Dissipative Evolutions in Quantum Mechanics

Roumen Tsekov

Jürg Fröhlich, Alessandro Pizzo

This note starts with a recapitulation of what people call the ``Measurement Problem'' of Quantum Mechanics (QM). The dissipative nature of the quantum-mechanical time-evolution of averages of states over large ensembles of identical isolated systems consisting of matter interacting with the radiation field is discussed and shown to correspond to a stochastic time-evolution of states of individual systems. The importance of dissipation for the successful completion of measurements is highlighted. To conclude, a solution of the ``Measurement Problem'' is sketched in an idealized model of a double-slit experiment.

Yuli V. Nazarov, Jeroen Danon

R. J. Wysocki

We propose a method of determining quantum evolution equations for dissipative systems. Our approach is based on the observations that (1) the equations of quantum mechanics for a Hamiltonian system correspond to the classical equations describing the evolution of a special statistical ensemble of copies of the system, and (2) the quantum equations can be determined if the corresponding classical equations are known and the satisfaction of the uncertainty principle for particle position and momentum is postulated. Namely, if these statements are required to remain valid for a dissipative system, the quantum evolution equations for that system can be obtained.

N. G. van Kampen

Giuseppe Vitiello

The formalism of Quantum Mechanics is based by definition on conserving probabilities and thus there is no room for the description of dissipative systems in Quantum Mechanics. The treatment of time-irreversible evolution (the arrow of time) is therefore ruled out by definition in Quantum Mechanics. In Quantum Field Theory it is, however, possible to describe time-irreversible evolution by resorting to the existence of infinitely many unitarily inequivalent representations of the canonical commutation relations (ccr). In this paper I review such a result by discussing the canonical quantization of the damped harmonic oscillator (dho), a prototype of dissipative systems. The irreversibility of time evolution is expressed as tunneling among the unitarily inequivalent representations. The exact action for the dho is derived in the path integral formalism of the quantum Brownian motion developed by Schwinger and by Feynman and Vernon. The doubling of the phase-space degrees of freedom for dissipative systems is related to quantum noise effects. Finally, the role of dissipation in the quantum model of the brain and the occurrence that the cosmological arrow of time, the thermodynamical one and the biological one point into the same direction are shortly mentioned.