Inaugural lecture of the

Workshop on "The Chaotic Universe"

given by the

Nobel Prize Ilya Prigogine

when the *City of Pescara* has bestowed on him
the

honorary citizenship.

The arrow of time

Ilya Prigogine

**©**IcraNetwork

Introduction

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.

The determinism in nature

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.

The problem of time in physics and in philosophy, towards reconciliation.

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.

Reversibility and irreversibility in nature

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.

The irreversibility and the basic physics’ laws

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.

Irreversibility, bifurcations and history

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?

Science as history of nature

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.

The irreversibility in the example of deterministic Chaos

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.

Irreversibility and thermodynamical 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.

The mathematics of time

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, example in field theory and in elementary particle physics

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.