# Probabilistic extensions of classical and quantum dynamics

Symposium organized by Ilya Prigogine and Ioannis Antoniou from July 4 to 10, 1996.

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**Participants**

Luigi Accardi (Università di Roma, Italy), Ioannis E.Antoniou (Instituts Internationaux de physique et de chimie Solvay, Université Libre de Bruxelles, Belgique) Vasili Basios (Instituts Internationaux de physique et de chimie Solvay, Université Libre de Bruxelles, Belgique), Viacheslav Belavkin (University of Nottingham, UK), Viatcheslav Belyi (ULB), Gopalakrishnan Bhamathi-Sudarshan (University of Texas, Austin), Mario Castagnino (Universidad de Rosario & Buenos Aires, Argentine), Pierre Coullet (Laboratoire de physique théorique, Sophia Antipolis, Valbonne), Maurice Courbage (Université de Paris VI), Ludmila Dmitrieva (St Pertersburg University, Russia), Dean Driebe (Ilya Prigogyne Center for statistical mechanics and complex systems, Austin, USA), Manuel (Manolo) Gadella (Universidad de Valladolid, Spain), Edgard Gunzig (ULB), Karl E.Gustafson (University of Colorado, Boulder, USA), Hiroshi Hasegawa (Ilya Prigogyne Center for statistical mechanics and complex systems, Austin, USA), Yuri Kuperin (St Persburg University), Roberto Laura (Universidad de Rosario & Buenos Aires, Argentine), Fernand Mayne (ULB), Yuri Melnikov (St Pertersburg University, Russia), Baidyanath Misra (Instituts internationaux de physique et de chimie, Solvay, ULB), Gonzalo Ordonez Ilya Prigogyne Center for statistical mechanics and complex systems, Austin, USA), Roberto Passante ( Consiglio Nazionale delle Ricerche, Palermo), Boris Pavlov (St Petersburg University, Russia), Andrei Pereverzev (University of Texas, Austin, USA), Tomio Y. (Tom) Petrosky (Ilya Prigogyne Center for statistical mechanics and complex systems, Austin, USA), Ilya Prigogine (Instituts internationaux de physique et de chimie, Solvay, ULB), George Pronko (Institutes for High Energy Physics, IHEP, Protvino, Russia), Linda Reichl (Ilya Prigogyne Center for statistical mechanics and complex systems, Austin, USA), Zdzislaw Suchanecki (Instituts internationaux de physique et de chimie, Solvay, ULB), George (Ennackal Chandy) Sudarshan ((University of Texas, Austin, USA), Shuichi Tasaki (Institute for Fundamental Chemistry, Sakyo-Ku-Kyoto, Japan).

**Summary**

*“Time is **a **Child Playing **Dice, **the **Kingdom belongs to **a **child. “, *Heracletus, Fragment n° 52

It was beyond the expectations of even the most орtimistiс observers that only two years after the last symposium, here in the Treilles, about “Time, Chaos and Resonances”, such a burst of creativity would take place. If something prives the success of an approach this is its power to invoke creative responses following up. From this point of view, the approach out lined by Ilya Prigogine and his collaborators two years ago, proved to be very successful.

The participants gathered here in 1996 to discuss, and plan their future research activities in the light of the recent deve****lopments in the foundations of physics. Indeed the theme of the conference was the *“Extension of **Classical and **Quantum **Mechanics **for **Unstable Systeтs.” *Thirty on scientists, participa****ted in the conference. Their fields of interest and expertise span the area of Theoretical Physics, thеir countries of origin span the globe.

All found themselves at home in the Treilles, thanks to the warm hospitality of the organizers and the natural beauty of the scenery. The conference lasted for seven days (July 5th till July 10th, 1996) .

**Report**

*Time **and Stochasticity in Physics and Philosophy*

The opening adress “*The **А**rrow of **Time **and the problem of the **two cultures” *was given by Ilya Prigogine, in the morning of Friday July 5 th. Ilya Prigogine, who co-organised the meeting with Ioannis Antoniou, is the Director of the International Solvay Institutes for Physics and Chemistry in Brussels, and Director of the Center for statistical Mechanics and Complex Systеms in Austin, Texas. In order to introduce the subject as well as the theme of the conference, he started by highlighting the important turning points of earlier research in nón-equilibrium statistical mechanics. Не explained the suссess of dissipative non-equilibrium structures and the early theory of subdynamics, gave useful hints for a final attack on the problem of irrevеrsibility. Не also gave a detailed account of why irreversibility demands a re-evaluation of our understanding of the fundamental laws of nature. The relation between the time-reversible, mechanistic, paradigm and the observed irreversible, innovative reality, as we know it, is an ages old debate. It has divided the scientific and humanistic cultures. The nature of time is here a central element.

These debates draw their origins from the other, older and equally profound debate of determinism versus indeterminism. They also involve issues of free will and creativity. That is why the two cultures, the humanistiс and the exact sciences, both are concerned with the debate. Tracing back this long history of ideas, Prigogine took us to a mental journey starting from antiquity and reaching our times.

Prigogine started with the classics, narrating how Plato’s dialogue “Parmenides” is dealing with the question of what is the “Olon” (whole, complete), this which is Being and Becoming, the “One and Absolute” of the platonic tradition. In this dialogue which is thought of as a classical exercise in dialectics with the original meaоing of the word, as he pointed out, we encounter for the first time the question of Time and its philosophical implications. The dialectic opposites of Being and Becoming serve here to describe the transcendental nature of “Olon” which is God for Plato. Plato places the question of time in central position and build around this his metaphysics. We will see in the following, how the question of time imposes itself in a central position in physics too. Continuing the lecture, he pointed out that in fact we have received a long heritage from Leibniz and Descartes and Newton where for the first time western thought conceptualizes the idea of the Law of Nature. The supreme example is Newton. The formulation of his Laws of Motion in *Principia, *and what followed, newtonian physics with its great successes mark the birth of science. It was inherited to us through the last three hundred years.

The example of newtonian laws, which are time reversible and deterninistic sets the floor for any further discussion. All other investigations of the laws of nature submit to these two premises. Classical physics world view is an atemporal one. Time is just a parameter and implies no evolution. The des****cription is time reversible and nature acquires the specifications of a good clock. God is the absolute clock-maker. Deviations from determinism seem only to be associated to concepts such as “chance”, or “accident” which are from the start anthropomorphic. Einstein, with his Theory of Relativity gives a more fluid description of time, still determinism and reversibility remain at the basis of the laws of nature.

Not even quantum mechanics challenged the time reversible formulation of the fundamental laws of nature. Although quantum mechanics goes beyond the traditional physics by introducing the concept of wave functions which determine probability amplitudes, nevertheless it remains a time reversible and deterministic theory ruled by Schrodinger’s equation. In the conventional Hilbert Space Formulation Irreversibility can only be introduced by “measurement” through the action of an observer.

The formulation of the Laws of Nature has always been the primary goal of Theoretical Physics, since its birth almost four centuries ago. This scientific field, fundamental for ail exact sciences, has witnessed quite a number of revolutions, always in the grand scheme of newtonian guiding principies. We now witness a third major scientific revolution. Complexity and Non-Integrability challenge the mechanistic world-view and lead to a probabilistic and temporal orientation of physics. This last scientific revolution happens, at the end of this century, in our time.

Yet, one should keep in mind the parallel developments during this century of scientific revolutions. These happened in statistical mechanics, chemistry and biology which kept on pointing to another picture of the universe. The physics of large systems, be it on statistical mechanics thermodynamics chemistry, or biophysics, аll are equipped with a privileged direction of time. The manifestation of the arrow of time as the Second Law of Thermodynamics dictates, is deeply rooted in their experimental and theoretical foundations. The concept of Entropy reveals that diere is a problem of time in science. How can the arrow of time arise from the time reversible laws of motion?

What is now often called “Complexity Theory” stemmed out frоm recent developments in non-equilibrium statistical mechanics, dynamical system theory and the theory of computation.

Its implications touch all areas of exact sciences. The research in Complexity associated with biology, astrophysics, image processing even cognitive and social sciences experiences a flourishing. Its unique interdisciplinary nature and its cultural implications make this last scientific revolution the most challenging of this century.

Prigogine gave a historical overview of the solution of the problem of irreversibility starting with the дilетmа of Epicurus which led him to the concept of “dynamen”. Today the conventional interpretations associate irreversibility either to “approximations” or the initial cosmological conditions. As Ilya Prigogine pointed out both interpretations are questionable. The observation of dissipative structures shows that irreversibility cannot be due to approximations. The arrow of time has a constructive role. Initial conditions are not sufficient to explain that we observe today both time reversible phenomena (i.e. the two body problem, earth-sun) and irreversible processes (like heat conduction) . We have to extend the basic laws of nature to include both. We have to describe the dynamical ruts of irreversibility. This has been done in detail in subsequent talks (see specially the lecture by Dr Tomio Petrosky) . In this perspective the laws of nature acquire a new meaning. They no more express certitudes but possibilities and probabi****lities as appropriate to an evolving universe. The future is not given, it is constructed while is going on.

Just after the introductory lecture we listened to George Sudarshan. Не continued in an epistemological and philosophical tone, mentioning the particular relation that each civilization develops with the concept of time. Не presented to us the *Concepts of **Time **in **Indian Philosophy, *referring also to the other great Asian civilization, the Chinese civilization. George Sudarshan started by drawing the well known analogy of the platonic interpretation of the world of ideas, with that of the Brahmanic Indian tradition. In both, substance is timeless while the world of phenomena is ruled by time. Outside this world of illusions, outside “Мауа”, present, past and future are one. Of course in Indian tradition and philosophy, like in any other, they coexist different schools. A detailed account of the Kala-Chacra, the wheel of time was then presented by the speaker.

It is inside this Kala-Chacra body of philosophical thinking where one can encounter the eschatological aspect for time. As Brahma breathes just once our whole universe comes to birth and meets its death with the end of each breath. A detailed account of the other subdivisions of Time were presented. A comparison with the western ideas, starting from Egypt, Judaic and Christian concepts of time opened the door to an unexpected interesting discussion.

The morning session of the first day concluded with a dis****cussion. It was anticipated that our Japanese colleagues also should take the floor to describe their traditional views. Indeed, Tomio Petrosky and Shuichi Tasaki, both noted that in Japan, usually a more pragmatic aid less ideological approach is prevalent. The problem of time although of great importance in physics, holds no reference or relevance to the mainstream Japanese philosophical problematic. “We are more open than the western colleagues to all kinds of different approaches and ideas, as long as they withstand the scientific criteria and provide novel directions and results”, both of the participants emphasized. For example, the Cartesian dua!ism holds no special epistemological or metaphysical status in Japan. Finally Ioannis Antoniou gave an overview of the future directions that the probabilistic revolution in physics may take. Не started his comments with the Heracletian fragments, mentioned by Ilya Prigogine, and pointed out that these rest on a different kind of logic, other than the Boolean logic which is the logic of stable dynamical systems. Не concluded also that science is in front of a great and challenging era of transformation and new way of scientific thinking is emerging.

*Probabilistic Evolution **of **Discrete Systems*

The afternoon section of the first day was devoted to discrete dynamical systems like maps of the unit interval. Maps p!ayed an important role in the early days of chaos theory. They served as a paradigm for the study of chaos and many important concepts spurred out of these studies. Nowadays, the spectral decomposition of chaotic maps has demonstrated the role of intrinsic irreversibility inherent in unstable systems. The probabilistic description of these maps comes as the natural one since trajectories are excluded from the domain of their evolution operator. This is so, since in order for these maps to acquire a spectral representation one has to go beyond Hilbert spaces and investigate them in the context of more general spaces, the so called rigged Hilbert spaces.

A rigged Hilbert space or Gelfand Triplet, is a construction which takes into account not only the oscillating but also the decaying modes present in the dynamics. It provides the mathematical framework for the probabilistic aid irreversible description of unstable systems. The arrow of time manifests simply and directly, as the evolution operator splits into two semi-groups. Riggеd Hilbert spaces for discrete unstable systems have been constructed by Ioannis Antoniou, Zdsislaw Suchanecki and Shuichi Tasaki.

On these grounds, Dean Driebe presented how one, by considering symmetry arguments, can simplify the construction of eigenvectors and eigenvalues of symmetric one-dimensional caps. Не also presented his new results on maps with Jordan block structure in the spectra.

Later Hiroshi Hasegawa talked about certain simple models which enable one to understand *Non**–**stationary Chaos associated to Anomalous Diffusion.*

Consequently, Vasilios Basios presented a method for *Probabilistic **Control **of Chaos *developed in collaboration with Ioannis Antoniou and Francisco Bosco. The idea being here that if one abandons the task to control trajectories but rather focus on controlling the support of the measures for the probability densifies one can obtain a controllability condition for one dimensional chaotic maps. Problerns that are intractable in terms of trajectories, become tractable in the probabilistic framework, such as the time rеquired to achieve control.

Finally in the end of the afternoon section, Karl Gustafson presented his latest results on *De Rham maps and Information Dimension, *a special kind of one dimensional chaotic maps with infinite invariant measures, which pose great challenges in the mathematics of chaos theory. Не talked about the De Rham maps and proponed a certain selection criterion for their invariant measures based on Information Dimension. In addition, Gustafson gave an interesting historical background on De Rham maps. It turned out that they came up from questions about carving wood sticks, late last century.

*Stochasticity and Irreversibility **of **Classical Dynamics*

The second daу, Saturday’s, morning session was devoted to *Classical Dyпamics and Stochastic Processes. *The session started with the first lecture of Tomio Petrosky: *`Poincaré Reso**nances and **the Extension of **Classical Mechanics ; *who gave an overview of the probabilistic formulation of Hamiltonian Systems. Не pointed out the close relation of Роincaré resonances and non-integrability of many body systems. While most dynamical systems are not analytically integrable in the Poincaré sense, they can be integrated on the probabilistic (Liouville) level. We obtain in this way new probabilistic solutions of classical dynamics. Tomio Petrosky pointed out that this extension of classical dynamics has been verified by numerical simulations. In short Tomio Petrosky described the methods to obtain new spectral representations of the Liouville operator which do not reduce to trajectories and arе manifestly time asymmetric.

The next lecture of the day was given by Linda Reichl. She presented the work of her group on *The Effects **of **Symmetry Breaking **on **Stochastic Processes. *Her lecture outlined their new results on the study of cavities and tunnels where quantum effects and classical chaos interplay. Work on these models provides а deeper understanding of complex mesoscopic systems.

Just after the morning coffee break, Maurice Courbage discussed the *Intrinsic Randomness **of Kolmogorov **systems with Symmetry. *These systems include the classical chaotic systems with positive entropy production.

Next, Pierre Coullet gave a presentation of the role of the *Dynamical Nucleation **of **Metastable **States in **Nonequilibrium Systems. *Nucleation is a crucial step in the process of self-organization and occurs in systems far from equilibrium. The lecture was enriched with a lively presentation of nucleation in self organizing biological systems. Photographs of fossils with colorful pigment patterns could not easily be distinguished from his simulations, they both captured the beauty and complexity of biological diversity. This is one more example of the constructive role of irreversibility in nature.

Closing the morning section, Yuri Kuperin presented a *Hilbert Space Approach to **the **Spectral Analysis of **the Boltzmann **Operator. *He presented new results based on the Extension Theory of operators developed by Boris Pavlov and applied them to the Bokzmann operator.

*Stochasticity and Irreversibility **of **Quantum **Dynamics*

The evening discussions were dedicated to *Quantum Stochastic Processes, *with the opening lecture of Vladimir Belavkin on the *Stochastic Extensions of Classical and Quantum Dy**п**amics and Filtering Th**е**ory. *Quantum stochastic processes is a new rapidly evolving field. Quantum noise and filtering draws considerable attention in view of new technologicai applications. Vladimir Belavkin was awarded a few montos ago the prize of the Russian State for his contributions to this field. Links with the Brussels School discussed.

In the same spirit, Luigi Accardi presented his contribution in *Irreversibility and the Stochastic Limit of Quantum Theory *which generalized the Prigogine Van Hove Asymptotic Theory of Irreversibility based on weak coupling.

Boris Pavlov closed the evening session of the second day with the lecture on Quantum Systems with Markov Background. These Quantum Systems interact with random potentiels and describe important phenomena like Mossbauer spectroscopy.

On Sundаy the morning session was scheduled on *Quantum Systems. *Tomio Petrosky opened the session with *The Liouville Extension of Quantum Mechanics. *Quantum theory has a conflicting historical background. The starting point was the black body radiation corresponding to __equilibrium__ between matter and radiation. On the other hand the Rietz-Rydberg spectroscopic rule led to the accepted formulation of quantum theory which leaves no space for approach to equilibrium. We need an extension of quantum mechanics to include irreversible processes. This theory has now been obtained by Ilya Prigogine and Tomio Petrosky. It follows the lines of the extension of classical mechanics presented earlier at this conference. The basic quantity is now the density operator which can only be reduced to probability amplitudes in simple situations corresponding to “integrable” quantum systems. There is here a close analogy with classical mechanics. We obtain аgain new non-reducible spectral representations breaking time symmetry as evolutions split into two distinct semigroups. As in classical mechanics, quantum laws now describe possibilities than certitudes.

The extension of quantum mechanics leads to numerous predictions come of which where discussed by Gonzalo Ordonez. Gonzalo Ordonez and Tamio Petrosky showed that the extended quantum theory leads to new results even for simple scattering problems. Traditional (“S-matгix”) scattering theory is only valid in the asymptotic limit, as time goes to infinity. Now we obtained results for the first time some of which are quite unexpected, like change in the form of spectral fines, new resonance effects vanishing for large times. All these effects have been confirmed by elaborate numerical simulations. It is hoped that experimentalists will measure soon these effects in real world situations.

Next Roberto Laura talked about *Quantum Systems with Diagonal Singularities. As *well known, diagonal singularities played a central role in the pioneiring work of Prigogine and Van Hove on irreversible processes. What is the role of these singularities in the Bxussels-Austin approach? In simple situa****tions we can describe continuous systems without going through the box normalization procedure. This work was done in collaboration with Bannis Antoniou, Zdzislaw Suchanecki and Shuichi Tasaki.

In the second part of the morning session Ludmila Dmitrieva presented her collaboration with Ioannis Antoniou, Yuri Kuperin and Yuri Melnikov on *Resonances of Periodic Potentials, *a subject which is close to crystals or lattices in solid state physics. Interestingly enough the generalized spectral représentation gives additional important information related to the inverse scattering problem.

Then Roberto Passante talked on *Unstable States **in Quantum **Mechanícs. *This work performed in collaboration with Tomio Petrosky and Ilya Prigogine, shows that a perturbation leading to a lower ground state leads to the emission of optical photons whose energy is slightly superior to that of associated to the perturbation — the total energy being conserved. This is a new unexpected quantum effect showing that in these situations energy is only conserved for the system as a whole — a prediction which would be worthwhile to verify experimentally. The morning session of Sunday dosed with the talk of Fernard Mayné in *Three **Body Processes in the **Lee **Model.*

The last day of the conference was also dedicated to *Quantum **Systems. *The morning session opened with the lecture of Ioannis Antoniou, who is the Deputy Director of the International Solvay Institutes for Physics and Chemistry in Brussels, and co-organized the conference with Professor Prigogine. Ioannis Antoniou gave an overview on a number of research projects which are currently pursued and made pos****sible by the novel probabilistic approach for сотрlех systems. Ioannis Antoniou talked about the *spectral **Decomposition and **Extension of **Dynamics **of **Unstable Systems. *Не made clear the connection between instability, the problems of integrability, computability, control and the problem of isomorphism for chaotic complex systems. All these problems, been shown to be intractable in the conventional framework, acquire a natural probabilistic solution in the new extended formulation through the complex spectral decomposition of the evolution operators. Не also presented new methods and computational tools, which have been recently developed in the framework of the European Russian collaboration project coordinated by the Solvay Institutes.

The next lecture dealt with *The Logic **of **Extended **Dyпamics **of **Unstable Systems, *presented by Zdzislaw Suchanecki. The logic of a system dictates what kind of propositions or ques****tions, one can ask. It turns out that the logic of uristable systems, formulated on the probabilistic level is not the conventional Boolean Logic but a kind of fuzzy logic. Similarly the Logic of Quantum Systems with Diagonal Singularity is a Non-Commutative Logic which allows only finite combinations of propositions. This Logic is more general than the conventional Logic of Von Neumann’s Quantum Mechanics in Hilbert space.

The next lecture of the morning session was about *Gamow **Vectors and Rigged **Hilbert **spaces, *by Manolo Gadella. The rigged Hilbert space formulation has been shown recently to be the mathematical framework for the spectral analysis of chaotic systems. A number of models have been investigated in this way. Rigged Hilbert spaces, and Gamow vectors were developed by Arno Bohm and Manolo Gadella at Austin, Texas, in the eighties with the aim of describing unstable, decaying elementary particles. Consequently, Gupalakrishnan Bhamathi Sudarshan lectured on the subject of *Double **Resonances **and Jordan **Block Spectra. *Together with George Sudarshan, they have developed a method of analytic continuation in the соmрlех plane suitable for identifying resonances. She presented their latest results and a discussion followed on the interrelationships of this analysis with other techniques and approaches.

Then, the second lecture of Boris Pavlov followed. This lec****ture was about *Lax Phillips Scattering Theory’ аfter Lax and Phillips. *Не presented an extension to the classical theory of Lax and Philips which allows for the discussion of more соmрlех systems, non smooth boundaries, even chaotic scattering.

Right after this, Yuri Melnikov presented *Resonance Scattering **on **Graphs. *Graphs are discrete systems so far studied with Topological methods. Yuri Melnikov presents the response of Graphs as a scattering process.

*Evolutionary Aspects at **the **very small and **the **very largе Scale*

The afternoon session of the last day of the conference was dedicated to problems on *Quantum Field **Theory and **Cosmology. *As Ilya Prigogine has repeatedly argued in the past, the arrow of time manifests in all areas of natural sciences. We all share the same direction of the arrow of time because we are part of thе same universe. The arrow of time is not an artifact of our theories but a cosmological fact. Contemporary cosmology, leaves us often with а deterministic and mechanical universe, ignoring the fluctuations and irreversibilry.

Baidhanath Misra opened the session with a lecture on the *Relativistic Time Operator and Chronons. *He introduced a time operator for the Klein-Gordon equation in field theory. He showed that the time operator leads naturally to “Chronons”, the elementary units of time. In this way time is quantized. The characteristic unit of time is around 10^{23 }sec. and corresponds to vacuum fluctuations.

Then, Shuichi Tasaki lectured on the *Second Quantization of Decay Modes and White Noise Analysis. *Using Hida’s formulation of white noise he proposed ways to deal with quantum fluctuations, specially arising in cosmological models. George Pronko presented a Relativistic Extension of Gamow Vectors. The last lecture was given by Mario Castagnino on *The Mathematical Structure of Superspace as a Consequence of Time-Asy**тт**etry. *The basic problem here was to reconcile the Timea symmetry with the requirements of general relativity.

*Towards the New Probabilistic Evolutionary Physics*

In the next day, concluding the dialogue and the conference, Prigogine referred to Bergson’s phrase *“Time **is invention **or **nothing’: *Science rediscovers time. This is a step towards the reconciliation of the two cultures dividing western thought. While some see in front of the “the end of science”, others see the emergence of a different, temporal approach which marks the beginning of a new era in science. The laws of nature no longer describe a static deterministic universe. Now they describe an innovating, creative, ever evolving, temporal universe. The new physics not only gives a more narrative picture of nature based on new mathematics but also provides new prediction algorithms.

*References*

Prigogine, La Fin des Certitudes, Temps, Chaos et des Lois de la Nature, Editions Cdilе Jacob, Paris (1 996) .

Prigogine and T. Petrosky, The Extension of classical Dynamics for ‘Instable hamiltonian Systems, to appear in International Journal of Computer and Mathematics with Applications. In press (1997).

Prigogine and T. Реtrosky, The Liоuvillе Space Extension of Quantum Mechanics. Ti appear in Adv. Cher. Phys., Vol. 99, p. 1-120, Eds. I. Prigogine & S. Riсе, John Wiley & Sons (1997).

Antoniou and S. Tasaki, Generalized Spectral Decomposition of Mixing Dynuniсаl Systems. lit. J. Quantum Chemistry 46, 425-474 (1993).

Bohm, M. Gadella. “Dirac Kets, Gamow Vectors, Gelfand Triplets”, Springer Lecture Notes in Physics 3, Springer Berlin (1989) .