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 is 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 has no answers for. 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 will 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 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 5-th 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 in 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". And 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 that can 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 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 good
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 re-thinked
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 goes wee, 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, and that has, as we shall see, so many common points
with the ideas hidden behind abstract mathematical symbols in the papers
by Igor and Grichka Bogdanoffs.
This vision has been formulated and published in 1987, in the
proceeding of the VI-th 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 of 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 1905-1912) 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
gluing together of a giant snow ball.
Gluing
the snow ball
Tragic
is, first of all, that the ball rolled by the crowd of the most talented
physicists of our time. 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 ball-conglomerate,
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 gluing back together the pieces that
have fallen out, 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 becomesto fall down from it. And, at the
same time, the more the ball itself veils the ball itself blocks the
view of the pushing crowd. Thus there are less and less 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 fashion. 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 by chance along the way.
Then I
went on to describe the new paths in physics - physics of complex and
self-organized systems :
New
Physics
At 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 taken from physics, informatics, and applied to systems imitating
biological systems. Slowly, and with difficulties, new horizons do
open. Computer experiments indicate new, yet unsuspected regularities.
New concepts are being formed. A virgin terrain is waiting for trailblazers.
Adventure is luring in. 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 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 the quantum theory itself?
2.2
The Quantum Seam of Life
We are
witnessing the 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 protein-nuclein 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. (Because)
The description goes on 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 action
of an even simplest electrical circuit consisting of a battery and
a lighting bulb. 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
protrude so far beyond the scope of his competence, it is only because
it seems to him that he is perceiving something essentially new. Something
thatdemans 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 has got to court, the problem of
interpretation of the "wave function of the universe" has 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
in a similar way as a continues pain gets with time duller and duller.
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 non-contradictory,
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 non-linear 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 device-intermediated 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, when a real
break of a paradigm is needed, there is a long lasting stagnation.
The few attempts at constructing a non-linear quantum mechanics [2,6,10]
did not bring 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 observation-measurement. 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 of such a " forced unification" have been, if one does
not count side effects, unfruitful. The same way the dualism energy-matter
may have a primitive character. Continuing the analogy : in the same
way as a gravitational field curves space-time [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 as gravitational
field was a worthy partner to classical matter. The same way as gravitational
field is local[f7]
in space-time, 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 [unequivocally??]conclude
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 will the paradox that Bohr is talking about 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 ruleŻ, or that " the algebra of observables of the
photon field has a non-trivial 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
- Bialynicki-Birula 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.: C*-algebres et geometrie differentielle,
C.R. Acad. Sc. Paris, 290(1980), 599-604.
- Haag, R., Bannier U.: Comments on Mielnik's Generalized
(Non Linear) Quantum Mechanics, Commun. Math. Phys.
60(1978), 1-6.
- 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),
221-256.
- 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 LBL-19144, 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. 341-375, Addison-Wesley,
Reading, Massachusetts, 1980
- Wheeler J.A.: The Computer and the Universe,
Int. J. Theor. Phys. 21 (1982),557-572
- 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 Mind-Body Question,
w: The Scientist Speculate, ed. I.J. Good, Heinemann, London 1962,
284-302
- Woronowicz S.L.: Twisted SU(2) Group, An example
of Non-Commutative Differential Calculus, RIMS
(Publ of the Res. Inst. Math. Sci. Kyoto Univ., 23), 1987, 171-181
Looking back
(Preliminary version)
Each time I read the text above, I am puzzled by the prophesy it makes.
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 our first joint paper for publication.
"Event Enhanced Quantum Theory", EEQT for hsort was born.
In this theory we brought together the quantum and classical worlds.
But the theory was still linear. Our equations were describing only
statistical ensembles and were unable to understand how Nature works,
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 more
from them, how to arrive at 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
sections, and finally I picked up 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. This
was a little yellow booklet, published evidently with small circulation
and printed with cheap poligraphy methods.... I opened it on a random
page .... and decided to borrow it, so as to examine what it was about.
I started reading. At first I could not understand much, yet I was
getting the impression that there may be some connection between EEQT
(the term "EEQT" did not exist yet at the time) and one of
the chapters of Davis' monograph. The monograph dealt with economics
processes such as stock market crashes. Following the stock market data
closely we notice that stock prices fluctuate, rise and fall, 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 continuous evolution. One way
to study these phenomena is by means of the catastroph theory. But Davis,
in one of the chapters, described a class of random processes that could
be used for modeling this class of evolutions. I did not get motivated
enough to study the whole monograph, but this particular chapter I began
to study with paper and pencil in hand. I was trying to understand the
definitions and the single theorem in this chapter. And I was getting
the feeling that this theorem was exactly what we needed.... It took
me several days to really 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 was this latter part that
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 this was needed in quantum theory. There was
also an additional difficulty: in quantum theory expectation values
of all observables are always 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.
This is the particularity of quantum theory that distinguishes it from
a classical theory: in quantum theory the class of observables is rather
poor! [ is 'limited', better?]} As the result of this particular feature
of quantum theory, we could apply 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 was the only random process that could do this.
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 the time I was working in Bielefeld with Philippe Blanchard,
on the "Quantum
Zeno Effect" (“a pot being watched cannot boil”),
we applied Davis' algorithm to this particular case. We submitted the
paper for publication in Physics
Letters A. But we were not boasting there about the non-uniqueness
of the process algorithm. Instead we described the process as "natural"
and "minimal", and it did not catch the attention of the referees.
In fact this was my intuition at that time: that the process is minimal
and natural. But, frankly speaking, if someone had attacked our work
asking for the precise reasons why we thought this process was minimal
and natural - we would have had a hard time finding a convincing answer.
At least then. Another year and another stimulus were needed for us
to put the dot over this "i"....
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