# Probability and Irreversibility in Quantum Mechanics

*Symposium organized by M. Gadella and I. Antoniou from 4 to 8 **July, **1**999*

Participants

ACCARDI Luigi (Universita degi studi di Roma Tor Vergata), ANTONIOU Ioannis (International Solvay Institutes, University of Brussels), BHAMATHI Gopalakrishnan (University of Texas, Austin), GADELLA Manuel (Universidad de Valladolid, Spain), GUSTAFSON Karl (University of Colorado, Boulder, USA), HASEGAWA Hiroshi (Ibaraki University, Japan), RIDA Takeyuki (Meijo University, Nagoya, Lapai), HORWITZ Larry (Bar-Ilan University, Israël), KARPOV Evgueni (International Solvay Institutes, Brussels), KTORIDES Christos (University of Athens), NOVOZMILOV Yuri (St. Petersburg University, Russia), ORDONEZ Gonzalo (University of Texas, Austin), PETROSKY Tomio (University of Texas, Austin and International Solvay Institutes, Brussels), POMEAU Yves (Ecole Normale Supérieure, Paris), PRIGOGINE Ilya (International Solvay Institutes, Brussels and University of Texas, Austin), PRONKO George (Instituto for High Energy Physics, Moscow region), REICHL Linda (University of Texas, Austin), SAITO Kimiaki (Meijo University, Nagoya, Japan), SI Si (Faculty of Information, Science and Technology, Japan), SMOLYANOV Oleg (Moscow State University, Moscow), SUBBIAH Suresh (University of Texas, Austin), SUCНANECKI Zdzislaw (International Solvay Institute, Brussels), SUDARSНAN George (University of Texas, Austin), TASAKI Shuichi (Nara Women’s University, Nara, Japan), WILLOX Ralph (University of Tokyo, Japan).

**Report **

The first talk was given by Ilya Prigogine. Hе presented a general overview of the role of irreversibility on out concepts of space time and on the formulation of dynamics. It is the outcome of collective work done by his colleagues over the last year (Antoniou, Hasegawa, Karpov, Ordonez, Petrosky and others). Hе started by reviewing the Misra-Prigogine-Courbage formulation of the Time Operator and of the Entropy Operator. Both are superoperators leading to *non-local *descriptions quite different from the pointwise description of classical dynamics (for a more detailed description of quantum mechanics, see the lecture by G. Ordonez). He next talked about situations where there exist probabilistic solutions with Broken time symmetry (chaotic maps). Special attention was devoted to the thermodynamic limit. It is shown that in this limit the probability density goes out of the Hilbert space and an intrinsically probabilistic description emerges. Thermodynamic systems as well as interacting fields require a new description outside of the Hilbert space. Irreversibility requires, therefore, new formulations of classical and quantum mechanics (see the presentations by Ioannis Antoniou and Tomio Petrosky). After a few remarks on kinetic theory (non markovian effects and causality) Prigogine discussed a problem flot solved by orthodox quantum mechanics: in the nature of dressed unstable states (see also the presentation by G. Ordonez). Of special interest are the kinetics of dressing. These lead to limits for Einstein causality both in quantum and classical systems and therefore to questions about the validity of Einstein’s causality of these existing superluminal effects. In conclusion, irreversible processes alter classical and quantum dynamics (to be described by density matrices in generalized spaces) and lead to questions about the validity of relativity when signals are produced by irreversible processes (Fermi two atoms model).

After the overview by Ilya Prigogine, Hiroshi Hasegawa presented in his talk, *Thermodynaтics of Simple Chaotic Systems, *a simple one-dimensional chaotic map for which the equilibrium density is an one-dimensional step function with the step at certain point that could be changed. When the parameter is changed the system goes out of equilibrium. A different thermodynamical behaviour of the system emerges, depending if the change in the parameter is slow (adiabatic) or not.

Next, Shuichi Tasaki presented his talk entitled Quantum *Nonequilibrium Stationary States. Hе *posed again the old problem consisting on the contradiction between the reversibility of the fundamental equations of microphysiscs with the irreversibility of the macrophysical world. To understand it he studies an one-dimensional lattice model of a conductor with impurities. Impurities in conductors produce resistivity which may in turn cause irreversibility in physical phenomena.

Ralph Willox in his talk *On the connection between тatriх integrals and integrable systems *presented a confection between the theory of integrable nonlinear partial differential equations and certain quantities arising in physically interested. He made use of a particular type of functions called the tau functions and used them to establish a bridge between matrix models and a particular type of integrable system (the Kadomtsev Petviashvili equation).

The next session was devoted to the study of several aspects of time dependent stochastic processes as a particularly interesting case of complex systems. In his talk entitled *Coтpleхity and Irreversibility in Stochastic Analysis, *Takeyuki Hida proposes a strategy to deal with stochastic complex systems. One starts from a simple stochastic process such as a Brownian motion or its time derivative, in this case the *white noise. *A linear superposition of the stochastic variables forming this simple process can be created easily in terms of an integral. This is what is called a *linear functional *of the simple process. Nonlinear functionals of simple process provide new stochastic processes with a higher degree of complexity. In order to measure the degree of complexity of a stochastic process, Takeyuki Hada proposes a concept of multiplicity which resembles the multiplicity of operators on a Hilbert space. The higher the degree of multiplicity, the higher the degree of complexity of the sуstem described by the process.

This method of constructing complicated processes starting with simple oves has several applications. The so called *Wiener **expansions *permit the study of unknown complex systems that permit a white noise input and has an output that can be compared with the output of another known system that has undergone the same input. *Frequence modulation *can be studied by using exponential functions of quadratic functionals of white noise. Finally information of complex systems can be obtained by studying their output under the assumption of white noise input data. Takeyuki Hida finished his lecture with a sufficient condition for the functional of a white noise to represent an irreversible process, which is closely related with its concept of multipliсity.

The next speaker Si Si started her talk, entitled *Random Irreversible Phenomena, *in the observation that X-ray emissions from black holes present a pattern which is very similar to the trajectories of stable stochastic process. The problem is now to obtain the input signal under the following optimization hypothesis: the output should be a Gaussian process. The strategy to obtain the solution is based on a concept introduced in 1949 by S. Bochner and called *subordination: *the observed data is a Gaussian process for which the time is the input process to be determined. This input is considered to be a Poisson process.

The session was completed after the talk by Kimiaki Saito entitled *Complexitt associated with infinite dimensional Brownian motions, *in which he analyzed stochastic processes regarding them as functionals of an infinite number of Brownian motions. This is also relevant for the study of the quantum Brownian motion. He described certain technicalities in the study of the infinite dimensional Brownian motion such as the infinite dimensional Laplace operator.

In his presentation *Probability, Geometry and Irreversibility in Quantum Мechanics, *Karl Gustafson recalled the Les Treilles meeting held in 1996, when Luigi Accardi delivered a series of afternoon informal talks on *Quantum Probability, *where he mentioned a generalization of the Bell inequalities. These inequalities lead to an experimental test (realized by Aspect in France) which showed a contradiction between the completeness of quantum mechanics and the validity of hidden variables theories. Karl Gustafson realized that Accardi’s inequalities were indeed a particular case of an operator trigonometry developed by him twenty years ago. The results predicted by the Gustafson operator trigonomеtry go indeed beyond Bell’s inequalities (in its more refined version due to Wigner) as he can propose a test to decide experimentally if the Hilbert space used in quantum mechanics should have a *real or complex structure.*

The talk by Tomio Petrosky about *quantum Decoherence and Time Symmetry Breaking *summarized a collaboration with Ilya Prigogine, Gonzalo Ordonez, V. Bassegov and Evgueni Karpov. Pure quantum states are often a linear superposition of other pure states. As we construct the corresponding density operator for these superposed quantum states, we obtain two terms. One is a miхturе of the pure states that made the original state by superposition. The other includes mixed contributions of these states only. Ву interaction with the environment, presence of resonances and other effects, these mixed contributions may vanish quickly with time. The mixed terms are responsible for many quantum effects that disappear as the mixed terms disappear. This is called decoherence. Controlling decoherence is vert’ important in the design of quantum computers. In addition, decoherence produces irreversibility in quantum processes and produces the appearance of classical effects in the microphysiсs. The process of decoherence may be described as a transition from a non Markovian process (memorу effects prevail) to a markovian process (in which memorу effects play no role). _{Ву }using the spectral decompositions of the Liouville operator, the technique developed bу the Austin-Brussels group, the authors provide a consistent description of the non Markovian process on a microphysical level without using nonphysical extra assumptions. Then, it is possible to estimate the scale at which decoherence occurs, show that is bigger than Zeno time (the time at which resonances start to decаy exponentiallу) and establish that the Pauli master equation has only a validity after this Zeno era.

The two next talks were presented by Gonzalo Ordonez. The first one entitled *Time Operator for the Friedrichs model *discussed various features of the Time Operator. The idea of Time Operator comes after the analogy between the uncertainty relations position-momentum and energy-time of Heisenberg. Pauli showed that if Time is to be considered as an observable, then the Time Operator must satisfy the canonical commutation relation with the Hamiltonian, which is forbidden by the semiboundedness of the spectrum of the Hamiltonian. Misra, Prigogine and Courbage showed that Time Operators should be defined in such a way that they fulfil the canonical commutation relation with the Liouvillian. Examples of Time Operators have been constructed by the group of Brus sels for certain chaotic maps, like exact maps, Kolmogorov maps or diffusion (Antoniou, et al.). In his second talk, Gonzalo Ordonez presented a Liouville space formulation of *dressed *quantum unstable states (or unstable particles) and showed that they correspond to an irreducible density matrix on quantum systems like the Friedrichs model with broken time symmetrу. Hе then showed numerical simulations corresponding to the time evolution from bare to dressed states (stable or unstabie) and to resonance scattering. In all cases there appear non locality and superluminal propagation in agreement with the recent work of Hegerfeld.

Tomio Petrosky presented two talks. In the first he considered again non localitу and superluminal effects. Considering the interaction between an oscillating and a classical field he showed that these effects appear also in classical dуnamics and lead to a violation of Einstein’s causality. The same is true for the interaction of two relativistic fields (model of Antoniou, Gadella, Pronko and Prigogine, 1998).

In his last presentation Tomio Petrosky emphasized the analogу between fields and thermodynamic systems. Interacting fields are dissipative systems which are flot Poincaré integrable. A spectral representation can be found for the Liouville representation outside the Hilbert space. Hе illustrated his talk using the A ? B+C model. Here again we can obtain the *dressed *A particle. New creation destination operators including dissipation can be defined and lead to a dissipative Fock space.

Next came Gopalakrishnan Bhamati who presented ber work in collaboration with George Sudarshan under the title *Jordan Block Spectra in Dynamical Мodels. *She discussed several models for unstable states like the Friedrichs-Lee, Cascade and rearrangement models. She showed how the Friedrichs-Lee model may present multiple pole resonances, which produces these Jordan Block structures, and the other models other kinds of resonance singularities like branch cuts.

The session ended with Yves Pomeau and his talk *Bose-Einstein Condensation in the Framework of Quantum Kinetic Theory *in which he retook an old interpretation of Ilya Prigogine to study the relation between superfluidity and nonequilibrium kinetic theory.

The morning session of the next day included two talks on waveguides on cavities. In the first one, Suresh Subbiah in *Nonlocality in quantum wаveguides, *described a two dimensional model of waveguides inside a rectangular сavity with two leads. This model can be еxaсtlу solved because it can be written as a Friedrichs model which shows resonances as branch cuts. Precisely the presence of these branch cuts mау be responsible for a superluminal scattering out of the саvity.

The second talk, presented by Linda Reich’ and entitled *Universality and Power Absorption in Open Quantum Systems *was intended to show evidence for universality in the scattering properties of chaotic electron properties. She studied the effect of quasibound states (resonances) on power absorption in certain open quantum systems. Among these systems, rectangular cavities, quantum billiards with a ripple, rectangular cavities with a ripple are considered with a special attention for the scattering on cavities. Pole resonances and branch cut resonances are found for the rectangular resonance and delay times and transition probabilities were obtained for the cavity with a ripple. She finished her presentation with several comments on the dynamical stabilization of electrons in high energy laser fields.

Then, Georges Sudarshan spoke on *Continuum Gamow Vectors. *Gamow vectors were considered as linear functionals on functions analytic on the complex upper half plane. In general, in the Liouville space, there is a continuum of such states which form an overcomplete set. Hе introduced an Age Operator, similar to the Time Operator discussed in previous talks and discuss the problem of statistics for unstable particles.

In his talk, *Friedrichs Model with virtual transitions, *Evgeni Karpov summarizes a joint work with Tirio Petrosky, Ilya Prigogine and George Pronko. The authors consider a harmonic oscillator interacting with a scalar field. The interaction is quadratic and depends on a form factor, and a coupling constant so that the model is equivalent to a Friedrichs model and can be solved exactly. After obtaining the annihilation and creation operators which diagonalize the total Hamiltonian, they found the corresponding vacuum which is obtained by means of a Bogoliubov transformation. The time evolution of the original vacuum and the original creation an annihilation operators (here called bare vacuum and bare operators) are obtained. Using the exact solution for the model, a previous result of Roberto Passante, Тоmiо Petroskу and Ilya Prigogine that the transition from the bare vacuum to the true vacuum leads to the emission of a photon, is confirmed. Therefore the transition from the bare to the classical ground state is an irreversible process. Resonance behaviour has also been studied in this model. His presentation closed the Tuesdaу sessions.

The first Wednesday session was devoted to the Liouville space with the talks *The spectrum of the Liouville von Neumann operator and extensions of Quantum Theory *by Ioannis Antoniou and *Decay and singular spectrum in classical and quantum theory *by Zdzislaw Suchanecki. The authors first corrected certain wrong statements that textbooks and research works disseminate on the properties of the Liouville operator. The point of departure was their result (with Stanislav Shkarin), according to which a Hamiltonian with a pure continuous singular spectrum produces a Liouvillian with an absolutely continuous spectrum, contrarilу to what was accepted up to now that the Liouvillian must have a continuous singular spectrum only. (This result may have dramatic consequences because it shows that quantum mechanics and, in particular, quantum scattering, has different properties as we lift it to the Liouville space.) They provided the correct formulas that relate the various types of spectra of the Hamiltonian with the various types of spectra of the corresponding Liouvillian. Finally, they provided a decomposition of continuous singular measures on the real line into decaying and non-decaying measures. The Fourier Transform of the decaying measures vanishes at infinity. This new decomposition of measures leads to an intrinsic decomposition of spectra and states into Decaying and Purely Non-Decaying. The Decaying states are exactly the mixing states describing approach to equlibrium.

One of the most awaited presentations was that of Luigi Accardi, who is an expert in stochastic methods in quantum mechanics. As he recognizеd in his talk, his work is addressed to « implement in quantum mechanics what Kolmogorov (with his formalization of classical probabilitу) did in classical mechanics ». This means that he and his collaborators are establishing a new mathematical probabilistic formalism called *Quantum Probability *that should include ordinary quantum mechanics as well as quantum field theory as particular cases. Hе started his talk by making a review of the concept of scaling and its consequences. Scaling mау relate a new theory with the old one (as for example we relate relativistic with classical mechanics by allowing the speed of light go to infinite), but it also тау perform the inverse operation. Scaling mаy establish a bridge between the new probabilistic developments and the ideas of intrinsic irreversibility developed bу Ilya Prigogine and coworkers. To illustrate it, he suggested the example of those systems depending on a small parameter, in which the van Hove *weak coupling limit *(the product of the square of the parameter with time is constant) is made possible. In these circumstances, it is possible to construct a quantum white noise for any interacting Hamiltonian, possessing the physical properties of the model. Hе finished his talk recalling the notion of stochastic limit, its relation to the general Prigogine program on the thеorу of irreversibility and the emergence and importance of interacting Fock spaces.

The evening session started with the Fock Medal Ceremony, an award from the Russian Acadеmy of Sciences to Professor Ilya Prigogine. After this cеremony, Yuri Novozhilov made an enlightening speech about the work and personal circumstances of Fock in the Soviet era. As a phуsicist, he worked on quantum mechanics and wrote an outstanding book on quantum mechanics, published in Russian in 1930, in which he already presented results which were « rediscovered » уears later in the West [like the Breit-Wigner formula for resonances, rediscovered by these authors six уears later]. Fock’s work on gravitation clarified in a unique way the relation between gravitation and relativity. Apart from these two main fields, Fock made occasional incursions on Geophysics, Radiophуsics and Optometrу.

The next speaker was George Pronko who in his *Quantum Мechanics of extended **relativistic objects *discussed verу general principles of quantum mechanics. The simplest object of this kind consists of two particles separated bу an one dimensional rod, but it may include more interesting objects like a model for an hadron in which three quarks are tied to a center by corresponding strings. A simple model of quantum relativistic extended objects was shown to be an analogue of the hydrogen atom in nonrelativistic quantum mechanics. The same model serves for a description of the spectrum of meson states.

Next, Oleg Smolуanov gave respective talks on *Schrёdinger Stochastic Equations and Related **Topícs *and *Functional representation of Fock spaces in quantum and classical dynamics. *In the former he presented several types of linear equations of first order in time derivative and discussed when this kind of equations have solutions. This depends strongly on the spaces in which the unknown function is defined. Hе presented an example in which a Hamiltonian system has no solution other than the trivial one, but the corresponding Liouville equation has more than one solution, thus connecting with the work of Antoniou and Suchanecki. In his second talk, he extends the comparison between Hamiltonian and Liouvillian systems to those having an infinite number of degrees of freedom. With Prof. Smolyanov the sessions on Wednesday were over.

The first two talks on the next dау were devoted to field theory. In the former entitled *A regularied formulation of gauge field théories in the continuum: successes and limitations, *Christos Ktorides recalled the difficulties to construct a consistent well fundamented theory for Quantum Chromodynamiсs (QCD). QCD presents problems like quark confinement which are very difficult and that painstaking steps have been made toward its proper solution. In this direction two different strategies have been proposed. In one of them, fields are replaced by lattices, statistical mechanical models are used in the lattice and then the distance between vertices go to zero. The other strategy is to play directly with fields as continuum systems. Both Stratégies cannot avoid the infinities so that regularization techniques are necessary. Christos Ktorides presented a scheme of nonperturbative regularization of gauge field theories.

Next, Manuel Gadella in his talk *Riggеd Fock spaces with applications to quantum field theory and irreversibility *presented a joint work with Ioannis Antoniou, Ilya Prigogine and George Pronko. A quadratic interaction between a local scalar field and a bilocal scalar field with continuous mass spectrum can be eхaсtly solved and shows resonance behaviour. The Gamow vectors for this resonance as functionals on suitable rigged Fock spaces are explicitly constructed.

Larry Horwitz presented a quantum version of Lax-Phillips theory, a theory for scattering of acoustic waves which has been extended to more general scattering problems, including quantum scattering. In his talk *Représentations of the Resonances of Quantum Field theoretical Lee-Friedrichs models in Lax-Phillips scattering theory, *he extends this formalism to a soluble relativistic model.

The last talk of the meeting was *Composite particles in quantum field theory *by Yuri Novozhilov. New approaches to description of composite particles in quantum field theory, which are suitable for the case of strongly bound systems, are presented. Explicit construction in terms of the constituents is avoided. The study is made through the path integral techniques in quantum field theory.