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Towards the theory of matter, geometry and information
by
Arkadiusz Jadczyk
....
Some history:
The first year
of my graduate studies had passed and I still had no idea of what my PhD
was to be about. Lopuszanski was an expert in
quantum field theory, the most advanced and the least understood branch
of theoretical physics. My attempts to understand it, to " really" understand,
had been unsuccessful. From long discussions with my advisor I got the
impression that I am asking questions for which he has no answers. Perhaps
no one can answer these questions ? I didn't know. I was getting desperate.
Many years later I realized that my questions could not be answered within
the standard paradigm of quantum theory, that the very foundations of
quantum theory needed to be changed. I didn't know then that in the future,
for my own work on the foundations of quantum theory I would get the Humboldt Award.
Not knowing
what the future would bring, with time passing so fast and with my understanding
of physics and mathematics progressing so slowly I was becoming seriously
afraid that I would never be able to find a worthy, unsolved problem,
and to solve it within the next three years. Fortunately help was underway.
Since 1965,
usually in February or March, the Institute of Theoretical Physics of
the University of Wroclaw was organizing, in the mountain resort of Karpacz,
the International Winter School of Theoretical Physics, each year devoted
to a different topic, with invited top expert lecturers from all over
the world. It was a real heaven for the young PhD student in distress.
In 1968 Lopuszanski
agreed to be the Director of the 5th School, and he chose as the topic
the "Axiomatic Approach to Quantum Field Theory and the Many Body Problem".
It was just right for me !
The term "
axiomatic" means in this context " mathematically rigorous, derived from
first principles". The point was ( and is to this day ) that Quantum Field
Theory, the theory was designed to completely explain the quantum world,
the mysterious conversion of light into matter and matter into light,
atomic spectra and nuclear reactions, etc. etc.  this theory that was
supposed to predict the results of all experiments with elementary particles
 this theory, when asked to obey the principles of Einstein's special
theory of relativity  was producing mathematically meaningless results.
No one was
able to succeed in constructing a mathematically reasonably behaving theory
starting from assumptions that physicists considered reasonable. It is
for these reasons that physicists borrowed the "axiomatic method" from
mathematicians.
The idea was
this : " If we are not able to construct a reasonable theory, let us assume
that it has been constructed. And let us make a list of the most important
properties that such a theory should have. These properties will be called
" axioms", and we will try to derive as much as possible from these axioms
by a rigorous process of mathematical deduction !"
Such an approach,
even if it looks like a good and attractive idea, has its dangers. Imagine
that somebody, having problems with constructing a car running on " free
energy" decides to study such an invention " axiomatically". So, our axiomatic
inventor lists the desired axioms :
1) It uses
no fuel ;
2) It produces
no pollution ;
3) The driver
is always safe.
These axioms
look good and one can start to derive consequences for the vehicle of
the future. But what if the only " model" that satisfies all three axioms
is the car that has no gas tank and that never moves !
The mathematician
will, of course, be satisfied. He has a " model" for his axioms. But the
physicist will say : " Your model is not really interesting to us. In
fact, it is trivial." And exactly the same applies to axiomatic field
theory : the only models that satisfy all axioms proved to be of no interest
to the physicists. Either they do nothing, or do something but not in
our world !
Even so, axiomatic
field theory, and later its more abstract version " algebraic quantum
field theory" allowed mathematical physicists and mathematicians in the
last fifty years to develop many interesting and highly non trivial concepts,
to achieve deeper insight into the relations between these concepts, and
to develop new powerful mathematical tools that have found applications
in other areas of physics, mathematics and engineering.
One such area,
that proved to be easier to tame than Quantum Field Theory was the so
called " Many Body Problem". An outsider could think that we are discussing
the problem of three, or four, or 100 planetary bodies interacting with
each other, isn't 100 already " many" ? But an insider knows that when
a theoretical physicist talks about "many bodies", he has in mind, in
fact, infinitely many bodies.
How can that
be, you may ask. Well, there are infinitely many integer numbers : .,
4,3,2,1,0,1,2,3,4 ,. and mathematics somehow manages to operate with
these, so there is also a way to deal with infinitely many bodies, provided
their interaction with each other is not too weird !
The mathematical
formalism of quantum field theory and of the many body problem is very
similar, almost identical. We have " Hilbert" space, we have " Fock vacuum",
we have " creation" and " annihilation" " operators", but in the case
of " many bodies" we do not assume that Einstein's relativity must be
obeyed and we avoid " photons". Photons are forbidden here  they are
replaced by more tamable " phonons"  quanta of vibrations of a crystalline
lattice.
The School
brought in to Karpacz experts in all these areas. While the German theoretical
physicist, master mind of algebraic methods, Rudolph Haag was lecturing
on solved and unsolved problems of quantum fields, the French mathematical
physicist David Ruelle and the Danish theoretical
physicist NM Hugenholtz were deriving from scratch the mathematical schemes
of Quantum Statistical Mechanics  which was in fact a different term
to denote the same concept : Many (that is infinitely many !) Body Problem.
In the evenings,
over a glass of wine, I could listen to their private conversations in
small groups and hunt for " open problems". Once in a while someone would
say : " it would be nice if we knew this, or that", and I would note it.
And then, after much hesitation, and with my body trembling, I would dare
to ask " my own" question, and watch the face of the Master  will he
laugh at me ? Will he call my question " trivial" ?
I was lucky.
When I asked my own questions, in both areas, quantum field theory and
statistical mechanics, I received the same answer : " We don't know. Why
don't you try to find it out ?"
And so I knew
what to do in the coming months. I was going to bring to Haag and Hugenholtz
and to Ruelle the head of the Gorgone, to find answers to questions that
they didn't know. I wanted to find out about the properties of the " vacuum"
state in both statistical mechanics and quantum field theory.
The meaning
of the " vacuum" is somewhat different in the two cases, but in both cases
we believe that it is a state of the highest possible symmetry. Indeed,
if there is " nothing" at all, then however we translate or rotate in
space this " nothing"  it will be the same " nothing". And if we wait,
however long, to see if something happens out of nothing, we will never
see anything happening. These were the " natural" assumptions that were
put as the axioms defining the vacuum state in both theories, in quantum
field theory and in quantum statistical mechanics. Were these the right
assumptions ?
Good enough
to deduce interesting properties and, at the same time, not too restrictive
ones?
And then comes
the difference between the two cases : in quantum field theory, when thinking
about the " vacuum", we think about the state with no " particles", with
the lowest possible " energy". In quantum statistical mechanics we usually
want to describe states of a given " temperature", that is not necessarily
" absolute zero" or " zero Kelvin". Such a state, as uniform as possible,
and yet " vibrating" so as to have a definite temperature, would usually
have an infinite energy  because infinitely many " particles" (or, as
physicists would say more precisely : infinitely many degrees of freedom)
are vibrating. The very concept of " energy", in this case have to be
rethought and redefined.
The first thing
to do was to learn all that there actually was to know about the subject.
It took me nine months to write my first "paper". In October 1968 I sent
it for publication in a prestigious journal specializing in my area of
research : "Communications in Mathematical Physics". Rudolph Haag was
the Editor in Chief. It took another year to reply to the referee's objections,
make changes, to add new references. The title was "On the
Spectrum of Internal Symmetries in the Algebraic Quantum Field Theory."
Three months later I submitted to the Communications my second paper :
" On Some
Groups of Automorphisms of von Neumann Algebras with Cyclic and Separating
Vector." And again almost a year passed before it appeared in print.
My time was running out, my stipend would last only till September 1970.
I had to pass through a series of exams, collect referee reports and finally
defend my dissertation.
On March 4,
1970 my Ph.D advisor, Jan Lopuszanski wrote a letter to Daniel
Kastler as follows :
Professor D. Kastler
Centre de Physique Theorique 31,
Chemin Joseph Aiguier 13
Marseille 9e
France
Dear
Professor Kastler,
Excuse
me that I trouble you once more with the problem of Dr. Jadczyk. I am
glad to inform you  en passant  that few weeks ago Mr. Jadczyk got
his ph.d. degree summa cum laude. It is already a pretty long time ago
we got your kind and warm letter concerning the visit of Dr. Jadczyk
at your Institute. It takes a very long time to arrange such things
in Poland after one gets an official invitation from abroad. Therefore
we are anxious to learn from you in what stage things are at present
to make tentative plans for Dr. Jadczyk for the future. We would very
much appreciate to hear from you at your convenience. Looking forward
to hearing from you
Yours
sincerely
Jan Lopuszanski
I spent in
Marseille three months  October till December. I had my ph.d. title and
I could choose my new avenue of research according to my own preferences.
Starting October 1, I was officially an assistant professor, which gave
me seven years for pursuing my own research, whatever direction I would
chose, which, if all went well, would culminate in the next academic degree
available : doctor habilitatus.
Being totally
free to choose the subject, no pressure whatsoever, I chose a direction
that I felt was neglected but important : " geometry
of indefinite metric spaces". What I have chosen to research during
my stay at CNRS was an introduction into the studies of geometry of spaces
that have infinitely many dimensions, studies by using algebraic rather
than analytic methods. This was an introduction to the area of research
that was later on generalized even further and became, as we call it today,
" non commutative geometry."
And here I
would like to jump into the future and sketch a vision that I have developed
over the years. This vision has been formulated and published in 1987,
in the proceeding of
the VIth Symposium on Bioelectronics at the Catholic University of Lublin,
Poland. I began with describing the situation of the fundamental research
today which goes, as I believe, along a road leading to a dead end:
Dead end
What causes
that a physicist, specializing in mathematical methods of high energy
physics, who attempts each year to spend a part of his time in Hamburg
or Geneve  the two largest European high energy centers, comes now
to Lublin for this Symposium?
There are
several reasons. One of them, perhaps the most essential one, is recognizing
the fact that high energy physics, physics of elementary particles,
physics which up to this time was considered fundamental, came to a
dead end. The last essentially new ideas were the special and general
theories of relativity (Einstein 19051912) and the theory of quanta
(Bohr, Schroedinger, Heisenberg, Dirac 1913 1928). These theories,
based on diametrically different concepts of reality and mathematical
formalisms, are today as far from each other as they were sixty years
ago. In spite of an enormous effort it has not been possible to build
a coherent theory that would include relativity and quantum theories.
Physicists that were actively participating in these efforts have been
experiencing tides of hopes and disappointments  with a somewhat regular
period of about 10 years. The last unfulfilled hope was in "multidimensional
supergravity". Actually almost all the capital has been invested into
"theory of superstrings".
Till the
end of the ten years period there remains yet five years. What will
be the next "hit"? Observing the development of events from a certain
distance, what strikes the observer is the tragic of the situation.
It is seen that we are not witnessing the building of a solid edifice
of science. Rather we witness the glueing together of a giant snow ball.
Glueing
the snow ball
Watching
the most talented physicists of our time roll their giant snowball of
a theory fills the present author with a sense of tragedy. They are
pushing it forward, not always in concert or the same direction, observing
with attention the gestures of coryphaeuses of science that are balancing
with great difficulty on the continuously moving top of the ball  from
where the goal appears to be seen.... But each successive"goal" proves
to be once more an ordinary piece of landscape.
With time
the glue ball, the ballconglomerate, becomes so heavy that  rolling
forward with growing difficulties  it starts to break into pieces under
its own weight. First class professionals, in a hurry, are glueing back
together the pieces that have fallen off and are sealing up the cracks
in the ball. Theoretically the bigger the ball is, the better the view
from its top gets. But, at the same time, the more difficult it becomes
to get to its top and the more dangerous it becomes to fall down from
it. And, at the same time, the bigger it gets, the more the ball itself
blocks the view of the pushing crowd.
Thus there
are fewer and fewer of those who know what they are doing. Furthermore
such knowledge is of little use for the careers of young scientists.
The governments of the richest countries are spending billions for this
"game". The new giant accelerator is being built. Today superstrings
are in vogue. Today there is money for superstrings. Thus one has to
work on superstrings. They call it "fundamental research". Physics of
high energies is thus building the foundation of foundations .... from
the glued snow and the litter that attaches to it by chance along the
way.
Then I went
on to describe the new paths in physics  physics of complex and selforganizing
systems :
New Physics
Before our
eyes do doors open leading from physics to biology. New physics is being
born  the physics of complex and organized systems. It is yet in its
swaddling clothes. It has not yet worked out its own methods. The methods
that are being borrowed from statistical physics and from physics of
condensed phase have proven to be insufficient already in the beginning.
And, is it still physics?
This mixture
of methods is taking elements from physics, informatics, and applying
them to systems imitating biological systems. Slowly, and with difficulty,
new horizons do open.
Computer
experiments indicate new, as yet unsuspected regularities. New concepts
are being formed. A virgin terrain is waiting for trailblazers. Adventure
casts its lure.
One of the
main mathematical objects in the theory of strings are "Virasoro algebras".
Today professor Virasoro is preoccupied with the theory of self organizing
neural networks. Most of his time he spends with a computer. He is not
an exception. Nevertheless the theory is only starting to crawl on all
fours. It is in the stage of simple classical models.
At the same
time the collective phenomena, and it is with such that we probably
have to do in biological systems, expose to us completely new dimensions
that accompany the quantum description. To what extent can we expect
here that taking into account quantum laws will prove to be essential
in explaining the riddle of life, if the physicists themselves are not
that sure about the methodological status of quantum theory itself?
2.2 The
Quantum Seam of Life
We are witnessing
breaking out of the ruling paradigm in biology, a paradigm according
to which life can and should be reduced to chemistry. Clearly describes
the new direction of attack W. Sedlak [11,
p. 13] When he writes: "Life can be thus described as a process of a
quantum nature, expressing the coupling of chemical reactions with electronics
phenomena in a proteinnuclein super conductive substratum." And further,
presenting his prognosis of the physics of life [11,
p.29]: "It must be a quantum physics (...). The description will be
extremely difficult, as it has to take into account at the same time
the two coupled quantum events  chemical and electronical ones. The
description addresses the quantum seam of the two processes. It is here
that life in its most basic instance is being played." It is clear that
chemistry alone is not sufficient to explain the action of even the
simplest electrical circuit consisting of a battery and a light bulb.
For this we have to go beyond chemistry and use the part of physics
dealing with radiation phenomena.
But can chemistry
together with physics, even quantum physics, suffice for making essential
progress in explaining biological phenomena, and in particular the phenomenon
of life? The Author of this text does not completely share the optimism
of W. Sedlak who seems to believe that even consciousness will be explained
through collective states of complex systems. Essential life processes
probably do take place at the "quantum seam". But is life only the sum
total of life processes? Or it is something more, a new quality? And
if so, then what kind of quality?
Quantum
Mechanics
The one writing
these words knows little, in fact next to nothing, about biology. He
is only a physicist who has spent most of his time on studying the works
of other physicists, trying to add his own little bricks to that which
is already known. And the remaining time he has spent meditating on
how little in fact we know and furthermore how uncertain even this little
that seems to be known is. Therefore he does not claim to understand
those things that are beyond the scope of his active research up to
this time. [f1]
Therefore if in spite of this he has the courage to speak about things
that extend so far beyond the scope of his competence, it is only because
it seems to him that he is perceiving something essentially new. Something
that demands attention.
Werner Heisenberg,
in a chapter "Conversations about the connection between biology, physics
and chemistry" of his book "Part and the whole" [7],
while reporting on his discussion with Niels Bohr, quotes him as follows:
"In natural
sciences it is always good policy to stay as conservative as possible
and to create new extensions only under the pressure of observations
that cannot be explained in any other way." The speculations presented
below (in Sec. 2 and 3) will be in an evident violation of this certainly
healthy principle. When deciding to publish them in spite of the above
the author has taken into account these words: "And Jesus beheld a
man working on the Sabbath, and He said to him, `Man, if you know
what you do, you are blessed, for you break not The Law in the spirit;
but if you know not, you are accursed and a transgressor of The Law."[1]
Quantum
Theory  today.
Quantum theory
has been formed in the first quarter of this century as the result of
a stubborn search for a new mathematical and conceptual apparatus that
would be able to explain the intricate spectra radiated by excited atoms.
Pretty soon thereafter the scope of applicability of quantum theory
was significantly expanded and it started to be applied (with considerable
success) to all the problems of the micro world. From the very beginning,
however its successes in predicting results of subsequent experiments
were accompanied by difficulties of a philosophical and methodological
character. As time progressed these difficulties were alternatingly
falling silent and then exploding to the surface again.
During the
last dozen years the discussion has been gathering strength again, and
this because of two reasons. First, when the problem of a possible marriage
between quantum mechanics and Einstein's general relativity gained importance,
the problem of interpretation of the "wave function of the universe"
appeared (as there were no "external" observers).
Second, progress
in experimental methods made it possible to experiment with single quantum
objects. This kind of experiment, as is the case with the cosmological
"experiment" with the unique Universe which we happen to inhabit, does
not exactly fit into the rigid scheme of the conceptual apparatus of
the standard quantum theory.[f2]
To characterize
the present status of quantum mechanics let us begin with stressing
out that, as long as we are not dealing with too strong forces and too
high velocities, this theory "works", and it even works wonderfully
well. There are so many physical phenomena that are so surprisingly
well described by quantum mechanics that those phenomena that somehow
escape the quantum description are classified as "curiosities" and are
set aside "for later". These successes of quantum mechanics are, from
the very beginning, accompanied by a kind of a "methodological shock"
 a shock whose force is getting weaker and weaker with time like a
continuous pain gets duller and duller as time passes.
This shock
was caused by the realization of the fact that quantum mechanics, as
it seems, is censoring our essential and logical questions. "Is electron
a particle or a wave?" This question is described as "having no meaning".
We could accept it. But also other questions: "What is an electron?"
"Which slit is it going through when it is not being observed?" "What
is it doing between two consecutive observations?" are also denied meaning.
Finally, when we are being told "It does not have any meaning" when
we ask "Why a given radioactive atom decays at this and not at some
other moment?" and "What happens during the decay? How does it go?",
then we start to suspect that the conceptual system of quantum mechanics,
even if logically noncontradictory, is, perhaps, not as perfect and
complete as it aspires to be.
In fact,
after some closer analysis, we realize that only those questions for
which quantum mechanics can provide answers are classified as "making
sense" ! But there are, as it seems, lots of other questions that make
sense, if only because Nature herself answers them every now and then.
In this way, after the initial excitement, we are beginning to perceive
defects in this "perfect theory".
At the same
time the experts cannot agree: Some argue that the theory is ok, that
what we are lacking is a "proper interpretation". On the other hand
those who blame the theory itself differ, often drastically, " in their
diagnoses and prescribed "cures". The Author tends to believe that the
main shortcoming of quantum mechanics is that "it has nothing to say
about individual events". At the same time the world seems to be woven
precisely out of such events.
Another weakness
of quantum mechanics is in the fact that it is not able to describe
the very process of a measurement.[f3] Therefore
quantum mechanics is not a complete theory. Niels Bohr, one of the major
founders, inspirations, and interpreters of quantum mechanics dogmatically
announced: "it is so, because it must be so." But we do not have to
follow Niels Bohr. We can choose to follow another great physicist,
E.P. Wigner, who, trying to find a solution to the aforementioned difficulties
of quantum mechanics concluded [19]:
"the postulate that the equations of motion of quantum mechanics cease
to be linear, in fact that they are grossly nonlinear if conscious
beings enter the picture."
In a "safe
middle" are the views of the Princeton physicist J.A. Wheeler. On one
hand he accepts after Bohr and Heisenberg, and even enhances it, this
part of the paradigm of the quantum mechanics in which the question
"what happens between two observation acts?" is qualified as meaningless.
He writes: "No elementary phenomenon is a phenomenon until it is a registered
phenomenon." [16]
But he also notices: "No element in the description of physics shows
itself as closer to primordial than the elementary quantum phenomenon,
that is, the elementary deviceintermediated act of posing a yes/no
question and eliciting an answer. Otherwise stated, every physical quantity,
every it, derives its ultimate significance from bits." [17
It from bit]
Quantum
mechanics  tomorrow
We are living
at the threshold of the century. From the sign of Pisces we are entering
into the sign of Aquarius. From the age of Steam and Electricity into
the age of Computers and Information. We have conquered, if only partially,
Light, and today the time comes for the Word.
At the same
time, it must be stressed, we have no other alternative than the well
established and tested scientific way and scientific method. But, with
even greater emphasis, we must realize that that the new tasks ahead
of us require true boldness in thinking and abandoning all prejudices.
The thoughts that follow are speculative. The predictions may come true
or not. The idea may prove to be right or wrong. The future will show
whether the weight have been correctly chosen, whether the impetus has
been exerted at the right moment and in the right direction, whether
the program that is sketched here will get, even if only in part, realized.
But first
of all, here are the goals : we would like to know what the life processes
and the processes of the mind consist of, and also what life itself
is and, in a further perspective, what consciousness is.
The main
thesis of the author of these speculations is this: for researching
and explaining almost all basic life processes it is necessary and sufficient
to use mathematics, physics, chemistry and biology, as they are developed
today, that is without the necessity of going beyond the present day
paradigm. The role of quantum physics will be here of a particular importance.
But if we will have to make the next step and to answer the question
: what is life itself and what is consciousness ?  then a real breakthrough
will be needed in two domains : we will need to rebuild ab initio
the quantum theory of complex systems and, on the basis of quantum mechanics,
to build a new theory unifying the processes of exchanging of
energy and information.
In the first
domain the progress is rather fast, on the border of mathematics, physics,
informatics, cybernetics, electronics and biology. In the second one,
where a real paradigm break is needed, there is a long lasting stagnation.
The few attempts at constructing a nonlinear quantum mechanics [2,6,10]
did not gather many followers. But, as this author believes, a theory
that is able to describe irreversible events, a theory that is able
to provide its own interpretation, must have a nonlinear character.
Einstein's general theory of relativity can well serve as an example
and as a template, as it is owing to the nonlinear character of this
theory that equations of motion for test particles follow automatically
from field equations, and need to be postulated ad hoc. We are still
waiting for the demystification of the " reduction of the wave packet"
 postulated in quantum theory as a discontinuous change of the state
of the system as a result of the act of observationmeasurement. Non
linearity of the theory seems to be here, as proposed by Wigner, indeed
necessary. But will it be also sufficient ?
The author
of these speculations believes that more is necessary. That what is
needed is a coupling between energy and information.
[f4]
But is it
not so that any exchange of information can be, after all, described
in terms of exchange of energy ?
In a sense
" yes", but in another sense, which is here more important, " no".
Again let
us take the general theory of relativity as an analogy and example.
There we have, from the very beginning, a division between matter and
geometry. The gravitational field represents geometry. Other fields,
taken together, represent matter.[f5]
Can one reduce matter to geometry, or geometry to matter ?
Until now
all attempts at such a " forced unification" have been, if one does
not count side effects, unfruitful. analoguously, the dualism of energy
and matter may have a primitive character.
Continuing
the analogy : in the same way as a gravitational field curves spacetime
[f6],
the information field may curve the state space. May
change the geometry of the space of quantum states. May enable the flow
of information and of energy through new channels. Now quantum
matter gets a worthy partner, just as the gravitational field was a
worthy partner to classical matter. The same way as gravitational field
is local[f7]
in spacetime, the information field is local in Hilbert space
where " near" means " similar". The geometry of the information
field must be, as we have said, a nonlinear geometry. Only
in this way can we explain the stability of structures, such as the
structure of life. With the phenomenon of life we can in this way, associate
a topological invariant (a kind of a vortex) in the nonlinear
field of information. Physical and chemical life processes
would be then controlled by a quantum feedback between information
and matter. And, when we speak about geometry, it must be noted
that it must be more than a classical geometry such as is sufficient
for the Einstein theory of gravitation. What is needed here is a kind
of quantum geometry. Such a geometry is today only in statu nascendi.
[5,20]
Light.
Niels Bohr
repeatedly stressed the paradoxical aspect of the quantum theory :
"On one
hand we are formulating laws that differ from the classical ones,
while on the other hand, in the domain of measurement and observation
we are using the classical concepts without any doubt whatsoever.
We have to proceed this way, because we have to use language when
we want to communicate our results to others. A measuring device serves
its purpose only when by observing it we can univocally draw conclusions
about the observed phenomenon, when there is a clear causal connection.
But when we observe an atomic phenomenon, we have to put a cut between
the phenomenon and the observer or his apparatus. Even if the placement
of this cut is to a large extent arbitrary, on the side of the observer
we have to use the language of classical physics, because we have
no other way to communicate our results."[7].
Quantum mechanics
of the future, a nonlinear quantum mechanics coupling together matter
and information that we are speculating about, has the aim of unifying
the two aspects, the classical and the quantum one, including the "
cut" between them, into a single formalism. In this the paradox that
Bohr is talking about will cease to be paradoxical and will become a
fact that has its counterpart in the theory.
Such a theory,
with its main idea sketched as above, must include in its framework
two completely different worlds : the quantum world and the classical
world or, making the cut in a different direction, the world of flesh
(matter) and the world of word (information). The differences between
these two worlds are so enormous that unifying them seems impossible
without a catalyzer. A probable candidate for such a catalyzer is light.
Why light
and not something else ?[f8]
It is not
easy to answer this question without going deep into rather difficult
formalism of quantum field theory. But, if only to give a taste of the
problems involved, let us begin with remarking that quantum electrodynamics,
that is the theory of interaction between light (i.e. electromagnetic
field of photons) with (electrically charged) matter, is struggling
with difficulties such as how to get finite values out of quantities
that appear to be infinite.
One can say
that the theory would be almost " perfect", were it not for two " catastrophes"
: The ultraviolet and infrared catastrophies.
The ultraviolet
catastrophe is not of interest for us right now, as it has to do with
extremely large energies, extremely small distances, and extremely high
temperatures. The infrared catastrophe, which appears at the other end
of the energetic scale, is related to the fact that every real physical
process involving electrically charged matter is accompanied by a "
photon cloud" consisting of very many photons of very small energy.
The total energy of such a cloud, when measured by an energy gauge scaled
so that the energy of the vacuum is zero, is infinite. Thus the term
" catastrophe".
The most
important fact for us is that the " shape" of this cloud can encode
classical information. (Using the language of quantum field theory we
may say that " coherent infrared states lead to continuous super selection
rule¯, or that " the algebra of observables of the photon field has
a nontrivial center, whose elements parameterize infrared representations").
For the sake of completeness one must notice that the " infrared cloud"
consists of photons of very small energy, and therefore of very large,
macroscopic (even cosmic) wave length.
Such photons
are, on one hand, undesirable, because of the energetic " infinities"
but, on the other hand, they seem to be absolutely indispensable for
the description of the information transfer during a quantum measurement
process. But also here the theory is only taking its first steps. [3,14]
Footnotes:
1 These words are a paraphrase of a paragraph of the introduction
to the book by A. Carel ,Man unknown" [4].
This book has played an important role in shaping the interests and
the views of the author of this text.
2 Unfortunately we can not devote to these problems,
which are at the heart of the modern picture of the nature, more place
than it is just strictly necessary for understanding this particular
lecture.
3 This is, in a sense, the same problem as the previous
one. Because one deals again with an " event", the event of the " registration"
of the result of the measurement.
4 There are reports that living organisms emit, at the
moment of their death, energy in the form of some kind of radiation.
It would be interesting to find whether at the same time we have also
emission of information.
5 By the way, this division is not rigid. In generalizations
of the theory of relativity, different versions of a " unified field
theory", electromagnetic field and other gauge fields belong to geometry.
But the duality matter versus geometry (when gravitation always belongs
to geometry) remains.
6 Such an information field could have certain functions
that are ascribed by R. Sheldrake to a morphogenetic field. [13]
7 Locality means that disturbances propagate through
the direct influence of the field in neighboring points.
8 W. Sedlak points out the special role of light in life processes
in his book " At the beginning, after all, there was light.
References
 The
Gospel of Nazirens, Ed. by Aklan Wauters and Rick Van Wyhe, Essen
Vision Books , 1997
 BialynickiBirula
I., Mycielski J.: Nonlinear Wave Mechanics, Preprint
1976.
 Buchholz
D.: private information.
 Carrel
A. : Man Unknown (New York: Harper & Brothers, Halcyon House,
1938)
 Connes
A.: Calgebres et geometrie differentielle, C.R. Acad.
Sc. Paris, 290(1980), 599604.
 Haag,
R., Bannier U.: Comments on Mielnik's Generalized (Non Linear) Quantum
Mechanics, Commun. Math. Phys. 60(1978), 16.
 Heisenberg
W.: Talk about the relation between biology, physics and chemistry.
In "The Part and the Whole", "Das Teil und das Ganze", Munchen 1969.
 Jung
C.G.: Synchronicity. An acausal Connecting Principle,
Routledge & Kegan Paul, London 1972.
 Lord
N.W., Girogosian P.A., Quelette R.P., Clerman R.J., Cheremisinoff
P,N.: Vychislitielnyje masziny budushchevo, Mir, Moskwa 1987
 Mielnik
B.: Commun. Math. Phys. 37 (1974), 221256.
 Sedlak
W.: Postepy fizyki zycia, PAX, Warszawa 1984.
 Sedlak
W.: Na poczatku bylo jednak swiatlo, PIW, Warszawa 1986.
 Sheldrake
R.: A new science of life. The hypothesis of formative causation, Paladin,
London 1981.
 Stapp
H.P.: Light as the foundation of being, Preprint
no LBL19144, Lawrence Berkeley Laboratory 1985
 Volkenstein
M.V.: Physics and Biology, Academic Press, New York 1982
 Wheeler
J.A.: Beyond the Black Hole, w: Some Strangeness in the Proportion; A
Centennial Symposium to Celebrate the Achievements of Albert Einstein,
s. 341375, AddisonWesley, Reading, Massachusetts, 1980
 Wheeler
J.A.: The Computer and the Universe, Int. J. Theor. Phys. 21 (1982),557572
 Wheeler
J.A.: Bits, Quanta, Meaning w: Problems in Theoretical Physics, A.
Giovannini, F. Mancini and M. Marinaro, (eds), University of Salerno
Press 1984
 Wigner
E.P.: Remarks on the MindBody Question, w: The Scientist Speculate, ed. I.J. Good, Heinemann,
London 1962, 284302
 Woronowicz
S.L.: Twisted SU(2) Group, An example of NonCommutative Differential
Calculus, RIMS (Publ of the Res. Inst. Math. Sci.
Kyoto Univ., 23), 1987, 171181
Looking
back
(Preliminary
version)
Each time
I read the text above, I am puzzled by its its prophecy. The Bioelectronics
Symposium took place in November 1987. In June 1990, in Florence, I sketched
a plan of research for the coming years. Philippe Blanchard of the University
of Bielefeld supported this plan and, a year later, in June 1991 in Bielefeld
we started implementing a part of this plan. One year later we sent for
publication our first joint paper. "Event Enhanced Quantum Theory",
in short EEQT, was born. We brought together, in this theory, the
two worlds: the quantum one and the classical one. But the theory was
still linear. Our equations were describing only statistical ensembles
and were unable to understand how does Nature work, how She decides at
which moment a given radioactive atom will decay, at which moment a photon
gets emitted or absorbed. Our equations, as it seemed to us, were correct,
and yet we did not know how to get from them more, how to get from them
a description of individual quantum systems.
The breakthrough
came next year, in June 1993, in fact by a pure accident (or, perhaps,
it was not so accidental ....). I remember this event distinctly. I was
randomly searching in the library of the university of Bielefeld, checking
one book after another, from different library sestions, and finally I
took in my hand a shabby little booklet with lectures by M. H. Davis on
"stochastic
control and nonlinear filtering", published in 1984 by the Tata Institute in Bombay. A little
yellow booklet, published evidently with a small and cheap poligraphy
methods.... I opened it on a random page .... and decide to borrow it,
so as to examine what it is about.
I started
reading. At first I could not understand much, yet I was getting an impression
that there may be some connection between EEQT (the term "EEQT" did not
exist yet, at that time) and one of the chapters of Davis' monogrph. The
monograph was dealing with processes in economics, such as stock market
crashes. Following the stock market data closely we notice that stock
prices fluctuate, grow up and fall down, and for long period of times
vary, on average, in a continuous way. And then, abruptly, there comes
a CRASH. A distinct and discontinuous change. After which we observe another
period of a continuous evolution. One way to study these phenomena is
via the catastroph theory.
But Davis,
in one of the chapters, described a class of random processes that were
fitting to this class of evolutions. I did not get motivated enough to
study the whole monograph, but this particular chapter I begin to study
with paper and pencil in my hand. I was trying to understand the definitions
and the theorem  as there was a single theorem in this chapter. And I
was getting the feeling that this theorem is exactly what we need....
It took me
several days to reallly understand what was going on. Davis considered
a piecewise deterministic random process, with a continuous evolution
interrupted by occasional jumps, and he provided a form of its infinitesimal
generator. This generator had a "differential" part, but it also had an
"integral" part  it is this latter part was responsible for "jumps".
It occured
to me, after having a better look at the formula, that the integral part
was similar to the extra term that we were adding to the Hamiltonian evolution
of a quantum system in our work on EEQT! But to see whether the idea was
correct, I had to replace the flat space used by Davis by a unit ball
in a Hilbert space  as it was needed in quantum theory. There was also
an additional difficulty: in quantum theory expectation values of all
observables are always being expressed in terms of a scalar product. These
expectation values are therefore, in quantum theory, always bilinear in
the wave function.
But to apply
the Davis theorem we would have to have at our disposal all possible functionals
of the wave function. That is the particularity of quantum theory that
distinguishes it from a classical theory: in quantum theory the class
of observables is rather poor! As the result of this particular feature
of quantum theory, we could apply the Davis' theorem only one way: we
could show that the random process described by Davis reproduced our equations
(so called "Master equation"), but we could not deduce, as Davis did,
that this is the only random process that could do that. And this was
a serious drawback, because the process algorithm described the behaviour
of an individual quantum system. And this was what we were looking for,
as this was exactly the part of the description that was lacking in the
standard quantum theory.
As at that
time I was working in Bielefeld, with Philippe Blanchard, on the "Quantum
Zeno Effect" (“a watched pot cannot boil”), we applied the Davis'
algorithm to this particular case. We submitted the paper for publication
in Physics
Letters A. But we were not boasting there about the nonuniqueness
of the process algorithm. Insetad we described the process as "natural"
and "minimal", and it did not catch the attention of the refrees. In fact
this was my intution at that time: that the process is minimal and natural.
But, frankly speaking, if someone would attack our work asking for the
precise reasons why we think this process is minimal and natural  we
would have had a hard time finding a convincing answer. At least not yet.
Another year, and another stimulus were needed for us to put the dot over
that "i"....
