical quantization of dissipative systems

The Heisenberg picture is not privileged for the canonical quantization of dissipative systems

Quantum Langevin systems

It is shown that the canonical quantization of dissipative systems, following a Langevin equation, reduce the physical applicability to particular approximations.

On the noise operator approach to quantized dissipative systems

Invoking complex classical coordinates and momenta a consistent Hamiltonian theory suitable for the quantization of dissipative systems has been developed previously. In another paper this formalism has been illustrated on the basis of a simple order parameter equation by means of density operator techniques. This quite naturally calls for a comparison with quantum noise operator techniques. The present paper is an attempt to satisfy these demands. Extensive use will be made of operator ordering techniques and quasi-classical Fokker-Planck equations. As before, a certain incompleteness in the extractable information is clearly exhibited. It will be observed that the two techniques do not produce similar results in a general dynamical state as a consequence of dissipation. However, in the stationary state and within certain approximations both methods do lead to identical conclusions for the order parameters statistics. It will be argued that within the present context in general noise operator techniques are to be favoured.

On the quantization of dissipative systems

A recent paper of Dekker on the quantization of dissipative systems is examined in some detail. It is argued that one can construct a large number of classical equivalent Hamiltonians for damped systems. These can be formally quantized according to Dirac's method, and the resulting equations are mathematically consistent, but yield different eigenfunctions for the same classical system. However, this procedure should be rejected on physical grounds. That is in quantum mechanics, unlike classical dynamics, the definition of the time derivative of a dynamical variable is unique, and is given by the commutator of the proper Hamiltonian (or the energy operator) and that variable. If the proper Hamiltonian is used for the quantization of a damped system, then the quantal equations are inconsistent for the cases where the rate of energy dissipation depends on the velocity of the particle. As an alternative approach to the quantal theory of dissipative phenomena, a generalization of the Hamilton-Jacobi formalism is considered, where the equation for the principle functionS, depends not only on the space and time derivatives ofS, but onS itself. This leads to a new class of damped systems in classical mechanics. The original Schrodinger method of quantization via the Hamilton-Jacobi equation has been applied to this class of dissipative systems, with the result that the wave equation in this case is a solution of a non-linear Schrodinger-Langevin equation. This formulation has no analogue in the Hamiltonian approach, since in the latter, the resulting wave equation is always linear.

On the statistics of quantised dissipative systems

The quantum statistical aspects of a simple classical order parameter equation will be discussed in view of a previously developed procedure for the quantization of dissipative systems in the Lagrange-Hamilton formalism. Regarding the lack of knowledge concerning the quantum nature of the physical interactions underlying the systems behaviour, however, one can of course only expect to obtain incomplete information about its quantum noise features. After quantising the classical equation of motion the density operator equation is derived.

A Note on the Quantization of Dissipative Systems

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.