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    Mr. Mayor, Authorities, Dear Colleagues and Friends: First of all, I am very sorry that I cannot speak Italian. I have always admired very much Italy, which is the land of the two cultures. Italy has produced outstanding scientists, as well as, outstanding artists and great philosophers.
    I feel very moved by your kindness and by the honour bestowed on me. I really don't think I deserve it because the problems in which I was interested would have been solved in the same way by anyone who would have followed the same path in his professional life. Perhaps my originality is that I came from philosophy and then to science, and not the inverse. Many people work in science and then they, in their older age, go to philosophy. My case is different, I was always interested in philosophy, as well as in archaeology, in music, and history. In all these fields the arrow of time plays an essential role. And that is the reason why I could never believe that the future is given. Plato stated that change is the origin of all philosophy, of art, of drama and still the problem of time, — of the arrow of time—, remains a problem which is still very controversial.

    I like to quote a letter of Einstein to Tagore. Einstein wrote, "If the moon would be asked why it follows its eternal path around the earth, he may answer that he is gifted with self-consciousness and that his decision was made once and for all." We smile, because we know that his path abides by Newton's Laws. Einstein asks that we should also smile when you believe that you act on your own initiative. Our initiative is simply an illusion, because there is no reason that determinism — which is found in nature — would stop in front of the human brain. In other words, man is an automaton. He may believe that he's free, but he is not free. It would be like we're in a movie. We don't know who was killed, we don't know who's the killer, but somebody knows it — the person who made the movie. In some sense every action, every part of our life, of the life of the universe, is already determined by the initial conditions as they were present in the big bang. Therefore, the pleasure of being invited to this beautiful ceremony, and my friendship with Professor Ruffini, would have been included in the information at the big bang. But that seems very strange, and I could never accept this view.

    But as I mentioned, the problem of time, remains very controversial. When I was young, I asked philosophers "What is time?" And all the philosophers answered that time is the most complex subject of human endeavour. It is the problem of ethics, of responsibility. On the other hand, when I asked physicists when I was young, I asked Pauli, I asked Bohr, they smiled and said, "The problem of time has been solved by Newton, with some changes introduced by Einstein. There is no point for a young man to enter into the study of time." But I am a very persistent person. I have had over my lifetime very few ideas, but I have continued to work on them for many years. In this sense I've followed the model of Einstein, who once said, "I have very few ideas, and when I have an idea, it is very difficult to get rid of it." So it is a fact is that for sixty years, I am working on the problem of time. What is the astonishing point in this persistence is that in spite of the tragedies of this century, in spite of problems in my life, I could continue for this long period, and I had the chance to have very excellent co-workers who helped me to clarify progressively this problem.
    The dichotomy between the philosophers' view on time and the scientists' view of time gave rise to a conflict between philosophy and physics. Hegel, Bergson, Whitehead, Heidegger and Sartre had only contempt for science: Science is giving a distorted view of the universe, as it does not include the idea of the arrow of time, which is a basic, existential dimension of human beings. That led finally to a kind of war of cultures, which is still going on. This war of cultures is illustrated by the recent article by Sokal, or the polemical book by Sokal and Bricmont. One example that Bricmont and Sokal give to discredit philosophers is the famous discussion between Bergson and Einstein, which took place in Paris in 1922. Einstein gave a presentation of his theory of special relativity, and Bergson expressed some doubts about it. It is true that Bergson had not understood Einstein. But it is also true that Einstein had not understood Bergson. Bergson was fascinated by the role of creativity, of novelty in the history of the universe. But Einstein did not want any directed time. He repeated often that time, more precisely the arrow of time, is an "illusion." So, these ideologies seem to be irreconcilable. Sokal and Bricmont use this confrontation to conclude that Bergson was crazy to provoke Einstein, and that philosophers should limit themselves to wisdom, to ethical problems, and not deal with science. But I believe that philosophy and science are connected; they're both expressions of human culture, and you cannot make philosophy without taking into account the science of your time, or do science without understanding what are the problems which are of interest to your contemporaries. I even feel that in some sense the philosophers and artists and writers have anticipated what is happening now. For example, Kandinsky or Duchamp repeat "Determinism cannot be true," and André Breton goes even so far as to state, "We should destroy laboratories because laboratories are giving us a false idea of men and of their existence."
    Curiously, this war of cultures is not only limited to philosophers, but it is also present in the writing of some scientists. For example, Steven Weinberg wrote, "Science should not interest the public because for the public, it is not important if relativity is right or wrong, or quantum mechanics is right or wrong. It should only be interesting to the public when you shall find the mechanism — the final mechanism — of the creation of the world and the definite formulation of the laws of nature." That is of course, not for tomorrow. Also I think that it is a paradoxical statement, because after all science requires the co-operation of society. If the results of science would be of no interest to the citizen, then how dare to ask the citizen to support science?
    In some way I see my own work as the work of reconciliation. I wanted to show that the direction of time can be inserted in the microscopic level of dynamics, and therefore, the famous dichotomy between the two cultures, between people like Heidegger and Einstein, loses its sense.

    We observe irreversibility on all levels of observation. There are simple irreversible processes such as physical chemical processes like heat conduction or viscosity. Every chemical reaction is an irreversible process. Each of our thoughts is in an irreversible process. We cannot conceive life without irreversible processes. And I believe you cannot conceive cosmology without irreversible processes. But how to introduce irreversibility into the basic laws of physics, that is a different problem. The two great theories of this century — quantum mechanics, and relativity — negate the direction of time. Therefore there are two tendencies. One tendency is to state that we introduce the direction of time by approximations which we introduce in the basic time-reversible laws of physics. Generally these are associated to some form of coarse-graining. Another version of the same tendency is to emphasize "decoherence". Decoherence would come from the influence of the external world. But what about the dynamics of the external world ? I think that both of these directions of thought are somewhat strange. To imagine that we introduce the direction of time through our approximations seems to be close to megalomania. We may consider that we are the children of time, the children of evolution, but it is difficult to imagine that we are the father of evolution. We would then be in a sense outside nature. But that is very difficult to believe. Also the idea that cosmology would be at the origin of irreversibility is very difficult to believe, because irreversibility appears today in some type of systems and not in others. For example, the two-body problem (as the earth, moon or sun) can be solved to a high approximation by time reversible laws. But, already the three-body problem introduces some aspects of irreversibility. If there would be a cosmological influence then it should likely act on all systems in the same way. Our problem is to distinguish dynamical systems which are reversible from systems which present irreversibility.
    My point of view is that we have to incorporate the direction of time by extending the formulation of classical and quantum mechanics. There are many types of systems, as I mentioned already: two-body systems, three-body systems; thermodynamic systems, in which the number of particles is very large, the volume very large, and the concentration, finite. And what we should show is that for some type of systems, we have reversible behavior, and for others we can incorporate irreversibility in the basic expressions of classical or quantum mechanics. It is curious that this point of view was received with great skepticism. Some part of this is due to the fact that the mathematics which one has to develop to incorporate the direction of time, is a much more difficult mathematics, and it could not be done before the recent developments in spectral analysis, to which I shall come back to in a few moments. But there is another aspect: the deterministic point of view was also a point a view that you can control everything by changing the initial conditions, so that science would lead to certitude. In contrast, introducing time irreversibility, we introduce also — as I shall explain in a few moments — probability. We come to the end of certainty, but the end of certainty means the possibility of novelty, of evolution.

    Now, let me go a little deeper into the subject. First, let me explain why I was so convinced that we have to introduce irreversibility in the basis of physics. My starting point was thermodynamics. I know of course that thermodynamics is a phenomenological science. As everybody knows, the basic law of thermodynamics is the law of increase of entropy. Now the interesting point is that systems which are close to equilibrium and systems which are far from equilibrium, react in a quite different way to perturbations. When you perturb a system close to equilibrium the system goes back to equilibrium like when you perturb a pendulum. The reason is that there are extremal principles in thermodynamics. For example, the entropy is maximum at equilibrium, if you perturb the equilibrium, you lower the entropy, and the system reacts by coming back to the maximum of entropy.
    The reason why I received twenty-two years ago the Nobel Prize was that I had shown that far from equilibrium, the stability with respect to perturbations is generally lost. Far from equilibrium we have bifurcation points coming from the non-linear character of the equations of evolution. Then you have many possibilities, many branches, and these branches lead to appearance of new space-time structures. Now this was for me an very important point and fortified my point of view that the direction of time associated to the increase of entropy, has a very important constructive role. Indeed, irreversibility can no more be considered as an artefact because we observe all these new space-time structures. I shall not go into details. These questions are today very well known and studied in many laboratories. Far from equilibrium we can see oscillating reactions, or a kind of new non-equilibrium crystallography, associated to what is called the Turing structures, and we can also have chaotic situations in which trajectories diverge exponentially in time.

    So there is a large number of new phenomena which are associated to irreversibility, and appear only in systems far from equilibrium. Now, there are two aspects which I want to emphasize. The first I already mentioned, that because of these new structures we cannot say they come from our approximations. The second point is that because of the appearance of these structures the role of probability becomes evident. In front of a bifurcation, you have many possibilities, many branches. The system "chooses" one branch; if you repeat the experiment it may choose another branch. The choice of a branch is associated to probability. In other words, the future is not given. Once I had obtained these results, I wanted first to see if they are not giving some insights in other domains of knowledge. For example in biology my student who is now very well-known, Jean-Louis Deneubourg, has made very nice experiments which impressed me very much. Imagine an ant nest, a source of food, and two bridges. You see that after some time all ants are on one bridge. Should you repeat the experiments, they may be on the other bridge. The mechanism is again an autocatalytic mechanism because each ant encourages the other ants to be on the same bridge. This is a very simple example of a bifurcation in biology. Also, human history is full of bifurcations. When we went from the Palaeolithic Age to the Neolithic Age due to the fact that humans could explore the resources of vegetation and of metallurgy, we may consider this as a bifurcation; even as a bifurcation with many branches, because the Chinese Neolithic is different from the Middle East Neolithic or the Latin American Neolithic. There are of course elements in common, and a visitor from Egypt would find the towns of the Aztecs very similar to his own with the temples, the palaces, the squares, and so on. In fact, one can probably say that every time we find a new resource, material resources like coal, or immaterial resources like electricity, the world is reorganized, and we have a bifurcation. In the present, the world is changed by the information technology. This technology has grown at an unexpected rate.
    This leads me to make a few remarks. When you compare the situation of humans now with the beginning of the century, there are some positive elements, in spite of the fact that it was a terrible century. Still, there are today more people reading, hearing music, and so on, so there are some positive aspects. We have lost our Eurocentric attitude, also the distinction between the social classes are much smaller than they were at the beginning of the century. Of course we don't know what will be the positive aspects of the information society. Again my friend Deneubourg mentioned to me that there are 12,000 species of ants, and some live in very small nests of a few hundred ants, and some are members of very big nests, counting millions of ants. Now it's interesting to see the difference of behaviour. In small nests every ant behaves independently, but in the large nests, there are collective motions, the intelligence is in the collectivity. And what is curious and somewhat frightening is that in the large ant nests, many of the species are blind. And I have found to some extent a similar behaviour in traffic flow. When you are in a dilute traffic flow, you follow your inclination, you follow a program which is more a less similar to the program which you would follow if you would be alone on the highway. However, if a critical concentration is achieved, you have a bifurcation and you go to a new branch in which you are pushing the others and you are pushed by the others, and that is what I called collective motion. Now, what will be the result of the information society? Will this lead to a collective form of life, or will it make the life of the average citizen more rich, more varied, and lead to an increased quality of life?

    Let's come back to the problem of time. As I mentioned, on all levels of observation we see a history — a cosmological history, a biological history, a geological history. It seems that you can only understand the structures which are around us from a historical perspective, and this historical perspective corresponds to a succession of bifurcations. So science today emphasizes the narrative element, becomes a history of nature: I would like to say, it becomes something like a novel, or a story of 1001 Nights in which Scheherazade tells a story to interrupt and tell an even nicer story, and so on, and here we have cosmology which leads to the story of matter, then to life, and to man.
    This view corresponds to a big change in comparison with the type of traditional description which science has presented us of nature. The ideal of classical science was more a geometry as it is realized to a supreme degree in Einstein's general theory of relativity. Now science becomes more a narration and the world appears more like a construction; a construction which is going on since the big bang, and still continues. I shall not go into the cosmological considerations, but I believe that the main point of the big bang is the entropy explosion. In fact whatever our detailed view on the pre-universe situation is that is was something like a quantum system in the ground state in which the entropy is vanishing and you have only virtual particles. Whatever the mechanism of the big bang, it leads to "real" matter which can live longer times, to excited states and to the creation of entropy. Therefore I believe that the big bang can be associated to an explosion of entropy. Of course many aspects of the big bang are still obscure.

    Now let me come to the main point of my lecture, which is how to incorporate the direction of time into dynamics. The mathematical and physical basis of our approach were definitely clarified only about five years ago, first in the frame of chaotic deterministic maps. I would like to mention that my colleague Dean Driebe has published recently a very nice book, Fully Chaotic Maps and Broken Time Symmetry, (Kluwer Academic Publishers, 1999) devoted to this subject, and you can find all of the mathematical aspects in this book, so I can be rather short. The simplest example is the so-called Renyi map. You multiply by two a number between zero and one and you every time after each operation take away what is above unity. Then you can show that two initial conditions differing as slightly as one wants give different trajectories. Now this can be described by "Newton's equations" which in the Renyi map reduce to xn+1 = 2xn (mod1). But the interesting point is that for all deterministic chaotic systems, there exists another representation for ensembles in which the central quantity is probability. The evolution operator of the ensembles can be analyzed in terms of probability, and not in terms of trajectories. How is this possible? In a famous paper Koopman has shown that as long as you remain in the Hilbert space of "nice," square integrable functions, the probabilistic description, and the description in terms of trajectories or wave functions in quantum mechanics, are equivalent. But that is only true as long as you remain in the Hilbert space. This space is a kind of generalized vector space in which there is a norm and a scaler product. But how then does it happen that you obtain here a new result in which probability cannot be reduced to trajectories, and in addition, gives broken time symmetry? It is because the evolution operator can be analyzed as it is done in quantum mechanics into eigenfunctions and eigenvalues, but the eigenfunctions are now generalized functions (called also distributions). The extension of the evolution operator outside the Hilbert space leads to a different formulation of the laws of physics, which incorporate the time symmetry breaking and in which the basic quantity is probability. Of course deterministic chaos is only a simple example; these conclusions apply to other situations, and especially to thermodynamic systems.

    Thermodynamic systems, as I mentioned already, are large systems in which the number of particles tends to infinity, the volume tends to infinity, and the ratio number of particles of a volume remains finite. Now it is well-known already for equilibrium thermodynamics that this leads to new phenomena, like phase transitions. If you would take a system of let's say 100 particles, there would be no well defined melting point or freezing point. The existence of these singularities is due to the thermodynamic limit, (number of particles N® ¥ , volume V® ¥ , N/V finite mean energy) and what we have shown is that irreversibility is again due to the limiting process involved in thermodynamic systems. Indeed one can show that even if you start with Hilbert space, the persistent interactions which are going on in a thermodynamic system, kick the system outside the Hilbert space. The mathematics is very simple and presented in a paper together with Tomio Petrosky (Extension of classical dynamics: The case of anharmonic lattices, I. Prigogine and T. Petrosky, in Gravity Particles and Space-Time, eds. P. Pronin and & G. Sardanashvily, World Scientific, Singapore, 1996). The evolution operator L (the Liouville -von Neumann operator) for particles has real eigenvalues inside the Hilbert space, but in general complex eigenstates (associated to relaxation time) outside the Hilbert space. That is in a somewhat simplified way, the origin of irreversibility.

There is a mathematics of time. The situation is somewhat similar to gravitation which needs non euclidean geometry. We obtain outside the Hilbert space a probability distribution which can no more be expressed in terms of trajectories, and this limiting process leads to time symmetry breaking. We obtain two semigroups, one dealing with the evolution from the past to the future, the other dealing with the evolution of the future to the past. Of course we have to make the choice of one of the semigroups. In a sense this situation is similar to the problem of matter and anti-matter. There is a symmetry between matter and anti-matter, but our universe is made mainly by matter and anti-matter is only the temporary result of high-energy experiments, at least so far as we know. And here also we see again that the universe is less symmetric than we thought. The classical view was that the direction of time does not exist, that past and future playing a symmetrical role. Now we see that is not true, that the time symmetry is broken in large systems. This means of course that Newton's equations or Schrödinger's equations are not valid in the thermodynamic limit. That doesn't mean that classical mechanics or quantum mechanics are wrong; that means only that the formulation of classical mechanics or quantum mechanics has to be modified for this class of dynamical systems.

    The extension of classical mechanics or quantum mechanics to thermodynamic systems applies also to field theory. In field theory the system is described by an infinite number of variables. A free field is an integrable system; it is again a system which is time symmetrical but the situation changes radically with interacting fields. If we take free fields such as the Klein-Gordon field or the Dirac equation, we would have occupation numbers which would be constant. But this is not what is observed. An excited atomic state falls down to the ground state and there exists a large number of resonances in elementary particle physics. These aspects illustrate the role of irreversible processes on the most fundamental level which we have at present (excluding the string theory which still remains a controversial subject). Again the spectral analysis of interacting fields outside of the Hilbert space lead to semigroups with complex eigenvalues which are responsible for the appearance of unstable particles with finite lifetimes. The conclusion that for interacting fields we have to go outside the Hilbert space was stated by Dirac at the end of his life. He wanted to avoid Schrödinger's equation as the Hilbert space is killed by the time evolution.
    These statements are also of importance for the understanding of dressed excited states as introduced in spectroscopy. This is a problem which remains unsolved in quantum mechanics as emphasized by Dirac and Heitler (see Quantum Electrodynamics). Dressed ground states can easily be defined, for example for a hydrogen atom, the dressing being formed by virtual photons. But to define excited dressed states or dressed unstable particles we need a clear distinction between virtual and real photons which is lacking in the Hilbert space.
    Note that in the Hilbert space a decaying state contains both an exponential part, but also at the beginning, a non-exponential part associated to the so-called Zeno time and for long times the so-called "long tails." So this cannot be an unstable state because the decay of unstable states should have a universal behaviour. If it would present deviations from the exponential law it would be possible to distinguish young elementary particles or atomic excited states from old ones. But that would violate the indiscernability condition of quantum mechanics; it would lead to a catastrophe. There are no unstable states which have a purely exponential decay in Hilbert space. You have again to go outside the Hilbert space and there in the space of the density matrixes you can define states which have purely exponential decay and an energy distribution which satisfies the uncertainty relation (More precisely the Zeno time is associated to the formation of the unstable state and the long tails with its interacting with the decay products).
    Excited states or unstable particles are in our view well defined density matrices with broken time symmetry.
    We come to the conclusion that the physics of populations is not the same as the physics of individual trajectories or individual wave functions. This is a situation well known in sociology because the behaviour of a population cannot be reduced to the behaviour of individuals. The same appears to be true for very important classes of systems like thermodynamic systems or interacting fields. This is of course a result of some interest for cosmology, because cosmology is concerned with large systems. Therefore cosmology and chaos have something in common : what is in common is that they have both to be described by functional analysis outside the Hilbert space which has been developed by eminent mathematicians such as Schwartz and Gelfand.
    Many people have suggested that the basic description of nature would contain probabilistic stochastic elements. This is what we see now. But the stochastic elements are not introduced by hand they are the outcome of our approach. I may add that the theory can be easily extended to relativistic situations (Relativistic Gamow vectors, I. Antoniou, M. Gadella, I. Prigogine and G. P. Pronko, J. of Mathematical Physics, 39, 1998). In this paper we studied the relativistic theory of two interacting fields. Of course general relativity is more complicated but it is also full of irreversible processes, as for example, related to the emission and absorption of gravitons. Therefore in general relativity our approach should play a role, but that is a problem for the future.

Conclusions

    Let's come to the conclusions. What is the concept of nature to which we've come? The Newtonian model of reality was that of an automaton. Still we have a great difficulty to believe we are an automaton. The concept of nature in quantum mechanics corresponds in some sense to the opposite view, as "reality" would be associated to the transition from "potentialities" to "actualities" due to our measurements. That means that the observer would be responsible for reality. This is also difficult to imagine. Then we would play a central role in the creation of reality.
    In our theory, measurement has lost all its subjective aspects. For thermodynamic systems there is no wave function and no collapse. Also concerning cosmology, in our approach we cannot speak about the wave function of the universe because it's again a large system and you can only speak about density matrices and probabilities. Therefore we have a different version of the concept of nature which contains probabilities and therefore contains the possibility of novelties, and novelties are the condition to speak about a history of nature.
    I believe that the 21st century will probably be a century of exploring the mechanism of "becoming." It is indeed rather sad that even if you can imagine that cosmology, or the origin of life are associated to successions of bifurcation, we know very little about the mechanism of bifurcations. We may safely assume that everything in our universe is evolving in the same direction of time: rocks evolve in the same direction, stars, galaxies, supergalaxies, all objects evolve in the same direction. We age all together. We can only conclude that our universe seems to be ruled by a semigroup with broken time symmetry. It is an open world in which the direction of time plays a central role.
    Well, I've come to the end of my lecture. I want to thank you very much for the honour which you've bestowed to me, and for the friendship which I enjoyed during my stay.