1

COMPLEXITY AND

RIGOROUS EXPERIMENTAL LOGIC

Antonino ZichichiINFN and University of Bologna, Italy

CERN, Geneva, SwitzerlandEnrico Fermi Centre, Rome, Italy

Pontifical Academy of Sciences, Vatican CityWorld Federation of Scientists, Beijing, Geneva, Moscow, New York

PRESENTED AT

International Conference on “Quantum [un]speakables” in Commemoration of John S. Bell, International Erwin Schrödinger Institut (ESI), Universität Wien (Austria), 10-14 November 2000

27th Sessions of the International Seminars on Planetary Emergencies, Erice (Italy), 24 August 2002

40th Course of the International School of Subnuclear Physics, Erice (Italy), 6 September 2002

Pontificia Academia Scientiarum, The Vatican, Rome (Italy), 8-11 November 2002

The joint Session: 6th Course of the International School of Biological Magnetic Resonance; 2nd Workshop on Science and Religion of the Advanced School of History of Physics; 10th Workshop of the International School of Liquid Crystals; Erice (Italy), July 2003

28th Sessions of the International Seminars on Planetary Emergencies, Erice (Italy), August 2003

31st, 32nd and 33th Course of the International School of Solid State Physics, Erice (Italy), July 2004

42nd International School of Subnuclear Physics, Erice (Italy), August - September 2004

International School on Complexity, 1st Workshop on Minimal Life, Erice (Italy), December 2004

INFN-Alice Meeting, University of Catania (Italy), January 2005

Trinity College, Dublin (Ireland), February 2005

Department of Physics, University of Padova (Italy), March 2005

43th Course of the International School of Subnuclear Physics, Erice (Italy), 6 September 2005

Italian Physics Society (SIF) XCI Annual National Congress, University of Catania (Italy), September 2005

Desy, Hamburg, 22 November 2005

INFN Eloisatron Project “The 1st Physics ALICE Week”, Erice (Italy), 5 December 2005

50th Anniversary of INFN Bologna - ALICE Week, Bologna (Italy), 21 June 2006

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COMPLEXITY AND

RIGOROUS EXPERIMENTAL LOGICAntonino Zichichi

ABSTRACT

The Logic of Nature allows the existence of Science (the asymptotic limit of Simplicity) and of History (the asymptotic limit of Complexity). We will review the present status of what we know in the Reductionistic achievements i.e. the Standard Model and its extension which predicts GUT (the Grand Unification Theory), the existence of the Superworld and the resolution of the quantum-gravity problem via the powerful theoretical structure of RQST (Relativistic Quantum String Theory). This is done in order to understand the basic features which allow Complexity to exist. In fact Nature shows structures which are considered as being complex and the general trend is that, in order to understand their roots, we must go from Reductionism to Holism whose frontier is Complexity. But “Complexity” is ill-defined; nevertheless people speak of “Complexity” as a source of new insights in Physics, Biology, Geology, Cosmology, Social Sciences and in all intellectual activities which look at the world through the lens of a standard analysis in terms of either Simplicity or Complexity.

We will see that the necessity to abandon the Reductionism in favour of the Holistic approach is the existence of phenomena whose Laws and Regularities have apparently no links with the Fundamental Laws of Nature from which they originate. We define these phenomena AFB (Anderson-Feynman-Beethoven). The other reason to introduce the Holistic approach, and therefore Complexity, is the appearance of Unexpected Events, apparently marginal, but producing Enormous Consequences. We define them UEEC events.

We will show that these two basic elements, AFB and UEEC – which are at the origin of Complexity with its Holistic consequences permeating all our existence, from molecular biology to life in all its innumerable forms up to our own, including History – do exist at the Fundamental Level. The proof comes from the sequence of totally unexpected discoveries which brought us to the Standard Model and its extension. The conclusion is that, if we want to understand Complexity, the only way is to continue with the Reductionistic analysis of the phenomena, no matter the Mass-Energy and Space-Time scales involved.

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TABLE OF CONTENTS I INTRODUCTION 7II THE FOUR DEFINITIONS NEEDED AND COMPLEXITY 8

III.1 THE BASIC DISTINCTION BETWEEN EWRL (HISTORY) AND EBUS (SCIENCE) 11

III.2 THE PROBLEM OF “WHAT IF?” 12III.3 A REMARK ON THE EVOLUTION OF LANGUAGE, LOGIC AND

SCIENCE 13III.4 THE RIGOROUS BASIS OF EVOLUTION 15IV THE SYSTEMS WHICH ARE CONSIDERED EXAMPLES OF

COMPLEXITY 17V THE ABILITY OF LEARNING AND THE INTERACTION WITH THE

ENVIRONMENT IN THE STUDY OF COMPLEXITY 25VI A FEW WORDS ON THE MATHEMATICAL DESCRIPTIONS NEEDED 27VII DO PROPERTIES COMMON TO COMPLEX SYSTEMS EXIST? 29VIII LET US TRY TO UNDERSTAND THE ROLE OF THE NUMBER OF

ELEMENTS 29IX THE WHOLE OF OUR KNOWLEDGE 31X DOES COMPLEXITY EXIST AT THE ELEMENTARY LEVEL? 37XI THE PLATONIC SIMPLICITY FOR SPACE AND TIME IS VIOLATED 41XII THE STANDARD MODEL AND BEYOND: HOW WE GOT IT 41XIII A FEW EXAMPLES OF DEVIATIONS FROM PLATONIC SIMPLICITY 44XIV THE PLATONIC GRAND UNIFICATION 56XV THE PLATONIC SUPERSYMMETRY 62XVI THE WORLD COULD NOT CARE LESS ABOUT THE EXISTENCE OF

THE SUPERWORLD: THE MOST RECENT EXAMPLE OF ANDERSON-FEYNMAN-BEETHOVEN-TYPE PHENOMENON 63

XVII THE INCREDIBLE STORY OF THE ISOSPIN 69XVIII WILL COMPLEXITY EXIST IN THE FUTURE TEN CHALLENGES?

[see Appendix 9] 77XIX THREE CONCLUSIONS AND THE FINAL ONE 78

APPENDIX 1: Language Logic and Science 83APPENDIX 2: Totally Unexpected Discoveries: A Personal Experience 111APPENDIX 3: Elements of Scientific Rigor in the Theory of Evolution 143APPENDIX 4: A recent Discovery on the Two Levels of Language 157APPENDIX 5: The 62 powers of Ten are not Science Fiction 163APPENDIX 6: We came from a Black-Hole and we are still in a Black-Hole 167APPENDIX 7: The World, Thinking and Language: From Plato to the Vienna

Circle to the Present 177APPENDIX 8: Yukawa Goldmine. From his particle to the QGCW 193APPENDIX 9: The Ten Challenges of Subnuclear Physics [see § XVIII] 217APPENDIX 10: Chaotic Strings 271APPENDIX 11: The Status of Reductionistic Achievements at the time of

Majorana 275REFERENCES 285

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I INTRODUCTION

The real world is characterized by two basic features, which are one on the opposite side of the other: Simplicity and Complexity. It is generally accepted that Simplicity is the outcome of Reductionism, while Complexity is the result of Holism. The most celebrated example of Simplicity is Science while the most celebrated example of Complexity is History. Science can be considered as the asymptotic limit of Reductionism and History as the asymptotic limit of Holism. A common characteristic to both Science and History is Evolution. In fact Science can be identified with the Evolution of our Basic Understanding of the Laws governing the World in its Structures (EBUS) and History with the Evolution of the World in Real Life (EWRL).

The greatest achievement of Science is the Standard Model, and its extension; this is the result of four centuries of Galilean research work based on Reductionism, i.e. on the identification of the simplest elements in the study of Nature. The Standard Model allows to identify the fundamental constituents of matter and the Fundamental Laws of Nature; its extension predicts the Grand Unification Theory (GUT), the existence of the Superworld and the resolution of the quantum-gravity problem via the powerful theoretical structure of RQST (Relativistic Quantum String Theory).

But Nature shows also structures which are considered as being complex; according to the present knowledge of these structures, the general trend is that in order to understand their roots we must go from Reductionism to Holism. In other words, Reductionism and Simplicity seem to go together, as well as Holism and Complexity.

The existence of Complexity in the real world emerges from two experimentally well-established basic elements: 1) the Anderson-Feynman-Beethoven-type phenomena (AFB) i.e. phenomena whose Laws and Regularities ignore the existence of the fundamental Laws of Nature from which they originate; 2) the Sarajevo-type effects, i.e. Unexpected Events of quasi irrelevant magnitude which produce Enormous Consequences (UEEC).

Purpose of this work is to show that the two experimental basic elements which motivated the birth of Complexity, (AFB) and (UEEC), exist also in Science (EBUS). It turns out that Complexity in the Real World exists, no matter the Mass-Energy and Space-Time scales considered. To prove that

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Complexity exists at all levels, including the fundamental one, it is necessary to study: i) the experimental evidence which called for the existence of Complexity; ii) how we have discovered the Standard Model including its extensions; iii) what would be the Platonic version of the Standard Model and of the Superworld: i.e. what would be the ideal Platonic Simplicity. From the existence of AFB phenomena and UEEC events at all scales three working conclusions follow and a final one.

II THE FOUR DEFINITIONS NEEDED AND COMPLEXITY

Let us introduce four definitions; two refer to Evolution and two are specific to experimental Observations. The first two for Evolution allow to put History and Science on a comparative ground. In fact:

1) History is the Evolution of the World in its Real Life: EWRL;2) Science is the Evolution of our Basic Understanding of the Laws

Governing the World in its Structure: EBUS. The other two refer to experimental Observations. 3) At different scales of our World, different phenomena take place,

with different laws and regularities and no apparent link between them. These laws and regularities are mathematically described by Effective Theories. We call these phenomena Anderson-Feynman-Beethoven (AFB) events.

Let us mention a few examples. Beethoven and the Laws of Acoustics.Beethoven could compose superb masterpieces of Music, without any

knowledge of the laws governing acoustic phenomena. But if these laws were not valid, Music could not exist.

The living cell and QED.To study the mechanisms governing a living cell, we do not need to

know Quantum ElectroDynamics, QED, despite the fact that all these mechanisms are examples of purely electromagnetic processes.

Nuclear Physics and QCD.Proton and neutron interactions appear as if a Fundamental Force of

Nature is at work: the nuclear force, with its Laws and its Regularities. These

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interactions ignore that protons and neutrons are made with quarks and gluons. Nuclear Physics does not appear to care about the existence of Quantum ChromoDynamics, QCD, despite that all phenomena occurring in Nuclear Physics have their roots in the interactions between quarks and gluons. In other words, protons and neutrons behave like Beethoven; they interact and build up Nuclear Physics without knowing QCD, i.e. the laws governing the interactions between quarks and gluons. This AFB event is the most recent and better known in terms of experimental measurements and of rigorous mathematical description.

The existence of AFB events is the first reason why Complexity is considered the be at work in the study of natural phenomena. In fact at different scales of the real world, different phenomena take place; the difference being of such a nature that the phenomena at a given scale take place with a total absence of any apparent link with the phenomena which exist at other scales.

4) Sarajevo-type Events., i.e. Unexpected Events with Enormous Consequences, UEEC.

No-one predicted the “Sarajevo-tragic-event”: one man assassinated one day in Europe. This event caused the 1st World War. Many other events of this nature have given rise to what is now called virtual History, which will be discussed in the next chapter. These types of events are well known in History.

This is not the case in Science where it has not been sufficiently emphasized that the greatest steps forward have all been totally unpredicted, like the Sarajevo-type Event. We will return to this point in Appendix 2. Here we would like just to quote few examples. No one was able to predict the existence of Radioactivity and no-one predicted the existence of the “Strange Particles”. Both these unexpected events had enormous consequences.

These two cases are examples of experimental discoveries but Sarajevo-type Events occur also in theoretical physics. During nearly half of the past century (20th), many physicists believed in the existence of Axiomatic-Field Theory. The “perfect” example was QED. We had in Erice the most formidable School on this topic, whose Director was one of the most prominent leaders in the field: Arthur S. Wightman.

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The theoretical “prediction” was that the mathematical description of all Fundamental Forces of Nature had to be QED-like.

Then the theoretical difficulty discovered by Landau came that if we want QED to be valid, no matter the value of energy being very very large, then QED must be “trivial” i.e. its coupling must be zero. After a few other decades of experimental and theoretical work the “prediction” turned out to be wrong, since QED is the only “Abelian” force of Nature.

We now have the Effective Theories which means mathematical-ad-hoc descriptions of the Real World at the various Energy Levels. This means that the prediction based on the believe that all Fundamental Forces of Nature would have been described by the same mathematical formalism of QED had to be abandoned: this result has been achieved via a series of Sarajevo-type Events.

The conclusion is that we do have Sarajevo-type Events and Beethoven-type Phenomenology in History EWRL as well as in Science

EBUS. “Complexity” has never been defined in a rigorous way; nevertheless

people speak of “Complexity Science” as a source of new insights in Physics, Biology, Geology, Cosmology, Social Sciences and all intellectual activities which look at the world through the lens of a standard analysis where AFB phenomena and UEEC events call for a rigorous understanding.

The first point in the game for “Complexity” is to understand the Anderson-Feynman-Beethoven-type Phenomenology: why the laws and regularities which govern “one level” of the real world cannot be understood from the knowledge of the laws and regularities that govern the other levels.

The second point is to understand the reason why UEEC events do take place.

A detail which is also coupled to Complexity is to explain how order can appear spontaneously (S. Kauffmann, “Origins of Order”, Oxford University Press, New York, 1993).

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III.1 THE BASIC DISTINCTION BETWEEN EWRL (HISTORY) AND EBUS (SCIENCE)

Let us now consider the mathematical description of EBUS = Science and of EWRL History.

We have seen that: The Sarajevo-type Events take place in History  EWRL, and produce Unexpected Events with Enormous Consequences UEEC.

The Sarajevo-type Events take place also in Science EBUS, and produce Unexpected Events with Enormous Consequences UEEC.

The basic distinction between History EWRL and Science EBUS is in the ability we have to describe Science EBUS using Rigorous Mathematics, while no one is able to describe History EWRL using Rigorous Mathematics.

The mathematical description of Science EBUS allows predictions to be made and these predictions can be submitted to experimental verifications which need to be reproducible.

For the other type of Evolution, History EWRL, there is neither prediction nor verification.

Suppose we were able to introduce Effective Theories as we do in Science, i.e. Mathematical models to describe a posteriori the Past. In this case the result would be that we would be able to make predictions in the field of History EWRL. And this means that we would predict the future. Some of these predictions would be proven by the facts that they do not follow expectations while some others will do so, as it happens with Science (EBUS).

In Science EBUS the mathematical description of the real world at different scales in all fundamental variables is a central part of the game.

In History EWRL the difference which exist in Time and Space for the various occurrences could be attempted to be described using mathematical models. No-one has attempted to do so since the mathematical modelling of what has been going on in History EWRL has never attracted the interest of people; the reason being “tautology”; i.e. its a priory classification in the field of “Complexity”. In other words Science EBUS is simple, History EWRL is complex.

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III.2 THE PROBLEM OF “WHAT IF?”

The existence of Sarajevo-type Events in History has brought the specialists on these studies to point out that another History could have existed “if” some Sarajevo-type Events had not occurred. This is the origin of the so-called “Virtual History” which has its roots in “what if?” On the following Table 1 I have reported a list of “what if?” in History EWRL and in Science EBUS, respectively.

“WHAT IF ? ”

Table 1

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While in History EWRL no-one can work out the consequences of a given “what if?”, in Science EBUS this is easy and it can be argued that, in any case, even if Rutherford had not discovered the nucleus, somebody else would have done it in some other Laboratory. This is not the case for History EWRL, since there is no reason to believe that if Hitler had not been born, somebody else would have done what he did. These are just two examples which illustrate a point of departure produced by “what if?”

In other words, while ABF and Sarajevo-type Events both exist in Science EBUS and History EWRL, the consequences of “what if?” would be on equal footing for Science EBUS and History EWRL only if the future for all of us would be under the control of a fundamental rigorous logic, as it happens to be for Science EBUS. Here is the point where the asymptotic limit of “Simplicity”, EBUS (Science), and the asymptotic limit of “Complexity”, EWRL (History), manifest a clear difference, despite the fact that both History and Science enjoy the privilege of Evolution.

III.3 A REMARKS ON THE EVOLUTION OF LANGUAGE, LOGIC AND SCIENCE

We will see in chapter IX that the greatest achievements of our intellectual activity are Language, Logic and Science which will be fully discussed in Appendix 1. These three intellectual activities share common properties, the most important one being evolution. On the other hand there are many Languages, many logic structures but only one Science since, out of all possible logical structures only one has been chosen to build the world were we are.

The present understanding of the role of Language in our intellectual activity brings us to the following four points:

1) To establish if all Languages have the same origin. To this problem are connected the points 2 and 3 which are related to the fact that Languages can be classified into “primitive” and “optimal”.

2) There are primitive Languages whose power of expression cannot cover the philosophical concepts of the great thinkers like Plato, Aristotle, Confucius, just to quote few cases.

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3) Once this level is overcome, all Languages have the same power of expression; nevertheless they continue evolving. And here the problem of their evolution comes in. Why are Languages still evolving, given the well established fact that each one has reached the maximum, optimal, level of expression?

4) Can “Evolution” be the way to distinguish the three achievements of our intellectual activity previously discussed: Language, Logic and Science?

In fact while the Evolution of Language does not produce any basic new knowledge, this is what happens with Logic and Science. Logic starts with Epimenide (nearly three thousands years ago) who discovered that there are statements which can neither be classified “true” or “false”. This “indecibility” in Language was brought into the field of the Arithmetic in 1931 by Kurt Gödel, with his famous discovery.

In the field of Arithmetic the evolution has some basic stones like the discovery by Pitagora of the irrational numbers, the discovery of George Cantor of the Logic of Infinite sets and, most recently, with the discovery of the so-called (Omega) numbers.

In fact the concept of number starts with the simple act of our intellect, that of counting stones, trees, and then abandoning the real objects. A fundamental step in the Logic of Arithmetic was achieved when the ratio of lengths of different geometrical figures were studied. This is how the first irrational (algebraic) number, 2, was discovered by Pitagora and this is how the first irrational (trascendent) number, the ratio of the length of a circle by the diameter of the same circle, was discovered in Euclidean geometry. It took nearly two thousands years to prove that the “irrationality” of is basically different from the “irrationality” of 2, as proved by Ferdinand Lindemann in 1882. The existence of the numbers needed the electronic computers to be discovered. The present understanding of the power of Rigorous Logic, after 3000 years of evolution, is very different from the one known at its origin.

With Evolution of Science things go even more drastically different. We recall that our definition of Science is the Logic of Nature. When this intellectual game started in the 1600 with Galileo Galilei no one knew that this Logic was there. Galilei is in fact the first to realize that, if we want to uncover the Logic of Nature, the only way is to implement an experiment, to be sure that

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its result is reproducible and to incorporate it in a rigorous mathematical formalism.

The Evolution of Science is characterized by the big steps being due to “totally unexpected discoveries”. Appendix 2 is dedicated to this topic. Let me quote the oldest example: for 2000 years mankind was convinced that the forces are proportional to speed, but Galilei discovered that the force is proportional to acceleration: a totally unexpected discovery. Another formidable example of totally unexpected discovery is due to Lorentz.

It took nearly two hundreds years of discoveries in electricity, magnetism and optics, to discover that “Space” and “Time” cannot be both “real”. This came out as a result of the property which possess the four Maxwell equations, which represent the rigorous synthesis of decades and decades of experimental discoveries which, following the lesson by Galilei, were expressed in a rigorous mathematical formalism by Maxwell. It was the geniality of Heinrich Lorentz to discover that these equations had the property of being invariant when subject to a mathematical transformation now known as Lorentz transformation where Time and Space cannot both be real.

The conclusion is that Evolution in Logic and Science produces very different results than Evolution in Language. But, one of the most typical conquest of Language is History, the asymptotic limit of Complexity, which we have already in chapter III.2 confronted with the asymptotic limit of simplicity Science, on the basis of the condition “what if?”. We recall that History has been defined as EWRL and Science as EBUS, both being characterized by the basic process called Evolution. Let us investigate the rigorous basis of it.

III.4 THE RIGOROUS BASIS OF EVOLUTION

Despite the fact that Complexity is ill-defined, there is a general belief that evolution is a property of complex systems. It is therefore necessary to study its rigorous basis; we will discuss this in Appendix 3. Here we want to point out the need for a rigorous definition of the property of Evolution, since everything in the real world evolves: even one of the most elementary example of the real world, the electron, evolves in Space and Time. In fact Evolution means, first of all, the study of what can happen if we evolve along the two components of the

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fundamental structures of our world: the Space and the Time. We do see with our eyes three space-dimensions and we do fill, with our sensorial elements, the existence of Time. The inert matter can move in Space and Time, thus evolving in its structure; an example is the growth of crystal structures. Also the living matter evolves in Space and Time. The vegetable forms of life move very little in Space (such as a tree or any other form of vegetable life shows) while animal forms of life (such as birds and fishes) are able to move in Space much more. Concerning Evolution in Time, the vegetable life can span time ranges of millennia (like very old threes show) while the animal life has smaller interval of time ranges for its evolution. Given this observable properties of Evolution in Space and Time of the various forms of matter, the most exact knowledge of Evolution achieved so far refers to the inert matter.

The first big step in this field is due to Dirac who discovered the mathematical formalism needed in order to describe the Evolution of the first elementary particle ever discovered, the electron, once the condition that Space and Time not both being real is rigorously taken into account. This gave rise to a totally unexpected result. The fact that the existence of the electron implies the existence of the antielectron. More generally the existence of a particle needs to existence of its antiparticle: i.e. an object with has exactly the same properties (mass, spin, lifetime) but opposite “charges” as we will see in chapter XIII (and especially Figures 18 and 19). The lesson we get from the rigorous study of the Evolution of Space and Time of the elementary particle called electron is that not only the concept of antiparticles come in but the new horizons of antimatter, antigalaxies and antiworld. So, in the field of “inert” and very elementary forms of the real world, the Evolution in Space and Time produces results that no human intellect had ever been able of imagining the existence of. When we go from this extreme example of Simplicity in the inert matter - the electron - to the opposite extreme example of Complexity in the living matter - our species - the result is that the Biological Evolution of the Human Species is below the third level of Galilean Science. As mentioned before, a detailed analysis of the scientific rigour needed to study the theory of evolution in all forms of matter is reported in Appendix 3.

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The analysis of the rigorous properties needed to identify the “Evolution” of a given system is a further proof that Science and History are at the two opposite extreme points in terms of Simplicity and Complexity. And now the problem arises: do phenomena exist between these two extreme examples of Simplicity (Science) and Complexity (History)? The answer is yes.

IV THE SYSTEMS WHICH ARE CONSIDERED EXAMPLES OF COMPLEXITY

There are a number of phenomena in the real world which have, as a common characteristic, the fact that macroscopic effects appear simple, despite the existence of complex microscopic phenomena which the macroscopic effects cancel. This happens because the microscopic fluctuations average out when larger scales are considered and the averaged quantities satisfy, for example, classical continuum equations.

The standard case of this is Hydrodynamics; here atomic fluctuations average out and the classical hydrodynamics equations emerge.

But there are phenomena where the fluctuations persist out to macroscopic wavelengths and the fluctuations at all intermediate scales are important. Examples of these phenomena are the turbulent fluid flow (turbulence in the Atmosphere), the critical phenomena like phase transitions, the magnetic impurities in non magnetic materials which generate the so-called Kondo Problem, the virtual phenomena in elementary particle physics which produce the Renormalization Groups Equations (RGEs).

These equations have many applications in phenomena which are far away from the study of the physics originated by the virtual effects. For example the percolation, the electron localization or conduction in random media, the structural transitions and the “Lifshitz” critical point, the problem of interfaces between two phases, the critical behaviour or ordering in random systems such as dilute magnets, spin glasses and Systems with random external fields. All these phenomena represent fields of our knowledge where, if it was not for the discovery of the RGEs, Complexity could have been considered to be of relevance.

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This is certainly not the case when the problem of life and of Systems like the Social ones are considered. We will now put together examples of both classes; those where Complexity persist and those where Complexity has been explained in terms of new mathematic tools. It is interesting to have a general view of the Systems whose properties could have been interpreted in terms of Complexity if the discovery of the way out had not occurred, since this could shed light on those Systems where Complexity is still considered to be of great relevance. It could very well be that the reason why some systems are considered to be complex is that we have not been able to discover the correct mathematical formulation appropriate for their description.

• Turbulence as an Example of Complexity in many Areas. Turbulence is not only important in the Atmosphere to help the weather prediction, but in a wide range of contexts such as the motion of submarines, ships and aircraft, pollutant dispersion in the earth's atmosphere and oceans, heat and mass transport in engineering applications as well as geophysics and astrophysics. Turbulence often occurs in conjunction with other features such as rotation, magnetic field and particulate matter, so the knowledge of the subject is useful to a number of closely related problems such as interstellar dust, energy transport and planetary magnetic fields. The problem is also a paradigm for strongly nonlinear systems, distinguished by strong fluctuations and strong coupling among a large number of degrees of freedom, and so even distant areas such as fracture – perhaps even market fluctuations – will benefit from a better understanding of turbulence. The subject is a particularly useful paradigm of a wide variety of non-equilibrium systems because the equations of motion are known exactly and can be simulated with precision. However, the complexity of the underlying equations has precluded much analytical progress, and the demands of computing power are such that routine simulations of large turbulent flows has not yet been possible. Thus, the progress in the field has depended more on experimental input.

A particular aspect of turbulence, namely wall-bounded flows, represent a field of great relevance for advanced research in theoretical, experimental and numerical simulation areas. Research in the field of turbulence with high Reynolds number flows, has been characterized by a slow rate of

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progress. For example, while wall-bounded flows present a wonderful array of problems in turbulence, we have lately been bogged down by some controversies that do not seem to have been resolved to everyone’s satisfaction. These controversies have been further complicated in the light of several recent experiments on zero pressure-gradient turbulent boundary layers (in USA, Sweden, Holland, and Australia), in the Princeton “superpipe” facility, and in carefully planned Atmospheric Surface Layer measurements on the salt-flats of Utah, that have yielded ground-breaking results and have substantially modified the consensus about the so called log-layer constants for the mean velocity profile, and the scaling laws behaviors of the turbulence intensities. New computations (the largest ever carried out in scientific numerical simulation) with moderately high Reynolds number direct numerical simulations (from Spain, USA and Japan), have also given new insights into the three-dimensional structure of these flows and brought into question the role of the inner/outer layer interaction in wall-bounded flows.

Thus, while many such issues remain to be explained, important new work has been appearing and new data are being acquired at a steady rate. The present goal is to be able to understand what the data say and develop a solid framework on which to build further developments, define new experiments and identify critical theoretical issues. Turbulence can be a great paradigm to deepen our knowledge on Complexity because the equations are well known and every idea can be checked precisely, unlike many other parts of the real world were Complexity is considered relevant but the mathematical formulation is far from being understood. A special case where Turbulence attracts the interest of the great public is Weather Prediction: i.e. Turbulence in the Atmosphere.

• The Turbulence in the Atmosphere. Global air circulation becomes unstable thus producing eddies on large scale, such as thousands of kilometres. These eddies breakdown into smaller eddies, which breakdown into even smaller eddies until chaotic motions are excited in all length scales down to millimetres, where viscosity damps the turbulent fluctuations and no smaller scales come into play until atomic scales are reached [H.A. Rose and P.L. Sulem, Journal Physics 39, 441 (1978)].

Suppose we want to have an Atmospheric flow simulation covering all length of scales of turbulence. This means that we should reach the millimetre

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scale. Since the atmospheric flow deals with quantity of air extending over many thousands kilometres in the horizontal plane and ten kilometres in altitude, the total number of grid points would be in the range of 26 powers of ten and this goes far beyond the power of any computer (at present).

• The Critical Opalescence. A phenomenon called “critical opalescence” is produced by the strong light scattering from micron size “bubbles of steam” and “drops of water”. In this case water and steam become “milky”. Contrary to every day’s life, when bubbles of steam and drops of water are clearly different [at room temperature and pressure], when the temperature is 374º C and the pressure 218 Atmosphere, the difference in density between water and steam goes to zero and the system is called to be at the “critical point” [R.C. Weast, CRC Handbook of Chemistry and Physics, 62nd ed. (1981)].

Away from the critical point, surface tension makes small drops or bubbles unstable; in fact the surface tension between the two phases vanishes only at the critical point where water and steam become indistinguishable and allow them to both exist with dimensions in the micron range. At the “critical point” of the water-steam phase transition, bubbles of steam and drops of water exist at all size scales, from macroscopic visible sizes down to atomic scales.

• The Kondo Problem. Magnetic impurities in non magnetic metals. It happens that electrons of all wave lengths, from atomic dimensions up to very large scales, interact with the magnetic moment of each impurity in the metal [P.W. Anderson, Journal Physics C3, 2346 (1970)].

• The Virtual Phenomena. The starting point is QED, as defined by Dirac, Fermi, Heisenberg, Pauli, Jordan, Wigner [J. Schwinger, ed., Quantum Electrodynamics, Dover, New York (1958)]. The solution of QED was worked out as perturbation series in “e0”, the bare charge” of QED. The QED Lagrangian contains two parameters: “e0” and “m0” (m0 being the “bare” mass of the electron). The basic ingredients of QED are, the electron (with bare mass and charge, e0, m0) and the photon. Only in lowest order does one find

e0 = em0 = m ,

where e and m are the physical values of the mass and the electric charge of the

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electron.When virtual effects are taken into account, it was found in the thirties

that

e and

m

are all infinite, due to integrations over momentum that diverge in the small-distance limit (large momentum limit).

In the late 1940s it was found that all divergences were of two types: charge-like and mass-like. A natural solution was to replace the “theoretically divergent value of the physical mass and charge” with the experimentally measured values of the same quantities. No matter the effect to be measured, the experimental results were in perfect agreement with the theoretical calculations, provided that the change of parametrization, from the Lagrangian parameters

“e0”and

“m0”

to the measured quantities”e”

and “m” ,

was at the same time followed be rescaling the electron and the electromagnetic fields.

There are many reparametrization of QED that eliminate the divergences but use finite quantities different from

“e” and

“m”to replace

“e0”and

“m0” .

Stueckelberg and Petermann discovered that transformation groups could be defined which relate different reparametrization. They called these

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groups “groups de normalization”. This is the origin of the “Renormalization Groups”.

• The General Difficulty. The above mentioned classes of phenomena share the same general difficulty: they involve very many coupled degrees of freedom. In fact it takes many variables to characterize a turbulent flow or the state of a fluid near the critical point. Analytic methods are most effective when functions of only one variable (one degree of freedom) are involved. Computers can extend the capability of theorists, but normal methods of numerical integration are difficult when dealing with 10 integration variables. Partial differential equations become extremely difficult when the number of independent variables is larger than three.

Montecarlo and statistical averaging methods can treat cases with up to 106 variables, but the problem here is to show convergence versus the computer time needed.

The Renormalization Group approach is a method for solving problems where many length scales come in.

The Renormalization Group approach is to integrate out the fluctuations in sequence, starting in a small scale and then moving to successive larger scales until fluctuations on all scales have been averaged out.

• Evolution and Life. The study of our remote past shows us the famous GPBE problem: the 5 Great Periods of Biological Extinction. These 5 Periods are connected with the emergence of the first biological species in this satellite of the Sun, about 3,5 billion years ago. This went on with the unicellular organisms for about 3 billion years, when complex unicellular life exploded 550 million years ago during the Cambrian period.

It is at this point in time that modern phyla were formed together with other forms of animal life that left no descendants; once again UEEC events show up. Nevertheless in the study of living organisms two problems emerge. One is to understand evolution; the other one is to explain the origin of life. There is a school of thoughts which considers that the way out to understand these “complex” problems is Holism, not Reductionism. Let us start with the simplest form of evolution, the chemical one.

It should be pointed out that there are two classes of “chemical evolution”. One which is “lifeless” and the other which produces “living organisms”. This class of chemical evolution processes is called “organic

20

evolution”. It has recently been suggested [G. Matsuda, Acta Med., Nagasaki, July 2005] that the primary factor of organic evolution is the “self-selection” of the organism itself, and the evolutionary factors due to competition with other species and symbiosis with other species appear to be secondary. In any case in the field of “chemical evolution” the big problem is the transition from inert matter to living matter.

Concerning the origin of life from inert matter, it was once thought that micro-organisms arose by the process of decay and even that vermin spontaneously developed from household rubbish. Controlled experiments using sterilized media by Louis Pasteur (1822-1895) and others finally disproved this belief. These experiments demonstrated that Abiogenesis, i.e. the origin of life from non living matter, has no scientific foundation.

Then the problem is to explain the origin of life. The present theory dates the origin of life back to 3.5 billion years ago when a soup made of organic matter (Ammonia, Methane, Hydrogen and Weather vapour) underwent a process of chemical evolution, using the energy from the Sun and electromagnetic storms, to combine into ever more complex molecules, such as Aminoacids, Proteins and Vitamins. In all these processes the interaction with the environment had a dominant role and eventually self-replicating nucleic acid, the basis of living matter, developed. The very first organisms may have consisted of such molecules bound by a simple membrane. And here the problems arises: what is the minimum amount of matter which can give rise to a living organism? This problem is called “minimal life” and should be one of the many topics where Complexity is considered of great relevance.

Let us assume that the transition from inert to living matter is explained thanks to Complexity. The next problem is the evolution of living matter. Here comes Charles Darwin (1809-1882) who proposed the theory of evolution of organic matter. The theory of Darwin [“The Origin of Species”, 1859] postulated that present-day species have evolved from simpler ancestral types by the process of natural selection which operates on the various examples of the same population.

There were two major unresolved problems: how the variations in a population arose and how they were maintained. The answer to these two problems came from Gregor Mendel (1822-1884) who discovered the Laws of

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Inheritance and produced what is now known as the theory of neo-darwinism. This theory makes use of the modern knowledge based on genes and chromosomes to explain the source of the genetic variation upon which selection works.

No matter what the details can bring to. The point is that the field of Evolution and Life is considered the paradigm of what Complexity can produce in the real world. In fact these are problems where AFB phenomena and UEEC events do show up. Thus Complexity and Holism are supposed to play a fundamental role and to allow a full understanding of the burst of biological creativity and extinctions. But now the great novelty is that Life and extinction are no longer considered a privilege of Biological structures. They are considered examples of what happens in many other fields such as the field of multinational corporate entities. According to this School of thought Life and extinction is not a privilege of biological structures. In the volume “The Transformation of Corporate Control” [Harvard University Pres, 1990] Neil Fligstein points out that 33 of the top 100 US companies in 1912 remain in the list in 1979. Biological living organism are like multinational corporate entities: both fallible and mortal. Apparently, each year, more than 10% of all US companies disappear.

Life and extinction exist all over the activities we can think of and extinction appears as being due to Sarajevo-type Events. Purpose of “Complexity Science” is to understand these phenomena and the reason why no one is able to make predictions on their occurrence. For example in the world of Economy and Politics there are phenomena such as the demise of the blue-chip companies Enron and World Com., the failure of Coca-Cola’s “New Coke” in the 1980s, the collapse of URSS which no one was able to predict as well as all other examples some of which appear in Table 1 of chapter III.2. Complexity Science should allow to understand why this lack of prediction does take place: i.e. to understand the UEEC events.

• Chemical Systems and Social Systems. Some features of the Social Systems have been understood, or claimed to be understood, in terms of Chemical Systems created by continually adding and removing certain

22

chemicals. These chemical structures cannot exist at thermodynamic equilibrium conditions.

In order to create these structures, the system must be driven far from chemical equilibrium and the structures emerge “suddenly”, not gradually, while one moves far enough away from equilibrium. The entropy of these systems is lower than their equilibrium counterparts. They feed on low-entropy ordered systems and expel higher-entropy by-products, thus continually producing entropy in the world around.

V THE ABILITY OF LEARNING AND THE INTERACTION WITH THE ENVIRONMENT IN THE STUDY OF COMPLEXITY

When we study Systems composed of living beings, like ants, birds and people, the “basic” elements of each system are not “molecules” but living beings. These “constituents” have the ability to learn and the property of interacting with the environment.

The ability to learn represents an important qualitative property which distinguishes the variety of Systems where we think that Complexity is at work. In fact there are different levels of Learning.

For example Systems made with ants, birds and other living beings, not endowed with reason, represent examples of Systems whose components have a power of learning at a level which is below that of the human species. Of all living beings this is the only one endowed with “Reason”.

When analogies are searched for in the various Complex Systems composed of living beings we should not forget that Social Systems are made with components (men and women) whose ability of Learning is higher than all other forms of living matter.

A parameter which enters in the properties of a Complex System is the time scale necessary to change the System as a whole. There are Systems whose components have zero ability of learning and negligible interactions with the environment. Question: How the ability of Learning and the interaction with the environment can affect the properties of a Complex System?

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Figure 1

Brain Activity

BiologicalPhenomena

The SubnuclearUniverse

The Behaviour of Financial

Markets

The Nuclear Universe

The Critical Opalescence

Turbulencein the

Atmosphere

Earthquakesand

Seismicity

Human GenomeImmune System

Internet Network

Traffic Flux

Brain NeuralNetwork

Social andEconomic Systems

Self-Gravitating Systems

Cosmological Structures

COMPLEXSYSTEMS

Spin glasses

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In Figure 1 we report an example of variety of Complex Systems which go from problems dealing with inert matter (Earthquakes and Seismicity, the Virtual Phenomena, the Critical Opalescence) to problems dealing with living matter (Biological Phenomena, Social and Economic Systems, Brain Neural Network). The structure of contacts on the World Wide Web (WWW) is an example of Complexity among the most recent ones.

VI A FEW WORDS ON THE MATHEMATICAL DESCRIPTIONS NEEDED

Let us assume that all phenomena, no matter if they belong to biology, physics, chemistry, economy, politics, society, can be described by a system of differential non linear equations strongly or weakly coupled. Meteorology is a field where all phenomena taking place in the atmosphere of our Earth are mathematically described by 3 differential non linear equations strongly coupled.

The Unification of all Fundamental Forces of Nature is the field where all phenomena taking place from the smallest structure of matter to the extreme frontiers of the Universe are mathematically described by 3 differential non linear equations weakly coupled.

In fact, no matter what is the System we are dealing with; if we want to express its evolution using a mathematical description, we need two basic conditions: differential and non linear mathematics. Differential since anything happening as a function of some variable quantity must take into account even the smallest change of the quantity being considered as variable; “non linear” mathematics, since the evolution of the System depends, not only on the conditions where the System exists, but also on the System itself.

Differential non linear mathematics (erroneously called by some fellows Chaos Theory) tells us that the System of coupled differential non linear equations has no analytic solution. The only way out is numerical computation. It is this the only possibility we have in order to know what predictions the mathematical structure is able to give. It is found that very small changes in the initial conditions can have enormous consequences. Once the analysis is made of the time needed for the computation, the answer can be that

25

in some cases there is no way of determining the evolution of the System but that of watching its development. Let us quote an example.

At the beginning of the XX Century Richardson had reached the same conclusions when he realized that the basic equations describing the motion and the thermodynamics of the masses of air which compose the atmosphere surrounding our Earth were known and the only difficulty was the time needed to perform the computation.

Having at his service 100 computing fellows the time needed to know the evolution of the atmosphere was so enormous that the “prediction” would have arrived after. In other words if London had sun or rain could be “predicted” after the sun or rain arrived in London.

The reason being the fact that the electronic computation had not been invented and the speed of the computing machines was very low indeed. In the 1950s John von Neumann, using his electronic computing technology, is the first man who was able to make “predictions” before the arrival of the meteorological events.

As mentioned above, Meteorology is an example of phenomena described by the mathematics of non linear differential calculus which implies the exponential dependence of the results. But there are phenomena such as the 5 GPBE and the contacts of the WWW which are described not by “exponentials” but by a “power law”: the frequency of an event falls away with the square of its size. According to the practitioners of these fields this result is not due to perturbations coming from outside, as it could appear to be the case for the Sarajevo-type Events, but from the intrinsic dynamics of the System which contains highly interconnected networks. Those working in the field argue that the reason for this to happen is the extreme conditions far from equilibrium that exist.

Let us assume that this is the case. If this is true, we should be able to “predict” in all fields where we would be able to separate the System from external events and to put under control the non equilibrium conditions existing inside the mathematical description of the System itself, the Sarajevo-type Events, not matter if this is History (EWRL) or Science (EBUS).

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Awaiting for this to happen, let us go back to the Systems considered examples of Complexity, some of which were reported in Figure 1. Let us try to find out if there are properties which can link the different examples of Complex Systems.

VII DO PROPERTIES COMMON TO COMPLEX SYSTEMS EXIST?

For example is there a minimum number, a threshold, of Fundamental Constituents above which the Laws of Complexity spring up in all these Systems? What is the effect of the ability of Learning and of the Interaction with the Environment in the properties of each Complex System? No one knows the answer to such a problem. Let us focus our attention on the different forms of Living Matter.

Experimental observations allow to say that a living organism can be made with only one living cell. It has been recently discovered that the smallest amount of matter which is able to reproduce itself has the dimensions of 100 nm (nm 109 metres). Nevertheless, experimental observations allow us to conclude as well that a Living organism can be made with many, very many, very very many, Living Cells. Question: can we conclude that the number of Cells plays a positive role: i.e. the higher the number of Cells, the better it is?

VIII LET US TRY TO UNDERSTAND THE ROLE OF THE NUMBER OF ELEMENTS

A Living organism (L) can consist of very many Cells (C). L and C are able to develop functions not having evident links with the properties of their Fundamental Structures.

The characteristics of C and L and the Fundamental Laws acting in C and L cannot be easily related to the properties of their Fundamental Constituents. What about the Number of Constituents?

There is a recent experimental finding where the number seems not to be relevant. The decodification of the Human Genome has shown that it contains only about 25,000 genes. Where is the seat of Human Species

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Complexity since the number of genes is similar to that of many other Living organisms? The number of genes does not seem to play a role as far as Complexity levels are concerned. What plays an important role is the network of interactions between Genes and Proteins. This interaction is responsible for the good functioning of a Cell. Research studies must then be shifted from Genomics to Proteomics.

Figure 2

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This brings to the problem of understanding the structure of proteins and the interactions between proteins and between proteins and environment (examples of Complex Systems).

The problem of trying to understand a Living organism produces the chain shown in Figure 2, where the ability of Learning and the Interaction with the environment goes down. The lowest level where the interaction with the environment is still relevant is that of the proteins. Below the proteins we have the Fundamental Constituents which bring us to the Standard Model where the Environment, as a Global Structure, can be totally neglected. But, it can be argued that in order to understand the origin of Complexity, we need to study the whole of our knowledge.

IX THE WHOLE OF OUR KNOWLEDGE

The fundamental quantities and their properties, needed to build up the Universe, together with the basic conceptual structure of all our intellectual activities are shown in Figure 3. Here are the basic elements.

The Universe could have existed and no Life; the fundamental quantities and the structures needed to build the Universe would have been exactly the same; Universe and Life could have existed but no Conscience; the next step is Creativity which could have not been there.

Finally Reason, which has produced the three greatest achievements of the Human intellect: Language, Logic and Science. This has been discussed in many Seminars and in Ref. [1], which is reported in Appendix 1 for the convenience of the reader. The time-sequence of Language, Logic and Science is shown in Figure 4.

It is thanks to Language that Permanent Collective Memory (PCM) exists. It has recently been discovered that what we call “Language” consists of two levels. The lowest one is the one needed in order to understand a “message” (i.e. a group of words constructed on the basis of appropriate rules). We can call this level “Language-understanding”. The other level is at a much higher degree of intellectual ability. It is the one which is needed in order to elaborate a “message”. Our species is the only one which is able to elaborate “messages”. This recent research work is illustrated in Appendix 4.

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Figure 3

30

THE TIME-SEQUENCE OF LANGUAGE – LOGIC – SCIENCE

Figure 4In Figure 5 we refer to Language at its highest level where its various

components become effective. The intellectual achievement, called Language, is due to the fact that

our species is endowed with Reason. The best definition of the activities which build “Language” can be

obtained by realizing that all these activities would exist even if neither Rigorous Logic nor Science had been discovered.

In Figure 6 the main achievements of Rigorous Logic are reported. All these achievements would exist even if Science had never been discovered.

In the following Figures (7, 8, 9) the point to notice is the vital condition which allows the three achievements to exist; i.e. “to be fascinating”

31

for Language, the “non-contradiction” for Logic and the “reproducibility” for Science.

SpokenLanguage

WrittenPCM

(Permanent Collective Memory)

All these activities would exist even if neither Rigorous Logic (Mathematics)

nor Science had been discovered

Figure 5

LOGIC Arithmetic Algebra Analysis Topology

Theory of Theory of Theory of Theory of numbers variables functions domains

(0, 1, 2, 3 …) (x, y, z, …) F (x, y, z, …) where(real numbers) functions

(0, 1) ; (0, 1) ; (0, 1, 2) ; exist.

All these activities would exist even if Science had never been discovered

HistoryPoetryLiteratureArts (Paintings, Sculpture)Music

CinemaPhilosophyEconomyFashion

All other intellectual activities

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Figure 6

Figure 7

Figure 8

33

Figure 9

One should realize that there are three levels of scientific credibility. They are illustrated in Figure 10 and quoted in Figure 3 as S1, 2, 3. In the same Figure 3 I have indicated a place where Complexity seems to show up (in the study of cosmological structures).

The three levels of scientific credibility:

First Level Second Level Third Level

Figure 10

Where there are experiments whose

results can be reproduced in the

laboratory.Example:

Discovery of the Fundamental Laws

Where it is not possible to intervene in order to reproduce a result.

Example:Stellar evolution

A one-off event.Example:

Cosmic evolution

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Let me say a few words on Creativity in Basic Science. Contrary to the other two cases, here Creativity implies the need to overcome the principle of the reproducible experimental proof.

And this means to check our imagination with the being who created the three Basic Structures (three families of elementary particles) and the three Fundamental Forces of the Universe (the electroweak, the strong subnuclear and the gravitational). As Isidor I. Rabi used to say, “he is smarter than all of us”.

Thus Creativity in Basic Science is the most difficult one, when compared to Basic Logic and Basic Language, since Basic Science means the Logic of Nature and this Logic has been chosen by a being who is “smarter than all of us”.

X DOES COMPLEXITY EXIST AT THE FUNDAMENTAL LEVEL?

To answer this question it is necessary to find out if AFB phenomena and UEEC events are present in the way towards the construction of the Standard Model which will be discussed in chapters XII and XIII.

In chapters XIV and XV we will consider the Platonic Structure of the Grand Unification and the Platonic Supersymmetry, respectively.

We will see that Complexity must be at a high level of departure from Platonic Simplicity in order to produce the Present Status of the Standard Model and Beyond.

In Figure 11 we show how the construction of the Standard Model produces a Tree Structure similar to those structures presented by the fellows working in the activities quoted in chapter IV, which are considered of “complex” nature. In Figure 12 we report the detailed sources of the Tree Structure in order to show that there are Sarajevo-type events and deviations from simplicity.

In Figures 13 and 14 there are some further relevant details on QFD, QED and QCD which are the 3 boxes on the bottom of Figure 12.

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Figure 11

36

Figure 12

37

Figure 13

Figure 14

38

XI THE PLATONIC SIMPLICITY FOR SPACE AND TIME IS VIOLATED

In a Platonic World where Space and Time were perfectly symmetric, they should have the same number of dimensions. If we denote with NS the number of Space dimensions and NT the number of Time dimensions, it should be NS = NT . In real life, the number of expanded dimensions is 3 for Space and 1 for Time. This is the first deviation from a Platonic Symmetry of Space and Time. Space and Time depart from Platonic Simplicity: NS NT. This is not all.

The most spectacular example of deviation from Platonic Simplicity come after nearly two centuries of discoveries in electricity, magnetism and optics with the great synthesis of the four Maxwell equations which allowed a deep understanding of the meaning and properties of light. The Maxwell equations had an unexpected and fascinating property: their invariance when submitted to Lorentz transformations. This invariance brought Lorentz to the discovery that Space and Time could not be both real.

The nature of Space and Time is such that these two basic ingredients of our existence cannot be both real. If Space is real, Time must be imaginary and viceversa. This is another deviation from the Platonic Symmetry of Space and Time.

To sum up: the number of Space dimensions is bigger than the number of Time dimensions, which is the minimum possible: one. They cannot be both real. One of them must be imaginary. The world appears to be very different from what could be its Platonic Simplicity.

XII THE STANDARD MODEL AND BEYOND: HOW WE GOT IT

This difference appears even more dramatically when we look at the Standard Model. The superb synthesis called the “Standard Model” is a part of a more general structure, where many problems are open. We call this structure “The Standard Model and Beyond”, “SM&B”. This Structure brings to the unification of all Fundamental Forces of Nature, suggests the existence of the Superworld and produces the need for a non-point-like description of Physics processes (the so-called Relativistic Quantum String Theory: RQST), thus opening the way to

39

quantize gravity. All this is summarised in Figure 15. For clarity we specify in Figure 16 three main problems “Beyond the Standard Model”. The dots indicate the problems whose nature is experimentally checked. The question marks are open problems.

Let us see how we got it.

40

PROBLEMS BEYOND THESTANDARD MODEL

MASS

Unification SU(3)SU(2)SU(1) m ~ 1016 GeV 3 2 1

mW ~ 102

GeV

p ≠ ∞ ?

m ≠ 0 SU(3)SU(2)WU(1)em

SU GUT 3 2 1

MSU GAP MGUT

1018 GeV ? 1016 GeV

Why so many ? 3 FAMILY What fixes the ratio of their masses ?

What fixes the quarks and leptons mixings ? Symmetry breaking Experimentally true

Figure 16

Why so small?

(mW

mpl)

Why not zero?(quark, leptons, EW Gauge Bosons)

HIGGS ?

Hierarchy SUSY ?

GUT

GAP

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XIII A FEW EXAMPLES OF DEVIATIONS FROM PLATONIC SIMPLICITY

We now discuss the greatest synthesis of all times in the study of fundamental phenomena. This synthesis is called the Standard Model. We will see that the basic achievements of the Standard Model could not care less about the existence of Platonic Simplicity. Platonic Simplicity is violated at every corner in the process of construction of the Standard Model.

We will devote our analysis to this point. The conclusion is that Complexity exists at the elementary level. In fact, starting from Platonic Simplicity, the Standard Model needs a series of “ad hoc” inputs. These inputs are the proof that at the elementary level there is experimental evidence for the existence of the two mechanisms: the AFB phenomena and the UEEC events.

A few cases where I have been directly involved are summarised in Figure 17.

The 3rd lepton, HL (now called ) with its own neutrino, HL (now called ),despite the abundance of neutrinos: e and .

Antimatterdespite S-matrix and C, P, CP, T breakings.

Nucleon Time-like EM structuredespite S-matrix.

No quarks in violent (pp) collisionsdespite scaling.

Meson mixingsV PS : (51º) (10º) 0 despite SU(3)uds .

Effective energy: the Gribov QCD-lightdespite QCD-confinement.

The running of 1 2 3 versus energy:the EGM effect, the GAP between EGUT and ESU, and the absence of the Platonic straight line convergence.

Figure 17

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Point 1. The divergences in Electro-weak Interactions and the Third Lepton, an example of deviation from simplicity, have been illustrated in Figure 13 (reported below for the convenience of the reader).

Note that for the Electroweak (EW) force, Nature has not chosen the simplest way out SU(2), but SU(2) U(1).

Figure 13

Point 2. The incredible series of events which originated with the problem of understanding the stability of matter is shown in Figures 18 and 19, together with the unexpected violation of the Symmetry Operators (C, P, T, CP) and the discovery of Matter-Antimatter Symmetry.

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The first deviation from simplicity started with the discovery by Einstein that mass and energy could transform each other. Since the dawn of civilization, it was believed that the stability of matter was due to mass: the heavier an object, the better for its stability. The Greeks were convinced that the stability of the columns for their splendid temples was due to the “heaviness” of the columns. When (1905) Einstein discovered that mc2 = E he could not sleep at night (Peter G. Bergmann testimony). If mass can transform into energy the world becomes unstable and we could not exist. Fortunately (for Einstein) in 1897 J.J. Thomson had discovered the “electron”. Since its charge is opposite to that of the proton the electric charge is enough to guarantee the stability of matter. In fact, charge conservation forbids the transition,

p e (1)

Einstein stopped to be worried and could relax; the world is stable, despite E = mc2. The only detail is that “m” must be understood as being the “mass” of an object, not its matter. Thus the basic distinction shown on top of Figure 18.

A totally unexpected event occurred a quarter of a century later, when (1930) Dirac realized that the evolution of the electron (the same particle which is in the atoms of our body) in Space-Time (with the property discovered by Lorentz in his investigation of the invariance of the Maxwell equations, i.e. they cannot be both “real”, one of them – either Space or Time – must be “imaginary”) brought the incredible conclusion that the antielectron must exist; i.e. a particle with the same charge as that of the proton. If this is the case, then the reaction

p e + (2)

can take place and the stability of matter is again gone. The world becomes unstable. Thus Stueckelberg invented another deviation from simplicity: the quantum number “baryonic charge” which (according to Stueckelberg) is conserved in nature. The reaction (2) is forbidden by this new conservation law. Life would have been by far simpler if the Greeks were right in their way of

44

explaining the stability of matter. Despite the complications described above, the story of the stability of matter does not end with equation (2).

Another deviation from simplicity was needed. This came with understanding that the word “charge” correspond to two fundamental physical properties. There are in fact two types of “charge”.

One is called “gauge charge”, and this is responsible for the existence of a Fundamental Force of Nature. There are in fact three “gauge charges”, the “electromagnetic”, the Subnuclear “weak” and the Subnuclear “strong”. The correspondent forces are described by the Gauge Symmetry Groups U(1) (for the electromagnetic forces), SU(2) (for the Subnuclear weak) and SU(3) (for the Subnuclear strong).

There is a further deviation from simplicity, the two Gauge Symmetry Groups U(1) and SU (2) are mixed and it is their mixing which produces the effective forces which allow our TV and radio plus all electromagnetic instruments to work and the Stars to function, thanks to the effective weak force which controls very well their level of burning. In fact, it is the strength of the weak forces (now called Fermi forces) which controls the amount of protons which transforms into neutrons (plus a neutrino and a positive electron) every second in a Star, thus allowing the “nuclear fuel” (the neutrons) for the Stars to exist. All we have said refers to the “gauge charges”, which are responsible for the existence of the fundamental forces.

But the stability of matter has to be there and, contrary to what Einstein thought in 1905, is not guaranteed by the existence of the electric charge: this is in fact a “gauge charge”.

In order to have the stability of matter, we need a totally different type of charge, called “flavour charge”. The number of these charges is 12, as illustrated in Figure 18. The incredible story is that they are four times more than the “gauge charges”; the reason why this is so, nobody knows at present. The answer could come from the Superworld.

The conclusion of the long way to understand the origin of the stability of matter is that once again, many Sarajevo-type events occurred and many deviations from simplicity, as reported in Figure 19.

45

Mass Matter

mi Mass Antimass m̄ i

i 1 (Intrinsic); i 2 (Confinement); i 3 (Binding)C m i = m

ii i 1, 2, 3

mi Qj Matter Antimatter mi

Q̄j

Qj Flavour Charges

j ( u d c s t b ) (1, 2, 3, 4, 5, 6)

( e e HL HL) (7, 8, 9, 10, 11, 12)

C miQj = miQ̄

j

i 1, 2, 3 ; J 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

Figure 18

From the Greeks who associated “stability” of matter with “heaviness” to our present understanding, the number of Sarajevo-type events is really impressive.

There are in fact seven decades of developments which started from the antielectron and C-invariance and brought us to the discovery of nuclear antimatter and to the unification of all gauge forces with all deviations from simplicity.

These steps are reported in Figure 19, which looks as complex and full of deviations from simplicity as a page of History (EWRL), despite being a page of Science (EBUS).

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THE INCREDIBLE STORY TO DISENTANGLE THE ORIGIN OF THE STABILITY OF MATTER

SEVEN DECADES FROM THE ANTIELECTRON TO ANTIMATTER AND THE UNIFICATION OF ALL GAUGE FORCES

• The validity of C invariance from 1927 to 1957.After the discovery by Thomson in 1897 of the first example of an elementary particle, the

Electron, it took the genius of Dirac to theoretically discover the Antielectron thirty years after Thomson.1927 Dirac equation [2]; the existence of the antielectron is, soon after, theoretically

predicted. Only a few years were needed, after Dirac’s theoretical discovery, to experimentally confirm (Anderson, Blackett and Occhialini [3]) the existence of the Dirac antielectron.

1930-1957 Discovery of the C operator [(charge conjugation) H. Weyl and P.A.M. Dirac [4]]; discovery of the P Symmetry Operator [E.P. Wigner, G.C. Wick and A.S. Wightman [5, 6]]; discovery of the T operator (time reversal) [E.P. Wigner, J. Schwinger and J.S. Bell [7, 8, 9, 10]]; discovery of the CPT Symmetry Operator from RQFT (1955-57) [11].

1927-1957 Validity of C invariance: e+ [3]; p̄ [12]; n̄ [13]; K20 3 [14] but see LOY [15].

• The new era starts: C ; P ; CP (*) .1956 Lee & Yang P ; C . 1957 Before the experimental discovery of P & C Lee, Oehme, Yang (LOY)

[15] point out that the existence of the second neutral K-meson, K20 3 , is proof

neither of C invariance nor of CP invariance. Flavour antiflavour mixing does not imply CP invariance.

1957 C.S. Wu et al. P ; C [17]; CP ok [18].1964 K2

0 2 KL : CP [19].

1947-1967 QED divergences & Landau poles.1950-1970 The crisis of RQFT & the triumph of S-matrix theory (i.e. the negation of RQFT).1965 Nuclear antimatter is (experimentally) discovered [20]. See also [21].1968 The discovery [22] at SLAC of Scaling (free quarks inside a nucleon at very high

q2) but in violent (pp) collisions no free quarks at the ISR are experimentally found [23]. Theorists consider Scaling as being evidence for RQFT not to be able to describe the Physics of Strong Interactions. The only exception is G. ’t Hooft who discovers in 1971 that the -function has negative sign for non-Abelian theories [24].

1971-1973 ; ‘t Hooft; Politzer; Gross & Wilczek. The discovery of non-Abelian gauge theories. Asymptotic freedom in the interaction between quarks and gluons [24].

1974 All gauge couplings run with q2 but they do not converge towards a unique point.

1979 A.P. & A.Z. point out that the new degree of freedom due to SUSY allows the three couplings , to converge towards a unique point [25].

1980 QCD has a “hidden” side: the multitude of final states for each pair of interacting particles: (ee ; p p̄ ; p; Kp; p; pp; etc. )The introduction of the Effective Energy allows to discover the Universality properties [26] in the multihadronic final states.

1992 All gauge couplings converge towards a unique point at the gauge unification energy: EGU 1016 GeV with GU 1/24 [27, 28] .

1994 The Gap [29] between EGU & the String Unification Energy: ESU EPlanck . 1995 CPT loses its foundations at the Planck scale (T.D. Lee) [30]. 1995-1999 No CPT theorem from M-theory (B. Greene) [31].1995-2000 A.Z. points out the need for new experiments to establish if matter-antimatter

symmetry or asymmetry are at work.(*) The symbol stands for “Symmetry Breakdown”.

Figure 19

47

Let us skip the Points 3 and 4 and move to 5. Point 5. The mixing in the pseudoscalar and in the vector mesons.

Figure 20

In the Physics of Mesons the totally unexpected result was the difference existing between the two mesonic mixing angles, pseudoscalar and vector: PS V. They should both be zero if SU(3)uds was a good Symmetry. The existence of Instantons was not known. They came after the discovery that PS V.

48

Point 6. Newton discovered that Light is the sum of different colours: QED. In QCD we have quarks and gluons interacting and producing Jets made of many pions pp + X as shown in Figure 21.

Figure 21

This is what Gribov defined: the QCD light. The Effective Energy shown in Figure 22 is at its origins, despite being totally unexpected.

49

Figure 22

The non-Abelian nature of the Interaction describing quarks, gluons and the Effective Energy has already been illustrated, with the deviations from simplicity, in Figure 13.

50

Figure 23

51

Point 7. The Unification of all Forces and the Supersymmetry threshold with its problems are reported in Figures 23 and 24 respectively.

Figure 24

During more than ten years (from 1979 to 1991), no one had realized that the energy threshold for the existence of the Superworld was strongly dependent on the “running” of the masses. This is now called: the EGM effect (from the initials of Evolution of Gaugino Masses).

To compute the energy threshold using only the “running” of the gauge couplings (1, 2, 3) corresponds to neglecting nearly three orders of magnitude in the energy threshold for the discovery of the first particle (the lightest) of the Superworld [32], as illustrated in Figure 24.

52

Let me now illustrate in Figure 25 the Platonic Grand Unification in the Convergence of the three Gauge Couplings 1 2 3 . The “Platonic” Simplicity would indicate the series of points making up the straight line as the ideally simple solution. The real solution is the sequence of points which totally deviate from the straight line.

This is just an example of comparison between the “Platonic” Simplicity and the “real world”, when we deal with the Grand Unification.

The points have a sequence of 100 GeV in energy. The last point where the “ideal” platonic straight line intercepts the theoretical prediction is at the energy of the Grand Unification. This corresponds to EGU = 1016.2 GeV. Other detailed information on the theoretical inputs: the number of fermionic families, NF , is 3; the number of Higgs particles, NH , is 2. The input values of the gauge couplings at the Z0-mass is 3 (MZ) = 0.118 0.008; the other input is the ratio of weak and electromagnetic couplings also measured at the Z0-mass value: sin2 W (MZ) = 0.2334 0.0008.

Figure 25

53

Talking about Supersymmetry, there is another important step: how we go from pure Theoretical Speculations to Phenomenology. This is not an easy task. The proof is given in Figure 26 where it is shown how many important properties of the physics to be described have been neglected by some authors (AdBF) whose claim was to “predict” the energy scale at which Supersymmetry is broken. In order to attempt to give such a prediction, there are at least five “details” to be taken into account, as reported in the last five columns of Figure 26.

Authors Input data Errors EC MSUSY CC UC TL MX TH EGMACPZ

[32, 5055]

WA 2 all possiblesolutions (24)

Yes physical Yes Yes Yes Yes Yes

Authors Input data Errors EC MSUSY CC UC TL

MX TH EGM

AdBF[56]

only oneexperimen

t

1 only onesolution

Yes Geometrical No No No No No

WA World Average Errors Uncertainty taken from all data (World Average) or from a single experiment EC Evolution of Couplings MSUSY Mass Scale assumed to represent the Supersymmetry Scale breaking CC Convergence of Couplings UC Unification of Couplings above EGUT TL Low Energy threshold MX Mass Scale at the breaking of the Grand Unified Theory to the

SU(3)xSU(2)xU(1) TH High Energy threshold EGM Evolution of Gaugino Masses

Figure 26

XIV THE PLATONIC GRAND UNIFICATION

54

Let us now move towards the Platonic structure of a Grand Unification, taking as basic points the Gauge Principle and the SSB (Spontaneous Symmetry Breaking) mechanism, which represent the conceptual structure of the Standard Model.

The simplest way is to have one and only one basic fundamental particle, B.

This particle must obey the very simple symmetry law which puts fermions and bosons on the same basis.

This basic fundamental particle can therefore exist either as being a boson BB or as being a fermion BF .

Let us consider first BB . The Fundamental Forces exist because a Basic Fundamental Boson BB exists.

Figure 27 illustrates the simple sequence which generates all known forces of Nature.

55

Figure 27

56

At the bottom of Figure 27 there is the QFD force, illustrated in Figure 28. The “Platonic” Simplicity suffers a further deviation.

Figure 28

In fact, we need to introduce many complications. The quarks and the leptons are “mixed”. This mixing is indicated by the index m, while the indices “u” and “d” refer to the two types of flavours (up-type) and (down-type) which are present in each of the three families: 1, 2, 3. There is a further complication. The two mixings for the “up” and the “down” flavours must be different.

In the case of the quark, this mixing is experimentally measured. In the case of the leptons, the experimental results are with nearly half

a century of delay, compared with the quark case. Mixing and violation of Symmetry Laws (for charge conjugation, C,

parity, P, and the product of the two, CP) are well-established in the quark case.In the leptonic sector, only future experiments will tell us if the same

Symmetry Laws are violated.

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There is no known reason why all these details – mixing of states and Symmetry Law violations – are needed. They have been experimentally discovered and show how many deviations from the simple “Platonic” structure are needed. So far we have developed the sequence of Platonic Deviations from Simplicity, starting from the basic fundamental boson BB .

We now show in Figure 29 the deviations needed from the Platonic Simplicity, when we start from the basic fundamental fermion BF . It has to be with “quark” and “lepton” flavours and have two flavours in each class (called Family). Total number of flavours, 12: 6 for quarks, 6 for leptons.

Why so many? The answer will probably come from the Super Space with 43 dimensions compactified into (3+1).

The quark sector interacts with two forces, QCD and QED, while the lepton sector interacts using only QED. The QFD force comes into play only after all the mixings come in. No one knows why all these deviations from the Platonic Simplicity are needed.

Figure 29

58

The bold symbols, QCD, QED in the column

BqF

indicate that the 6 quark flavours interact via these two forces. In the lower part of the same column, the “mixing” indicates that the quark states are no longer “pure” states. They are “mixed”; only these mixed states

(qmu )1, 2, 3 and (qm

d )1, 2, 3

interact via the QFD forces. The column below

BlF

has the same structure, but the “mixings” are not the same as in the “quark” column.

Furthermore, no one knows at present if the Symmetry CP is violated as it is in the quark case. This is why in the box CP there is a question mark. Another detail needs to be specified.

In the quark case, the CP Symmetry breaking, CP , has been experimentally established not to be via the basic Standard Model mechanism, SSB. A further deviation from simplicity.

In the leptonic case, we do not know if the CP Symmetry is violated. It could be it is. In this case it will be interesting to know if it follows the SSB mechanism.

All these question marks are evidence of further deviations from the simple Platonic descriptions of natural phenomena.

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A synthesis of the Platonic Grand Unification and the deviations needed is shown in Figure 30.

Figure 30

60

XV THE PLATONIC SUPERSYMMETRY

The Platonic Concept of Supersymmetry is schematically reported in Figure 31, where the basic point for a Platonic Concept of Supersymmetry is given; i.e. the only fermions with spin 1/2 allowed to exist would be the “gauginos”.

THE PLATONIC CONCEPT OF SUPERSYMMETRY

The Gauge Principle should generate aGauge Force Gauge Bosons

If NATURE was platonically SUPERSYMMETRICSupersymmetry Transformation should generate Gauginos

1st DEVIATION FROM PLATONIC SIMPLICITYOUR FERMIONS ARE NOT THE GAUGINOS

2nd DEVIATION FROM PLATONIC SIMPLICITYTHE FUNDAMENTAL FERMIONS ARE OF TWO DIFFERENT CLASSES: LEPTONS AND QUARKS

3rd DEVIATION FROM PLATONIC SIMPLICITYTHERE IS NOT ONLY ONE BUT THREE FAMILIES

OF FUNDAMENTAL FERMIONS

4th DEVIATION FROM PLATONIC SIMPLICITYTHE FUNDAMENTAL FERMIONS BECOME MIXED WHEN THE

WEAK FORCES ARE SWITCHED ON: MIXINGS EXIST

5th DEVIATION FROM PLATONIC SIMPLICITYTHERE ARE DIFFERENT MIXINGS

Figure 31

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If the only allowed fermions would be the “gauginos”, there would be no way to have quarks and leptons.

Our fermions are not the gauginos. A deviation is needed. And this is the first one. Our fermions are in fact of two classes: quarks and leptons. Another deviation is needed to introduce quarks and leptons. And this is not enough: one Family would not suffice. We need another deviation, the third one, in order to produce Three

Families.Once again this is not enough. We need a further deviation: the fundamental fermions became mixed

when the weak forces are switched on. This fourth deviation is followed by another one, the fifth: the mixing

of states in the quark sector and in the leptonic sector are different.

XVI THE WORLD COULD NOT CARE LESS ABOUT THE EXISTENCE OF THE SUPERWORLD: THE MOST RECENT EXAMPLE OF ANDERSON-FEYNMAN-BEETHOVEN-TYPE PHENOMENON

A few words on how is the world made. A flower, the sea, the air we breathe, the Moon, the Stars, the Sun itself, whatever we call World is made of Fermions and Bosons.

Fermions: (with “half-integer” spin like the electron). These particles – called quarks and leptons – are the “bricks”, more precisely, the tiny “tops”, of our material existence.

Bosons: (with “integer” spin like the photon). Particles having this “spin” value are the “glues” of the Fundamental Forces acting between “bricks”. The “glues” are also tiny “tops”.

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THE THREE COLUMNS

Figure 32

The upper panel of Figure 32 shows the three families of quarks and leptons, together with the respective values of their electric charge, Qe.

The lower panel shows the spaces of the so-called “subnuclear flavour” charges (with additively conserved quantum numbers).

Notice that the third family lepton was originally (1967) named HL

by the BCF (Bologna-CERN-Frascati) Group.

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Its partner HL, the existence of which was postulated in 1967 in the experimental proposal by the BCF group to search for heavy leptons at Frascati, would have been directly observed only recently.

THE THREE FUNDAMENTAL FORCES

Figure 33

The upper panel of Figure 33 shows a standard Lorentz Space-Time to  illustrate how gravitational Forces originate from the fact that, in any point (i x⃗ , t) of such a Space, a change of reference system is possible without altering any physical result.

64

In the lower panel the origin of the other forces, the [SU(2) U(1)] electroweak one and the [SU(3)] strong one, is sketched.

These forces derive from the possibility to operate in fictitious Spaces with one [U(1)], two [SU(2)] and three [SU(3)] complex dimensions, under the condition of invariance of the physics results. Why do columns have half-integer spin and why do forces have integer spin? Can columns with integer spin and forces with half-integer spin exist?

OUR WORLD HAS FOUR SPACE-TIME DIMENSIONS

yThree Space Dimensions

xOne Time Dimension z

DB Bosonic Dimensions DF Fermionic Dimensions DE Expanded Dimensions DC Compact Dimensions

Super Space

THE SUPER SPACE HAS 43 DIMENSIONS

DEB º (3+1 )º ( x⃗ ; it )

DCF º 32 ; DC

B º (9+1 )

DEB º (3+1 )

65

Figure 34Where could be the proof that we come from the Superworld?

The ashes of the Superworld (the so-called neutralinos) could explain the compactness of our Galaxy.

Figure 35

Neutralinos cannot aggregate into Stars since, being neutral, they lose little energy. This would allow neutralinos to remain in a sphere concentric with our Galactic centre.

Even though they aggregated into Stars, neutralinos could not emit light, like ordinary Stars do.

Fire needs the plasma of protons and electrons. This is why super Stars cannot exist.

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WHY DO WE NEED THE SUPERWORLDThere are fundamental reasons making the Superworld a need.

1) The two energy scales must be kept separate: 1019 GeV (Planck) and 102 GeV (Fermi).

2) The gravitational attraction of light must be prevented from being infinite. Otherwise we could see neither the light from Stars nor our light. The “gravitino” (Supergravity) allows the gravitational attraction of light to be finite.

3) Gravitational attraction is powerful but it cannot be infinite. We would be stuck to the Sun. Space would not exist between Stars and Galaxies. Cosmic expansion would not exist. In order to have a finite gravitational attraction, theories are needed in which the Euclidean concept of point is abandoned. The point is replaced by a string. No more Pointlike Theories but String Theories. These theories must be supersymmetric: the already quoted Supersymmetry Law (F B) must be valid in their mathematical structure. Otherwise “tachions” would appear. This is the origin of Relativistic Quantum String Theory (RQST).

4) If we want the Unification of all fundamental phenomena – the synthesis of which is provided by three “gauge couplings”, 1 2 3, running with the energy (Renormalization Group Equations) – the Supersymmetry Law (F B) must necessarily be introduced.

5) An interesting detail: no Scale-Supergravity is an Infrared solution of RQST. This might allow to understand the extremely small value of the Cosmological Constant.

6) Finally: why Three Columns and Three Forces? The answer to this question should come once we will be able from the compactification of the 43-dimensional Super Space to go to our present world with (3+1) Space-Time dimensions.

7) Problem. Supersymmetry does not show up at our energy scale. Hence the problem arises to compute the energy above which the (F B) Law starts to act. Thanks to the EGM

67

effect, this energy level is 700 times more accessible than thought so far.

XVII THE INCREDIBLE STORY OF THE ISOSPIN

The origin of the isospin and the impressive sequence of UEEC (Sarajevo-type) events, is shown in Figure 36. The global symmetry SU(2) borned in the study of the nuclear interactions ended up in the electroweak unification and in the discovery of QCD whose most important Feynman diagram is reported in Figure 37. This diagram allowed the understanding of the existence of the global flavour symmetries SU(2), SU(3) and the various regularities experimentally discovered but never really understood.

FROM ISOSPIN TO THE SUPERWORLD

(SU(2)) SU(3)uds SUSY

Effective Energy

PS VInstantons

TableTable

QCDEW

Mixing

3rd Lepton

Theoretical Foundations

PhenomenologyUnification and Gap

68

Figure 36THE ORIGIN OF THE ISOSPIN

IS IN THENON-ABELIAN NATURE OF SU(3)C

Figure 37

69

In fact, gluon do not carry any flavour and are coupled among them

with the same coupling, 3, which is present in all vertices where a - quark -

flavour (u d c s t b) emits or absorbs a gluon. The reason why all six quark

flavours have the same 3, coupling is granted by the fact that QCD must be

acting trough gluons. The fact that quark flavours exist has nothing to do with

QCD. The fundamental force is QCD and its quanta are the 8 gluons, coupled

with 3. The world of QCD could have been, there even if the quarks where not

existing the gluon-gluon interaction is at the origin of the illusion that the two

global symmetry groups SU(2) and SU(3) had a fundamental role in subnuclear

physics. This role belongs to QCD, more precisely, to the non-abelian nature of

QCD.

Another non-abelian force is the Fermi force which allows the EW

field quanta (±, 0) to interact each other with the 2 coupling. Once again, the

universality of the weak force is due to their non-abelian nature. The EW force

could be there without any need of lepton and quark flavours. This is not the

case for QED which is the only abelian force known, since photon-photon

interaction is not allowed as a direct vertex. In order to explain why all

fundamental particles share the same 1 (electromagnetic coupling) it is

necessary to put in the same multiplet leptons and quarks. Why leptons and

quarks of different families carry the same 1, needs to be explained via the

missing mechanism. If this mechanism was not there we could not explain why

(u, c, t) and (d s b) should have the same 1. For the leptonic flavour the story

repeats.

The first unexpected event was the discovery of a particle, as heavy as

the proton, but electrically neutral: the neutron. These two particles, the proton

and the neutron, act as a unique entity called the “nucleon”. The nucleon

generate a new force of nature called “strong nuclear force”. This force has a

unique “field quantum”, called -meson.

The mathematics needed to describe the global symmetry of the

nuclear force and of its “quantum” was believed to be the symmetry group

70

SU(2); the number “2” specifies the number of “complex” dimensions of the

“intrinsic” space where the symmetry group SU(2) operates.

In this “intrinsic” space the nucleon is a “fermion” while the -meson

is a “boson”. Since the nucleon is a fermion also in the Lorentz-space and the

is a boson in the Lorentz-space, the conclusion was that a correlation between

the Lorentz-space and the “intrinsic” space must exist: a boson must be iso-

boson and a fermion must be iso-fermion.

And now an unexpected event: the discovery of the existence of

bosons which are iso-fermions (example the K-meson) and of fermions which

are iso-bosons (example the baryon).

This Sarajevo-type event was followed by another totally unexpected

discovery. The nucleon is not the only source of the nuclear force: there are very

many particles carrying the nuclear charge.

The name adopted was baryons, for them all. Furthermore the -meson

is not the only field quantum of the strong nuclear force: there are very many

other field quanta called “meson resonances”.

The unique property of the “nucleon” as the source of the strong

nuclear force and of the -meson as the only quantum of the same force was

totally dismantled by the proliferation of baryonic and mesonic resonances.

This is still nothing if compared with the original explanation of the

small value for the mass of the and of the heaviness of the nucleon.

The reasons why the has the small mass which characterizes the

range of the nuclear forces are the two unexpected discoveries:

1) The existence of a global symmetry property: chirality;

2) The spontaneous symmetry breaking of this global symmetry.

We need to add another important and unexpected discovery.

The existence - as we will see later - of a non-abelian fundamental

force (QCD) acting between the constituents of the -meson (quarks and gluons)

and being generated by the gauge principle which does not destroy

chirality invariance.

71

The final step is a challenge to everybody’s imagination: how the logic of

Nature follows its way. In fact, we will see later that, the origin of the nuclear

forces is QCD, a gauge force whose local symmetry group is SU(3), the same

mathematical structure which has no link whatever with the way the global

(flavour) SU(3) group had been discovered, under the illusion that this was the

correct way to go, with the being there as a final confirmation of this logic.

The field quanta of the basic QCD force which generates the nuclear

forces is the “gluon”; the meson which is mostly made up with “gluons” - now

called ' - is as heavy as the basic building blocks of nuclear physics: protons and

neutrons.

We could not exist if the light meson, , was not there. For the -meson to

exist the steps of unexpected experimental and theoretical discoveries mentioned

above were (and are) needed. These discoveries have apparently no relation with

the problems of the time when the proton and the neutron were discovered.

Let us go back to the discovery of the neutron.

No one had the idea that the atomic nucleus needed a particle as heavy

as the proton but carrying zero electric charge. In fact the nucleus with more

than one proton could not be stable for the very simple reason that two protons

repel each other via the Coulomb forces.

When the neutron was discovered by Chadwick (1932) the problem of

describing the nuclear forces arose. The first attempts produced the collapse of

the nucleus since attraction between two particles ends up with a big crunch;

repulsive forces were needed in addition to the attractive ones. It took nearly

half a century to discover that vector mesons exist, but this was the result of

experimental discoveries. When the isotopic spin was proposed by Heisenberg

the symmetry group SU(2) was imagined to describe the nuclear forces: i.e. the

existence of a unique entity, the nucleon, which can either be electrically

charged (proton) or neutral (neutron).

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The source of the nuclear forces was imagined to be the nucleon and

the quantum of the nuclear field was thought to be a particle of mass between

the very light electron and the very heavy nucleon. This is why it was called

“meson”. The discovery of the pion (1947) by Lattes, Occhialini and Powell,

prompted Enrico Fermi to make the optimistic statement: «We have probably

understood all which is needed to explain the nuclear forces and the

electromagnetic forces».

In fact the electric charge of the electron was the source of the

electromagnetic field whose quantum was the photon; the nucleon, with its

nuclear charge, called “baryonic number”, was the source of the nuclear field,

whose quantum was the -meson.

The problem of explaining the source of the nuclear field became a

central one. The most interesting attempt was formulated by Yang and Mills

who studied the consequences of transforming the “global” symmetry SU(2)

(father of isospin) into a “local” symmetry. This produces a fundamental force

of vector nature but of zero mass. Since the nuclear forces have a small range of

action they much have a “quantum” with mass. How to produce this mass no

one knew.

Another UEEC event was needed and this came with the discovery

that superconductivity can be explained by introducing an imaginary mass in the

electromagnetic Lagrangian, thus producing the new phenomenon called

Spontaneously Symmetry Breaking (SSB) which provides the mass to the field

quantum.

This could finally explain the short range of the nuclear forces; the

difficulty being the lack of quantitative computation since these forces are

characterized by strong couplings.

The UEEC event, instead of giving the final solution to the problem of

explaining the finite range of the nuclear forces via the SSB mechanism, did

produce an enormous effect in a totally different field: that of the weak forces.

Here there was a desperate need of finding a field quantum much heavier than

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the one needed for the nuclear forces. In fact the range of the weak forces was at

least three orders of magnitudes smaller than the nuclear forces.

Thus Salam and Weinberg proposed to use the SSB mechanism in

order to generate the very heavy mass needed for the weak-field quantum. A

further complication was that the “local symmetry group” needed could not be

only SU(2) and in fact Shelly Glashow had already suggested the solution in

terms of SU(2) U(1). A totally unexpected experimental discovery

corroborated the Glashow’s proposal: i.e. the discovery of the “neutral weak”

currents by Lagarrigue and Musset at CERN.

The final step was to prove that the “gauge forces” generated by the

two groups SU(2) U(1) could produce, via the SSB mechanism, a massive

field quantum needed for the weak forces and a massless field quantum needed

for the electromagnetic forces, without problems of infinities. In more exact

words, the theoretical structure of the system had to be “renormalizable”. This is

exactly what ‘t Hooft and Veltman did. The final result was (and is) that the

original isospin group SU(2) thought to be the way out to solve the problem of

understanding the nuclear forces ended up of being of basic importance in

another field of the fundamental interactions: the field of the weak forces.

Another UEEC event was coming.

In the same year (1947) which marked the discover of the so much

wanted -meson, a totally unexpected event occurred in the Blackett Laboratory:

the discovery of totally unexpected particles, called “strange”. During the

following decades an enormous number of baryons and mesons where

discovered, some of them carrying the new “quantum number” called

“strangeness”. The original SU(2) group was clearly insufficient to describe the

new baryonic and mesonic states experimentally discovered.

In order to describe this unexpected novelty, called strangeness, a new

global symmetry group was proposed by Gell-Mann and Ne’eman: SU(3); this

new global symmetry group generated the idea that more fundamental particles

must be responsible for the existence of the multitude of baryons and mesons.

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These particles, considered mathematical entities, were called “quarks” by Gell-

Mann and “aces” by George Zweig. The global symmetry group SU(3) thus

acquired a physics basis with the three quarks (u, d, s):

up u

down dstrange s

The enormous number of baryonic and mesonic resonances gave the

impression that the global SU(3) flavour symmetry group was going to be the

final solution.

But another UEEC event was coming: the unexpected experimental

discovery that the proton is made of “pieces” (called partons by Feynman) but it

cannot be broken. According to Democritus, if an object is made of “pieces” it

must be possible to break it. To explain this paradox, a new force of nature -

now called QCD - was needed. And the symmetry group was - incredible but

true - the same one SU(3). The same but local (not global) and with three

fundamental charges, called subnuclear “colours”, which have nothing to do

with the three “flavour” charges quoted above: u, d, s.

These new “colour” charges are carried by the field quanta of the

QCD force which are 8 , not 1, as it is the case for QED.

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XVIII WILL COMPLEXITY EXIST IN THE FUTURE TEN CHALLENGES ? [see Appendix 9]

On many occasions I have presented and discussed the Ten Challenges [57] of Modern Science [reported for the convenience of the reader in Appendix 8]; they are summarised in Figure 38. The point to call attention to for future purposes is if Complexity will show up in the Future of these Ten Challenges.

THE TEN CHALLENGES1) The Physics of Imaginary masses: SSB2) The Physics of Direct Symmetry Breaking DSB; P , C , CP , T

: CPT , and Matter-Antimatter Symmetry3) Supersymmetry SUSY4) Non-perturbative QCD and the Physics of deconfined colour charges5) Anomalies and Instantons6) Flavour mixing in the quark sector7) Flavour mixing in the leptonic sector8) The problem of the missing mass in the Universe9) The problem of the hierarchy10) The Physics at the Planck Scale, the gap and the number of expanded

dimensions

Figure 38A remark on the challenge No. 10. To study the existence of the

microscopic expanded dimension is the experimental challenge which could shed new light on the meaning of the Planck scale, as illustrated in Figure 39.

Figure 39

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XIX THREE CONCLUSIONS AND THE FINAL ONE

Conclusion – 1We do not know what will be the final outcome of String Theory. What we do know is that all these attempts to understand the Logic of

Nature appear to be very “simple” when compared with the “Complexity” of the world we are part of.

We have shown in Figure 3 [reported here for the convenience of the reader as Figure 40] the various components of our knowledge which make up the apparently very different domains of our world.

Figure 40

The conceptual understanding of the constituents of the Universe after 400 years of Reductionism are shown in Figure 41. The key point is that we have reached this extremely “simple” understanding of the fundamental natural phenomena via an incredible series of ABF and UEEC (Sarajevo-type) events. Furthermore it should be pointed out that the apparently simple structure of the hypothetical goal, the Superworld, is full of problems to be solved and, once again, it has been obtained with a remarkable set of AFB and UEEC events.

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SUPERWORLD

F B

THREE COLUMNS AND

THREE FUNDAMENTAL FORCES

“Subnuclear Colour” Charges s ; t ; m ; E ; ; Q (generating the Fundamental Forces).

Space Time Mass Energy Spin Charges “Subnuclear Flavour” Charges

(responsible for the stability of Matter).

REDUCTIONISM

Figure 41

Conclusion – 2Present trend: from Reductionism to Holism. Is Holism the way to

understand the origin of Complexity? Is Big Numbers the way to understand the origin of Complexity?If this is so, what is the role of the Number of Elements? And what is

the Big Number needed?Big Numbers come from the ratio of Space, Time, Mass-Energy and

Action, as given below:

Space SUniverse

Planck Length@ 1062

Time t p

Planck Time=1033⋅p ´ 107 sec

10−43 sec=

=1040sec10−43 sec

@ 1083

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Mass-Energy MUniverse

M ν@ 1080GeV⋅102

10−4 eV=

=1091eV10−4 eV

= 1095

Action hUniverse

hPlanck= 10123

With these Big Numbers compete those coming from the Mathematical Model of Brain. According to some models, in order to produce new ideas the number of possible combinations is much larger than 10123 .

Conclusion – 3

Three points need to be put in evidence:

i) so far, no new fundamental forces have been discovered when our

Simplicity (the Standard Model and its extensions) has been expanded

to include the Complexity of the Universe;

ii) the Simplicity of the Standard Model needs many deviations which

are like the two basic ingredients of Complexity: the AFB

phenomenology and the UEEC events. This means that Complexity

exists at the fundamental level;

iii) if we try to formulate the simplest version of our rigorous logic to

describe Nature – the Platonic Simplicity – the result is that nothing in

the real world looks like it, not even the Standard Model.

FINAL CONCLUSION

We have shown that the experimental foundations for the existence of

Complexity are found in the Galilean study of the Logic of Nature. It is

therefore not correct to claim that holism is the right way to go in order to

understand Complexity.

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Therefore if we want to discover the origin of Complexity we must go on continuing the study of the Logic of Nature following the Galileian method. In fact during the ten thousand years of civilization, the world over, our ancestors did believe that holism was the right way to understand the world. But, up to the moment when Galileo Galilei had the right idea of disentangling a single problem out of the many others to be understood, no Fundamental Laws of Nature had ever been discovered.

In these past four centuries, after Galileo Galilei, the amount of knowledge we have been able to get on the structure of Matter and of the Universe has over passed all previous knowledge obtained by mankind since the dawn of civilization. This has to be ascribed to the credit of the reductionistic approach. Reductionism has allowed us to discover the fundamental roots of Complexity and this is the way to go.

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APPENDIX 1LANGUAGE, LOGIC AND SCIENCE

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APPENDIX 2TOTALLY UNEXPECTED DISCOVERIES: A PERSONAL EXPERIENCE

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APPENDIX 2TOTALLY UNEXPECTED DISCOVERIES: A PERSONAL EXPERIENCE

ANTONINO ZICHICHI

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APPENDIX 3ELEMENTS OF SCIENTIFIC RIGOR IN THE THEORY OF EVOLUTION

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APPENDIX 3ELEMENTS OF SCIENTIFIC RIGOR IN THE THEORY OF EVOLUTION

1 – PremiseThis Appendix is a coherent synthesis of my attempt to encourage our

colleagues in the biological Sciences to introduce the Galilean rules in their research work concerning evolution.

2 – The Three Levels of Scientific Credibility

The scope of this work is to lay out a rigorous, Galilean-type scientific foundation for the Biological Evolutionism of the Human Species (BEHS). Following Galileo Galilei we know that three levels of scientific credibility exist. Let me elaborate on the three levels, since the understanding of these levels is closely related to the scientific rigor that is needed in the description of the Biological Evolutionism of the Human Species, a field where Complexity is considered to be a dominant property.

The first is that which entails: (1) mathematical rigor as a fundamental referent in the formulation of a problem, (2) the invention of an instrument capable of carrying out the key experiment for giving an answer to the problem, and (3) the reproducibility of the result obtained. The reproducible result is one of the foundation blocks of Galilean Science. It is obvious that the result also must be expressed in mathematically rigorous terms, and it is this that permits the elaboration of a theory able to describe not only the reproducible result that is obtained thanks to the invention of the original instrument, but also to point out further experiments to be conducted with new instruments in order to put the new theoretical formulation to the scrutiny of further experimental tests. An example of present day frontier of Physics: the Superworld. We think that a description of the phenomena known so far requires a Space-Time with 43 dimensions: 11 bosonic and 32 fermionic. The elaboration of the mathematical

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structure that describes this reality has arrived at the conclusion that new particles must exist; we have dedicated the last decade to the search for these particles without being able to obtain any reproducible experimental proof.

The Superworld theory is an example in which there is mathematical rigor in the formulation of the problem but there is no reproducible experimental proof. Therefore it could be that the Superworld theory is not part of the Logic of Nature. This is what the years to come will tell. The Superworld is an example of first-level Galilean Science to the extent that the experimental tests are susceptible to direct control: in case of doubt it is possible to intervene by repeating the experiments and by inventing new instruments that allow us to overcome doubts that may arise in the course of data analysis for a particular experiment. An experiment that we are able to keep totally under control, here on Earth.

The second level of scientific credibility is that in which it is not possible to keep the experimental test under control. There is mathematical rigor in the formulation of the problem and there is the invention of new instruments for observing the effects searched for, but there is no direct intervention. An example: the theory of stellar evolution. In one part of the sky, we observe the birth of a Star. In another part, the shining of another Star born for some time. In yet another part, the death of yet another Star. Different observations of many Stars being born, of others that are living and still others that are collapsing, allow the elaboration of a theory of stellar evolution. There is mathematical rigor. Reproducibility is guaranteed by the observation of different examples of Stars as they are being born, during their lifetime and as they are dying. What is missing, however, is the possibility of direct intervention. In cases of doubt we cannot turn off or turn on a Star. We cannot change the characteristics of a particular star in order to scrutinize, through experimental tests, a finding that could be born from the theory of stellar evolution’s mathematical elaboration itself. This theory is strongly linked to the first-level Galilean Science. Example: in the theory of stellar evolution no astrophysicist could have imagined the existence of neutron stars. It was first

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necessary to discover neutrons here on Earth by conducting Galilean-type experiments at the first level of scientific credibility. It was the discovery of the neutron that permitted the elaboration of mathematical models that led to the theoretical hypothesis of the existence of neutron Stars. Quite recently, the observation of certain stellar phenomena has been interpreted as indicating the possible existence of “quark Stars”. The existence of this new class of particles, the quarks themselves, however, was discovered here on Earth by conducting Galilean-type experiments at the first level of scientific credibility. This is the link that should exist between the second and the first level.

Moving on to the third. This level of scientific credibility refers to phenomena that occur only one time. At first glance it could seem that the third level contradicts the notion of “experimental reproducibility”. This is not so. The third level does not in fact leave the first level out of consideration. An example of a phenomenon that happens only one time is that which is described by cosmic evolution.

The Cosmos has the Physics of pre-Big Bang as its initial phase. Then comes the Big Bang with Time intervals that range from billionths of billionths of billionths of billionths of billionths of a second (54 10-45 sec: Planck’s Time) to the Time needed for cosmic evolution with the energy of the vacuum (Alan Guth’s Time: 10-34 sec) to the evolutionary period in which – other than gravitational force – enter into play the Three Fundamental Forces (strong subnuclear, weak subnuclear and electromagnetic) of the so-called Standard Model with its three building blocks of fundamental particles, each of which is composed of two “quarks” and two “leptons”. The Time intervals in play for this phase of cosmic evolution are tenths of billionths of a second. And so one arrives at the few seconds necessary for making the Cosmos with the particles familiar to us (protons, neutrons and electrons) and finally the plasma of these particles in the sea of “photons” that lasts a few hundreds of thousands of years (according to the most recent data, the Time interval is 380 thousand years). At this point the Cosmos, made essentially of protons, electrons and photons, passes into the phase in which the Stars and the Galaxies are born.

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According to the most recent theories, it could be “Black Holes” (made with the very primitive form of matter which existed much before the one of the “Standard Model” particles) that act as nuclei for the formation of galactic structures in which stars are born. The duration of this phase of cosmic evolution is millions of years.

After 15 billion years we reach the present with ourselves, the Sun, the Earth, the Moon, the oceans, the mountains, the sunrises and sunsets, the Cathedrals, Michelangelo’s Pietà and the incredible detail that in this cosmic evolution there is, in addition to the inert matter, also the living matter, both vegetable and animal. Among the countless forms of living matter there is one and only one that is endowed with Reason. It is in fact thanks to Reason that it has been possible to discover Permanent Collective Memory, rigorous Logic and Science.

3 – The Evolution of the Universe: an example of the Third Level

Cosmic evolution is Galilean Science to the extent that it is formulated in rigorous mathematical terms and linked to the first level. From the pre-Big Bang on, everything is based on that which has been discovered at the first level. It is not possible to prove experimentally the reproducibility of cosmic evolution.

No one knows how to make a Big Bang to verify the details that we would like to put under experimental testing. We can only conduct experiments to understand what happens as we come close to the Big Bang. Today we have arrived at a tenth of a billionth of a second (10-10 sec). Keeping in mind that Planck’s Time lasts 54 10-45 sec, it is wise not to forget that a good 34 orders of ten separate us from the instant before inflationary expansion bursts forth. These 34 powers of ten are the measure of our ignorance in the rigorous knowledge of that which we call the “theory of cosmic evolution”.

This theory helps us to understand just how difficult the study of phenomena belonging to the third level of Galilean scientificcredibility is.

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4 – The Evolution in terms of Galilean Rigor and Experimental Reproducibility

To this third level, we repeat, belong all the phenomena that happen only one time, as it is the Biological Evolutionism of the Human Species.

Our species being the only form of living matter endowed with Reason, it is well to put the “theory of Biological Evolutionism of the Human Species” under the Galilean-type rigor.

There are those who say that this “theory” represents the frontier of Galilean Science. We would like this to be true. To accomplish this, however, it is necessary to establish for this theory a foundation in mathematical rigor and experimental reproducibility. Doing this requires an analysis that is attentive to the phenomenon called “evolutionism”.

Evolution exists at the level of elementary particles, at the level of aggregates made up of inert matter, and at the level of aggregates of living matter.

First of all, a clarification. While being studied, the phenomenon called “evolution” can reveal itself only in “Space-Time”. The first rigorous study of evolution at the level of elementary particles concerns electrons. It is not by chance that the electron itself is the first example of an “elementary particle” (discovered by Thomson in 1897).

Dirac, fascinated by the discovery of Lorentz that Space-Time could not be a real quantity but instead a complex one (if Space is real, Time must be imaginary, and vice versa), decided to study with rigor the evolution of the electron in Time and Space. This was how he discovered his equation.

The rigorous study of evolutionism at the level of elementary particles brought Dirac to discover a reality that no philosopher, no poet, no thinker of any epoch or civilization was able to imagine. This reality begins with antiparticles and brings us to the discovery of antimatter, antistars and antigalaxies to arrive at our world, which seems to be made up only of matter, stars and galaxies, without any antistars or antigalaxies. An experiment to be conducted in the year 2008 in the International Space Station will tell us if it is really true that in the course of cosmic evolution every trace of antimatter was

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broken down in order to build up a Universe, like the one in which we are living, that consists only of matter. If in our laboratories we had discovered that antimatter could not exist, the problem of a Universe made only of matter would not exist.

This is not so. The existence of antimatter was confirmed in a rigorously Galilean manner in 1965. Nevertheless, in the Universe there is probably no more antimatter.

It is possible to formulate in a mathematically rigorous mode the theory of cosmic evolution that cancels out antimatter at a certain point.

According to this theory of cosmic evolution, we are here thanks to the fact that, in the process of “cancellation”, a tiny fraction (one part in 10 thousand million (1010)) of matter prevailed over antimatter.

No one could say if this theory is that which corresponds to the cosmic reality of which we are a minimal part. The only certainty is that this theory will be scrutinized closely via Galilean-type experimental tests in the years to come.

Starting from the evolution of an elementary particle we have arrived at the problems of cosmic evolution.

This means that we have passed from typical structures of the subnuclear world (10-17 cm) to galactic structures that reach to the confines of the Universe (1029 cm); better still, if the inflationary evolutionism of Alan Guth is true, to even greater cosmic distances.

The theory of evolution in the study of inert matter, from the heart of a proton to the confines of the Cosmos, enables one to interlink within a single structure everything that happens in zones of space that are differentiated by at least 46 powers of ten. We have done this using the three levels of Galilean scientific credibility.

This is the most rigorous knowledge we have when dealing with the concept of the evolution of the fundamental structure of inert matter. Let us call this level number 1.

The Table below describes the details of this level.

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Table 1: EVOLUTION AND SCIENCE

LEVEL NUMBER ONE

I EVOLUTION IN THE FUNDAMENTAL STRUCTURE OF INERT MATTER:

I-1 Evolution in Space-Time of the lightest electrically charged lepton: the Dirac equation.

I-2 Evolution in the description of the elementary processes involving inert matter: the Feynman diagrams and the problem of Renormalization (i.e. no divergent results in theoretical calculations).

I-3 Evolution in the Universe and in its structure.

I-3-1 The Physics of the Pre-Big-Bang.

I-3-2 The Physics of the Big-Bang.

I-3-3 The basic structure of matter and of the Fundamental Forces in the Evolution of the Universe: from the Planck Scale to present day.

I-3-4 The origin of Galaxies and their distribution in Space-Time.

I-3-5 The origin of a Star and its evolution (Gravitational, Electroweak and Strong Forces).

I-3-6 The origin of condensed forms of cold matter (Planets, Asteroids, Comets and others).

The level number 2 refers to the evolution of the macroscopic structure of inert matter. This and the other levels are schematically given in the Table 2 below.

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Table 2: EVOLUTION AND SCIENCE

LEVEL NUMBER TWO

II EVOLUTION IN THE MACROSCOPIC STRUCTURE OF INERT MATTER.

II-1 The crystals.

II-2 Other forms of conglomerate matter and the understanding of their properties.

THE OTHER LEVELS

III THE TRANSITION FROM INERT MATTER TO LIVING MATTER.

IV EVOLUTION IN THE ENORMOUS VARIETY OF “NON-ANIMAL” LIVING MATTER.

V THE TRANSITION FROM “NON-ANIMAL” TO “ANIMAL” FORMS OF LIVING MATTER.

VI THE EVOLUTION IN THE ENORMOUS VARIETY OF “ANIMAL” FORMS OF LIVING MATTER.

VII THE TRANSITION FROM THE INNUMERABLE POSSIBILITIES OF NON-REASONING LIVING FORMS OF MATTER TO THAT OF LIVING MATTER WITH “REASON”.

VIII THE EVOLUTION OF THE SPECIFIC FORM OF LIVING MATTER CALLED “THE HUMAN SPECIES”.

IX THE DISCOVERY OF COLLECTIVE MEMORY, i.e. WRITTEN LANGUAGE.

X THE DISCOVERY OF LOGIC AND OF ITS MOST RIGOROUS FORM: MATHEMATICS.

XI THE DISCOVERY OF SCIENCE: THE LOGIC OF NATURE.

XII REFLECTIONS ON HOW IT HAPPENS THAT WE ARE THE ONLY FORM OF LIVING MATTER WITH “REASON”.

All these levels need to be fully understood before we reach the level where we need to think about how we happen to be the only form of living matter with “Reason” (level XII).

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In fact, the extraordinary characteristic of the world in which we live is that the Hardware is the same for all forms of matter: from the most elementary inert element (the electron) to the most advanced form of matter with Life and Reason (the Human Species).

The Table below (Table 3) illustrates the five points that represent the Hardware.

Table 3: THIS HARDWARE (i.e. OUR OWN) OBEYS THE FOLLOWING LOGIC

RGEs (Grand Unification).

Gauge Principle (hidden & expanded dimensions).

The Physics of Imaginary Masses: SSB.

Flavour Mixings & CP , T .

Anomalies & Instantons.

From the structure of a Proton (1017 cm)to the

extreme borders of the Universe (1029 cm).

Atoms, Molecules, Inert and Living Matter.

More detailed information on the Hardware is given in Table 4.

Table 4: DETAILED INFORMATION ON THE HARDWARE

RGEs (i (i 1, 2, 3); mj (j q, l, G, H)) : ƒ (k2).

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GUT (GUT 1/24) & GAP (1016 1018) GeV. SUSY (to stabilize mF/mP 1017). RQST (to quantize Gravity).

Gauge Principle (hidden and expanded dimensions).— How a Fundamental Force is generated: SU(3); SU(2); U(1) and

Gravity.

The Physics of Imaginary Masses: SSB.— The Imaginary Mass in SU(2) U(1) produces masses ( mW

; mZ0 ;

mq; ml ), including m = 0. — The Imaginary Mass in SU(5)SU(3)SU(2)U(1) or in any higher

Symmetry Group (not containing U(1)) SU(3) SU(2) U(1) produces Monopoles.

— The Imaginary Mass in SU(3)c generates Confinement.

Flavour Mixings & CP , T .— No need for it but it is there.

Anomalies & Instantons.— Basic Features of all Non-Abelian Forces.

Note: q quark and squark; mF Fermi mass

scale;l lepton and slepton; mP

Planck mass scale;G Gauge boson and Gaugino;k

quadrimomentum;H Higgs and Shiggs; C Charge

Conjugation;RGEs Renormalization Group Equations; P Parity;GUT Grand Unified Theory; T Time Reversal;SUSY Supersymmetry; Breakdown of Symmetry Operators.RQST Relativistic Quantum String Theory;SSB Spontaneous Symmetry Breaking.

The five basic steps in our understanding of nature. The renormalization group equations (RGEs) imply that the gauge couplings (i) and the masses (mj) all run with k2. It is this running which allows GUT, suggests SUSY and produces the need for a non point-like description (RQST) of physics processes, thus opening the way to quantize gravity. All forces originate in the same way: the gauge principle. Imaginary masses play a central role in describing nature. The mass-eigenstates are mixed when the Fermi forces come in. The Abelian force QED has lost its role of being the guide for all fundamental forces. The non-Abelian gauge forces dominate and have features which are not present in QED.

Since the Hardware is the same, the following remarks are in order.

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It could very well have been that the basic Hardware was there, but not Life itself.

It could have been that the basic Hardware and Life were there, but no Consciousness (free will).

It could have also been that the basic Hardware plus Life plus Consciousness were there, but no Reason.

These points are illustrated in Table 5. It happens that Reason is there with its three great achievements:

Language, Rigorous Logic and Science as reported in Table 6.

Table 5.

THE BASIC HARDWARE IS THESAME FOR ALL FORMS OF MATTER

Basic Hardwarebut no

Life

Basic Hardware and Lifebut no

Consciousness (free will)

Basic Hardware plus Life and Consciousnessbut noReason

Table 6.

REASON

LANGUAGE:Written Language

LaW

Permanent Collective Memory

RIGOROUS LOGIC Lo Mathematics

SCIENCE S1, 2, 3 The Logic of Nature

5 – Conclusion

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When we speak about evolution we should not forget the basic constituents of Galilean Science: mathematical rigor and experimental reproducibility. The Biological Evolutionism of the Human Species (BEHS) is below the third level of Galilean Science, as can be deduced when we compare this form of evolution with the evolution of the Universe. The Figure below is a synthesis of all I have said regarding the rigorous description of the concept called “evolution”. A full explanation of this Table is in Appendix 1. I have decided to show it here again in order to give an idea of how complex it is to describe “evolution” when we want to include all we think we know of the world where we live.

We would like to encourage our colleagues engaged in the study of biological evolution to follow our suggestions in order to reach the goal of bringing the Biological Evolutionism of the Human Species to the third level of Galilean Science, like cosmic evolution.(*) A. Zichichi “Language, Logic and Science” proceedings of the 26th Session of the

International Seminar on Nuclear War and Planetary Emergencies, Erice 18-26 August 2002 (World Scientific, 2003). Appendix 1 of the present Volume.

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APPENDIX 4A RECENT DISCOVERY ON THE TWO LEVELS OF LANGUAGE

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APPENDIX 4A RECENT DISCOVERY ON THE TWO LEVELS OF LANGUAGE

While unable to talk, dogs may have the requisites for understanding what is said to them. The proof was made public by a group of researchers, directed by Julia Fischer, from the Max Planck Institute of Lipsia. In a televised presentation, a Border Collie named Rico was shown to possess an exceptional ability to retrieve specified toys. The vocabulary employed was just over 200 words. This is the first step - the association of a word with an object.

The second consists in trying to understand how well Rico used what he had learned in the acquisition of new words. The technique is called “fast-mapping” and is used by children to associate new words with new objects. The experiment with Rico, carried out by German scholars of Anthropological Evolutionism, involved adding a new toy to the set already known to the dog and using a word that Rico had never heard before. The dog almost always chose the new toy, successfully associating the new word with the new object. This experiment proves that the “fast-mapping” mechanism is not a prerogative that is exclusive to our own species.

Another experiment consisted in finding out whether or not there were visual links between the words Rico received from his owner and the objects he had to choose. To do this, the German scholars decided to put ten toys in one room and Rico with his owner in another. If the words of the owner did have visual links to the objects being selected, Rico should not have known what to do because the toys were far away from the owner. In the distant room there were 10 toys. The owner told Rico to choose two. The experiment was repeated 20 times. Rico retrieved 37 toys corresponding to the correct names. Three toys were chosen mistakenly.

And now the proof of “fast-mapping” in the absence of “visible” links. Seven toys plus one new one were placed in the room. The owner used a new word to tell Rico to bring him that toy, the name of which Rico had never heard

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before; nor had ever seen. The experiment was repeated ten times. Rico made three mistakes.

And so we come to the most difficult experiment: putting Rico’s memory to the test. To verify the extent to which Rico was able to remember the link between the new word and the new toy, four weeks were allowed to pass by. To distract him, the new object was grouped with four known toys and four toys he had never seen. The experiment was repeated six times. Rico erred on three. This level of error (that of 50%) might seem high, but in fact corresponds to what a three-year old child is capable of.

Perhaps Rico is a very intelligent dog with a strong desire to learn, says Fischer. It should also be noted that dogs have been close to the human species for thousands of years and it is possible that in the course of their evolution they have inherited a heightened capacity for responding to verbal messages. There are, however, other forms of living matter that demonstrate the ability to understand, even without these roots. Another property of the canine species is that of having been selected by man to work with him, and it is possible that its brain has been subject to evolutionary processes that render this species especially sensitive to verbally-expressed messages.

Julia Fischer thinks that this discovery could help us come to understand why Language did not evolve in living species other than our own. The stakes at play are enormous. If a living species is capable of “understanding” a message, why is it blocked like this, content to “understand” without evolving towards the elaboration of messages?

The passage from “understanding” to “speaking” requires a special organization of the base-elements (neurons) of our cerebral machine. This special organization must be capable of producing a series of voluntary controls over the different parts of the body needed to produce the “messages”: vocal cords and associated structures.

The controls needed for “speaking” cannot be much different from those needed for moving the right pieces. Primates (the species of mammals to which we belong) have a formidable control over these parts, but it is not

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“gesticulative”: which is to say that they do not use them to express “messages”. Our “gesticulations” are linked to our “speaking”. Primates neither gesticulate nor speak, even though they have perfect voluntary control of the necessary mechanisms. This brings us to the conclusion that there must be an enormous leap of quality in the cerebral structures that enable the elaboration of “messages”. Why did this “leap” in quality not happen in any living species other than our own?

These results make us reflect on the uphill steepness of the evolutionary road that leads first to spoken and then to written Language, from which Permanent Collective Memory is then born. And then Rigorous Logic and finally, Science. No other form of living matter, despite millions of years of biological evolution, has arrived at the first stage: spoken Language. This experiment is the best response to those who believe they have understood everything about BEHS (Biological Evolution of the Human Species).

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APPENDIX 5THE 62 POWERS OF TEN ARE NOT SCIENCE FICTION

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APPENDIX 5THE 62 POWERS OF TEN ARE NOT SCIENCE FICTION

The 62 powers of Ten reported in chapter IV are not Science Fiction. Example: the LISA (Laser Interferometer Space Antenna) will measure distances between Free-Floating Gold Blocks held in Space by carefully controlled Electrostatic Fields in order to detect disturbances in Space caused by a passing gravitational wave produced at time TG = 1033 sec after the Big Bang.

We are now at 20 109 years after the Big Bang (BB); i.e. at  6  1017 sec, therefore

T (Now )

T G@ 6 ´ 1050

Gravitational Background Radiation (GBR) produced at TG = 1033 sec after the BB is like the Microwave Background Radiation (MBR) produced at the time of decoupling between Matter and EM-Radiation, when the Universe was 300 thousand years old;

TEM 3 105 years after the Big Bangi.e. TEM 1013 sec after the Big Bang.

The impressive effects on the Uniformity of the MBR, at the level of 105, refer to time distances of just 5 powers of Ten, in fact

T (Now )

T EM=¿

20 ´ 109 years3 ´ 105 years

@105¿ .

The two Background Radiations, one, Gravitational, occurred at TG

1033 sec, while the other, Electromagnetic, occurred at TEM 1013 sec; these two events represent the imprinting of the Cosmological Evolution over time scales which are in the Ratio

T EM

T G @1046

.

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Thus, when we speak about the Universe, we are dealing with Ratios of Time-Scales in the range approaching the extreme limits of 62 powers of Ten.

A few words on the structure of LISA: Three Spacecraft flying 5  106 km apart in an equilateral triangle formation:

The Gravitational Wave produces an effect of “squashing” and another of “stretching”, at 90° each other, as illustrated below:

The order of magnitude of each effect is in the range of few parts per ten thousand of billion of billion:

DXL

=10−9 mm5 ´106 km

=2 ´10−22

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APPENDIX 6WE CAME FROM A BLACK-HOLE AND

WE ARE STILL IN A BLACK-HOLE

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APPENDIX 6WE CAME FROM A BLACK-HOLE AND

WE ARE STILL IN A BLACK-HOLE

In the Table below we show the values of the matter density in our world today and in the world we came from, i.e. the Planck Universe.

Every The Worldday’s weWorld come from

Human Body Density Planck Density

1gr/cm3 1093 gr/cm3

Our World The Planck Universecm 1033 cmgr 105 grsec 1044 sec

Planck discovered [1] what we now call “the Planck Universe” because he wanted to know what would be the units of Length, Mass, Time and Temperature which retain their significance for all environments, terrestrial and human, or otherwise, thus being independent of special bodies or substances such as it is the case for our units of Length (centimetre), Mass (gram), Time (second) and Temperature (degree Celsius or Kelvin).

In his universal outlook of the world - independent of our restricted environment - Planck wanted that the fundamental units of Mass, Length and

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Time must depend only on the values of the Fundamental Constants of Nature: the speed of light c, the constant of action h, and the gravitation constant G.

It should be pointed out that the Boltzman’s constant k (which represents the “quantum” of entropy, or the minimum amount of “chaos”) is a quantity which converts the units of Energy into units of Temperature. This allowed Planck to have a fundamental value also for the Temperature. Here are Planck’s units:

Length = (G h /c3)1/2 = 4.13 1033 cmTime = (G h /c5)1/2 = 1.38 1043 secMass = (h c /G)1/2 = 5.56 105 grTemperature = K1 (hc5/G)1/2 = 3.5 1032 Kelvin

It is remarkable the way Planck considered these quantities: «In the new system of measurement each of the four preceding constants of Nature (G, h, c, K) has the value one». This is the meaning of measuring Lengths, Times, Masses and Temperatures in Planck’s units.

These quantities had a special meaning for Planck [1]: «These quantities retain their natural significance as long as the Law of Gravitation and that of the propagation of light in a vacuum and the two principles of thermodynamics remain valid; they therefore must be found always to be the same, when measured by the most widely differing intelligence according to the most widely differing methods».

When Planck was expressing his ideas on the meaning of his fundamental natural units there was neither the Big-Bang nor the cosmic evolution. In fact the very instant of the cosmic expansion is the Planck-Time and the corresponding density of the Universe is the Planck density.

In our world the density is dictated by the fact that we are made by atoms and therefore the basic quantities are the mass of the nucleon (mN 1024 gr) and the radius of the atom (108 cm) which is dictated by the electric charge and the mass of the electron.

Let us call the value of this “density” the “atomic density”:

ρatomic @M nucleon

Ratom3 @10−24 gr

(10−8 cm )3=10−24 gr

10−24 cm3 =1 grcm3

.

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The density of water is typical of the atomic density. In the above formula we neglect details like (4/3 ) in front of Ratom

3 to estimate the “atomic

volume”.When we go from water to lead the “atomic density” increases by an

order of magnitude. This is due to the increase in the mass of the nucleus by two orders of

magnitude

mPbnucleus

102 Mnucleon

and a correspondent increase by an order of magnitude in the atomic volume.The next possible density - many orders of magnitudes higher - is the

“nuclear” density: 1015 times greater than the “atomic” density. The reason being the value of the nuclear radius, which is of the order

of one Fermi-unit (1013 cm), i.e. five orders of magnitude smaller than the atomic radius.

Atomic and nuclear bindings are specific for a given Element of the Mendeleev Table and do not change when the amount of the given Element changes.

For both forms of matter, atomic and nuclear, the density does not change when the amount of matter increases: one ton of lead has the same density as one kilogram of lead.

For matter where the binding force is gravitational (without any other forces being involved), the density decreases when the amount of mass increases.

More precisely the density decreases with the square of the mass. This is the great discovery of Schwarzschild and is coming from the relation which exists between the Radius of a Black-Hole,

RBH ,

and the mass of the same Black-Hole,

MBH .

Here is the Schwarzschild formula:

RBH=2G M BH

c2 =K M BH @ 1. 5 ´ 10−28⋅cm⋅gr−1⋅M BH

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with G being the Gravitational Constant, c the speed of light and

K =

2Gc2

@ 1.5 ´ 10−28⋅cm⋅gr−1 .

When the mass is the value of the Planck unit given in the previous Table. i.e.

MPlanck = 2.2 105 gr,

the correspondent Radius of the Black-Hole is

RBHPlanck @1 .5 ´ 10−28 ´ 2. 2´ 10−5 ´ cm@ 3. 3 ´ 10−33 cm .

The Black-Hole Radius increases linearly with its mass value, as shown in Figure 1.

The density is given by the mass over the volume

ρBH=

M BH

V BH¿

M BH

(K⋅M BH )3

The result is that the Black-Hole density decreases with the square of the Black-Hole mass

ρBH=K−3¿M BH

−2

The remarkable fact is however that the extensions of the world we leave in is now (1028 cm) and – as shown in Figure 2 – its density satisfies the same relation of the world we came from, whose radius was 1033 cm and its density

ρPlanck≅54

×1093 gr × cm−3 .

In other words the Planck density and radius satisfy the Black-Hole condition exactly as the present day density of the Universe and its dimensions satisfy the Black-Hole conditions: we came from a Black-Hole and we are still in a Black-Hole.

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Figure 1

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Figure 2: The Figure shows the relation which exists between the value of the Black-Hole radius (RBH) and the corresponding density (BH), from the Planck scale to the Universe scale now.

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APPENDIX 4 - REFERENCES

[1] M. Planck, “Über Irreversible Strathlungsvorgänge”, S.B. Preuss. Akad. Wiss. 5,

440-480 (1899).

The problem of the Fundamental Units of Nature was also presented by M. Planck

in a series of lectures he delivered in Berlin (1906) and published as “Theorie der

Wärmestrahlung”, Barth, Leipzig, 1906, the English translation is “The Theory of

Heat Radiation”, (trans. M. Masius) Dover, New York (1959).

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APPENDIX 7THE WORLD, THINKING AND LANGUAGE:

FROM PLATO TO THE VIENNA CIRCLE TO THE PRESENT

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APPENDIX 7THE WORLD, THINKING AND LANGUAGE:

FROM PLATO TO THE VIENNA CIRCLE TO THE PRESENT

For over two thousand years, man has been studying the relationship that must exist between Reality, Ideas and Language. The Reality we live in and form part of, the Ideas that emerge from this reality and can span far beyond it, and finally, the Language we use to describe it all.

The most ancient school of thought devoted to this question was the Academy of Plato in Athens, the fulcrum of Greek Philosophy. The 20th century counterpart of this school was born with a group of physicists, mathematicians and philosophers in Vienna. This group met every Thursday evening for most of the twenties and thirties in a little room of the University of Vienna Mathematics Institute. This group was baptized the Vienna Circle in 1931, and its purpose was to study the relationship between scientific theory and objective reality. The Vienna Circle set out to scrutinize the greatest invention of all since Plato: the discovery of Science. Despite a widespread interest in evolutionism, Darwinism was not considered because it was not actually very scientific. Plato’s conception was of an objective reality, and Language served for the rigorous description of natural phenomena as objects of close study. This was the realm of scientific theories, among which Darwinism as a discipline lacking scientific elaboration could not be included.

The founder of the Vienna Circle was Hans Hahn (1879-1934), Professor of Mathematics at the University of Vienna and teacher of Kurt Gödel (1906-1978).

Because the Academy of Plato predated the discovery of Science, Language was the only tool it had. The Vienna Circle, however, incorporated Science into its debate on the relationships between Language, the World and Science. The discovery of Science had already had enormous consequences for our actual and potential perceptual abilities (with the advent of countless

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inventions that surpass our five senses) and our capacity to articulate these perceptions.

The Vienna Circle ended up reaching a conclusion in line with Galilean thinking, according to which the only meaningful assertions are those that can be confirmed by a specified method or algorithm.

This “verifiability principle” lies at the foundation of “logical positivism”, a label that would come to be identified with the Circle of Vienna philosophy. Not to be mistaken as extraneous, this detail in fact concerns one of the most fundamental debates in the philosophy of Science: is there a difference between the truthfulness of an assertion and its demonstrability? In other words, is it possible to demonstrate every proposition that is true? Demonstrability requires the existence of precise rules that can lead us to a conclusion of truthfulness in the final analysis. But can we be certain that such rules exist for all possible truths? Until 1931, all mathematicians would have readily agreed that such rules must exist for every possible truth. In 1931, however, Kurt Gödel demonstrated that it is not always possible to couple a truth with its demonstration. By doing this, Gödel destroyed what had until then been a universal convention – that there was no difference between truthfulness and its demonstration.

Gödel discovered that there is an unbridgeable gap between that which is true (within a logical system) and that which we can effectively demonstrate using the logical tools of that same system.

It is as if we imagined a collection of all possible recipes for producing Sicilian cannolis; this “bible of recipes”, however, would be forever incomplete. There would always exist at least one Sicilian cannoli that everyone could recognize as genuine and authentic, but for which no one would ever be able to write the exact recipe.

Gödel’s famous Theorem demonstrates limitations in the concept of demonstration itself. Our brief account presents Gödel’s finding - the greatest Logical conquest of all times - at the outset because of how it was born from the

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heart of the Vienna School; Gödel’s participation and the discussion topics themselves provided the intellectual nourishment for his discovery.

The principal objective of the Vienna Circle was to unify all domains of knowledge through the vehicle of Physics. This “reductionist” conception met some success in domains like Chemistry and Biology, and it was this same premise that inspired Otto Neurath (1882-1945) and Rudolph Carnap (1891-1970) to advance the audacious thesis that every proposition must be expressible in the Language of Physics.

Many members of the Vienna Circle were in touch with Ludwig J. Wittgenstein (1889-1951), author of Tractatus Logico-Philosophicus, and his ideas were at the centre of their discussions. Wittgenstein’s work was about the interaction between Language, Logic and observation of the world. According to Wittgenstein, Logic is necessary but not sufficient for describing reality. The role of Logic is to help us understand which realities are theoretically possible and which are not. For example: if the phrase “the velocity of light in a vacuum is constant” is true, then the statement that “the velocity of light in a vacuum is not constant” cannot be true. Logic is necessary, but - and this is the main conclusion of Wittgenstein - Logic is not sufficient. Language cannot express all of reality. Syntax can never completely eliminate semantics.

Shifting from Language to Mathematics, Gödel demonstrated that the operations of formal Logic can never tell us everything about all relationships within the completely abstract world of whole numbers, even though they are the simplest type of numbers imaginable.

These were the ideas that came into debate among the physicists, mathematicians and philosophers of the Vienna School, which helped Kurt Gödel become the greatest logician since Aristotle.

Analysis of the interrelationship of Language, Logic and Science came to the centre of attention from 1931 on as thinkers embraced Gödel’s extraordinary discovery. Tangible limits had been placed on human cognitive capacities for the first time in the history of thought. This contradicted the monumental Treatise by Bertrand Arthur William Russell and Alfred North

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Whitehead, the Principia Mathematica: three volumes that eliminated “language” by using sequences of symbols to express all of the assertions of Classical Mathematics. To demonstrate that 1 + 1 = 2, for instance, the exclusive use of logical symbols required hundreds of pages.

This was the price they thought necessary for meeting the challenge posed in 1928 by David Hilbert, the premier representative of mathematical thinking at the time, during the International Congress of Mathematicians in Bologna.

The fundamental question posed by Hilbert in his address in Bologna, at the most ancient and prestigious University of the world, was to determine whether or not it was possible to demonstrate all mathematical truths. Hilbert himself was firmly convinced that this was an obtainable objective: a system, rigorously logical and formally impeccable, that would axiomatize all Mathematics. The challenge to the mathematical community re-visited the list of 23 problems that this great German mathematician had presented in Paris in 1900 to mark the dawning of a new century(*).

Hilbert’s challenge met the most authoritative response in the work (1910-1913) of Russell and Whitehead, who believed that freeing Mathematical Logic from the use of Language was indispensable.

Hilbert was seeking a sort of “truth machine”, in which one could insert a proposition, push a button and receive an answer: true or false. Meeting the Bologna challenge would have shattered the link between what is actually true and what is logically demonstrable, a relationship that had provided a foundation for all mathematical thinking for centuries.

The nineteenth century had already revealed how conceptions of “points”, “lines” and “geometrical figures” that had been held valid for thousands of years did not, in fact, describe all of geometrical reality.

(*) The German Emperor was the only European Head of State to decree the

beginning of the twentieth century in 1900 instead of 1901, as all other Kings and Emperors did erroneously. It is not a coincidence that Hilbert was the mathematical advisor to the Emperor. As the reader might recall, the same error repeated itself at the beginning of the twenty-first century. (See A.Z. L’irresistibile Fascino del Tempo, il Saggiatore, 2000).

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In the early 1800s, two scholars of antique geometrical concepts, János Bolyai and Nikolaj Lobac̆ eskij, independently discovered the existence of “triangles” for which the sum of the angles can differ from the famous 180 degrees of Euclidean geometry. In “hyperbolic” geometry, for instance, the sum of the angles is less than 180 degrees, whereas in “elliptical” geometry it is greater than 180 Euclidean degrees. In the universe of “points”, “lines” and geometrical shapes, these non-Euclidean truths are just as “true” as those of Euclidean geometry. In Euclidean geometry, if given a straight line and one single point external to it, then there is one and only one straight line that is parallel to the given line. In the non-Euclidean geometries, alternatively, there can be an infinite number of parallel lines, or even no lines at all.

For the greatest exponent of mathematical thought of the times, the non-Euclidean geometries had shifted the issue from truth and demonstrability (the sum of the angles that differed from 180 degrees in non-Euclidean geometries was rigorously demonstrable, therefore true), to the mathematical universe (of which geometry is a part) and the real universe: the world we live in seems to respect Euclidean geometry perfectly. A poorly understood abyss had opened up between the real world and the worlds constructed out of axioms and precise rules, ones we call geometry, arithmetic, algebra, analysis and topology(*). These constructions should leave no room for paradoxes, yet logical paradoxes do exist.

Such paradoxes use the same logical structure as mathematical proofs, and Hilbert was concerned about the logical coherence of the mathematical enterprise. His perspective - “every defined mathematical problem must necessarily be susceptible to an exact solution, either in the form of an actual

(*) Arithmetic is the theory of numbers. Algebra, alternatively, is the theory of

relationships among variables, where a variable can be any number. Analysis is the theory of relationships among variable functions, including complex functions. Geometry is the theory of functions – scalar, vector, tensor, etc. – in a determined metric space. Topology is the theory of relationships among spaces with various properties, with or without various metrics, and with environments. A space that has a metric is, in fact, already constrained/enclosed. A topological space is a collection of points plus a collection of environments. This makes up a rich branch of Mathematical Logic called the proximity theory.

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true answer to the question posed, or through a proof of the impossibility of its solution”.

Hilbert challenged his colleagues to formalize all mathematical truths in a way that would exclude paradoxical assertions, like those that arise in ordinary Language and Logic, from Mathematics.

The paradoxes of the liar (Epimenides) and the barber (Russell), for example, are two assertions that are impossible to define as either true or false. Let’s look at them more closely.

First, the liar’s paradox invented by Epimenides: “I am Cretan, and I am telling you that all Cretans are liars”. This affirmation is neither true nor false. If we hypothesize that it is true and that Epimenides is telling a lie, then it cannot be true that all Cretans are liars - the affirmation turns out to be false. Having hypothesized that it is true, we concluded that it must be false. Now let us hypothesize that it is false: it is not true that all Cretans are liars, therefore Epimenides statement that all Cretans are liars is true. This means that all Cretans are in fact liars, and that the affirmation turns out to be true. We began with the hypothesis that it is false, but arrived at the conclusion that it has to be true

Moving on to the barber’s paradox, invented by Russell: “The village barber shaves all of the village inhabitants that do not shave themselves. Who shaves the barber?” In a similar fashion, if we hypothesize that the barber shaves himself then we must conclude that he does not shave himself, and vice versa.

Turning to Mathematics, we find that non-demonstrable mathematical truths do in fact exist. One famous example is associated with the schooldays of Carl Friedrich Gauss (1777-1855). To punish children who were too boisterous during lessons, the Professor would give them the task of adding up all of the numbers from 1 to 100, certain that this would keep them busy for a while. One student, however, found the result in just a few minutes. That student had divided the numbers from 1 to 100 into two groups - the first from 1 to 50, the

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second from 51 to 100. He arranged the groups into two rows of numbers, as follows,

1100101

299

101

398

101

. .. . .. .

. .. . .. .

¿

5051

101

¿

and recognized that the sum of 101 repeated for each and every pair of numbers. All that remained was to multiply the number 101 by the total number

of numerical pairs, which was 50, to find the answer for the Professor:

50 101 = 5050.

The method invented by Gauss works for any number. If the Professor had chosen any other number, N, instead of 100, what Gauss had done was equivalent to, first, dividing N in half:

N2 .

When N = 100, this gives 50 pairs. Then add 1 to N: when N = 100, this gives 101. To finish, multiply the two results:

N2(N+1 )=100

2 (100+1)=50 ´ 101= 5050

.

The formula works for both even and odd numbers. Let us try an even number for N: 6. We have three pairs,

62=

3,

each of which has the sum 6 + 1 = 7,

1 2 3 ¿6 5 4 ¿

7 7 7

and the formula works once again

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N2(N+1 )=6

2(6+1)=3´ 7=21

.

Now, let us test an odd number: 7. We have the following four couples of numbers (for an odd N we introduce 0 to balance the two groups):

0 1 2 3 ¿7 6 5 4 ¿

7 7 7 7

The formula works perfectly:

N2(N+1 )=(N+1)

2N=( 7+1

2 )7=82

´ 7=4 ´7=28.

The division into two groups, discovered by Gauss, is a demonstration of the following formula: the sum (expressed by the symbol ) of all numbers (indicated by “n”) from 1 to N:

∑1

N

(n )=12

N (N+1)

is equal to the product

N2 times (N+1).

The number N is arbitrary but given. The demonstration is not valid for any positive whole number, but only for a number that is chosen at will; being chosen, it is given. The proof of the general formula

∑1

N

n= N2(N+1 )

.

uses a principle called “mathematical induction”, which starts by verifying that the formula works for the case N = 1.

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∑1

1

n=12(1+1 )=1

.

Then it confirms that it holds for an arbitrary, but known and given, positive whole number. This step is used to logically deduce that it must hold for (N+1). In other words, if it holds for N = 1, then it must hold for N = 2. If it holds for N = 2, it must hold for N = 3, and so on.

This “mathematical induction” technique is used often, even though it is not an instrument of formal logical inference. It enables us to infer the result for an infinite number of cases (all positive whole numbers) from a finite group of conditions (the two cases: N = 1, and N that is arbitrary but given).

This is a fundamental arithmetical formula and all of its proofs employ “inductive” arguments analogous to the one just described.

Some philosophers think that “inferential” principles like “mathematical induction” should not be admissible as demonstration. If we accept this point of view, the fundamental arithmetical formula

∑1

N

n= N2(N+1 )

would not be demonstrable.But it is in fact true, as can be proven using any number N. Non-

demonstrable mathematical truths, therefore, do exist. In his address at the Bologna Congress, Hilbert challenged his

colleagues to formalize all mathematical truths in a way that would exclude non-demonstrable truths as well as the types of paradoxical assertions that can surface in ordinary Language, like the ‘liar’ and ‘barber’ paradoxes invented by Epimenides and Russell.

Three years after Hilbert’s speech at the Bologna Congress, a young ex-physicist, Kurt Gödel, transformed Hilbert’s dream in the discovery of the limits existing for an even more rigorous way of thinking, that of Mathematical Logic, which could not be disputed.

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The young ex-physicist from Vienna was not one of those who spoke of the Principia Mathematica without having even opened any of three volumes. Gödel used the Principia like a runway for launching an incredible supersonic airplane that, once in flight, observed from high above all of that which mathematical thinking believed it had “seen” from the ground.

Gödel was able to demonstrate that even the application of all of the tools of Rigorous Logic and “mathematical induction” together left some mathematical truths that still were not demonstrable. Contrary to what Hilbert had thought, the abyss between what is true and what can be demonstrated is un-closable. The argument had been liberated from the bindings and dangers intrinsic to language by the work of Russell and Whitehead, which had paved the runway by reducing everything to symbols and sequences of symbols

Gödel not only read the Principia, but also codified all the symbols and assertions of the formal logic structure used by Russell and Whitehead, assigning a single number to each assertion and every sequence of Arithmetic assertions that could be expressed through the Principia Mathematica formalism.

In this formalism there are “elementary logical signs” and “logical variables” linked by “signs”. Examples of “elementary logical signs” include the symbol “=”, which means “is equal to”, and the symbol ““, which means “exists”. A number is associated with each “elementary logic sign”. If there are twenty “signs”, the first twenty numbers: 1, 2, ... 20, are used to identify them.

There are three types of “logical variables” that are ordered hierarchically and linked to the precise role of the “variable” within a complex logical expression. The three types of variables are: “numerical”, “enunciative” and “predicative”. The “numerical” ones can only be assigned numerical values. The “enunciative” ones substitute for entire formulas. The “predicative” ones express properties like “less than”, “odd” and “even”.

This is the formidable invention of Gödel: all logical expressions and relationships of demonstrability of the Russell and Whitehead treatise can be

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written by linking these three types of “logical variables” with “elementary logical signs”. Gödel codified “numerical variables” with prime numbers, “enunciative variables” with the squaring of prime numbers, and “predicative variables” with the cubing of prime numbers.

Imagine we have a “formula” that consists of only numerical variables. Within this formula, each “numerical variable” is associated with a prime number, while each “sign” is associated with a number. If the total number of signs is twenty, the numbering of “numerical variables” must employ prime numbers greater than twenty for the simple reason that the numbers from 1 to 20 are already in use for the “elementary logical signs”.

Suppose that our “formula” consists of eight “elementary logical signs” and two “logical variables” of the “numerical” type. Each “logical sign” would be represented by a number from 1 to 20. The two numerical “logical variables” would be referred to by the first two prime numbers greater than twenty, which are

23 and 29.

Therefore our formula would be represented by a sequence of ten numbers that identify the formula without any possibility of confusion. Now, however, it is necessary to transform the sequence of ten numbers into one single number. To do this in a way that avoids identification problems, Gödel employs the first ten prime numbers, which are

2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

To obtain one single number, we simply multiply them together. Our formula is thus translated into the product of these ten prime numbers.

Logical Formula 2 3 5 7 11 13 17 19 23 29.

At this point, however, it is necessary to associate this succession of ten prime numbers with the sequence of ten numbers that identify our formula. To do this, Gödel uses each number of the sequence as an exponent. We indicate it with ni, where “i” assumes ten values, some of which repeat if the formula includes logical signs that repeat.

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Our formula is made up of ten numbers, of which eight (elementary logical signs) can be numbers from 1 and 20, and two (numerical logical variables), are the numbers 23 and 29, mentioned above. Our Logical Formula will correspond to the following number:

Logical Formula 2n1 3n2 5n3 7n4 11n5 13n6 17n7 19n8 23n9

29n10

in which each exponent n1, n2 ... n10 has a numerical value that indicates its role in the numeration of Gödel. The main point is that the Logical Formula produces a number that represents it.

The reader should not despair - we are trying to render comprehensible the greatest discovery of all time in logical thought. This discovery had its starting point in the monumental work of Russell and Whitehead, a work that the famous educator and philosopher John Kemeny described as “a masterpiece discussed by practically all philosophers but read by practically no one”. The numeration of Gödel associates one single, unique number to each affirmation and each sequence of arithmetic affirmations that can be expressed through the formal logic constructed by Russell and Whitehead in their Principia Mathematica. By making each formula correspond to a number, Gödel eliminated the use of common words in the Language of Mathematics once and for all.

“Hilbert's Dream” was to solve all Mathematical problems, and to do this required that all Mathematical problems be described solely with formulas, with no words. The Principia of Russell and Whitehead did precisely this and transformed “Hilbert's Dream” into a rigorous and monumental edifice. Godel then transformed the formulas of this monumental Work into numbers, thereby superceding “Hilbert's Dream” and relegating it to the past.

The realization of Godel's project invites us to ponder how the real

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world extends well beyond our human ideas and fantasies. Being of the world, these can capture a part of the picture but never all of it. On the other hand, we still need an analytical instrument that can help us distinguish the true from the false. This instrument is the Reason that has enabled us to discover three great things: Language, Logic and Science. As described in Appendix 1, it is thanks to Science that we have discovered the seven fundamental components needed to construct the real world as we know it and participate in it. These seven components suffice to describe everything that refers to inert matter, as illustrated in Figure 3 (pages 15 and 54).

Life extends beyond the confines of inert matter. When we introduce life into a description of the world, the first mystery to solve is the transition itself from inert matter to living matter. We do this in the Chapter that follows the next one.

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APPENDIX 8YUKAWA GOLDMINE

FROM HIS PARTICLE TO THE QGCW

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APPENDIX 10CHAOTIC STRINGS

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APPENDIX 10CHAOTIC STRINGS

Couple of years after my various presentations and discussions about Complexity at the elementary level I was very interested to know about the work by Christian Beck [1] who found that there are some scalar fields (also called “chaotic strings”) chaotically evolving in discrete Space and Time which seem to fix the Standard Model parameters as local minima in the vacuum energy landscape. This allows to select out of the many possible vacua, the relevant vacuum thus producing the solution to the problem of uniqueness of vacua in the description of the elementary processes which are at the roots of all phenomena where from the rest of the Universe emerges with its expanded (3+1) dimensions and all other regularities and symmetries we have been discovering during the past century.

The work of C. Beck would imply that “chaotic strings” can provide a theoretical argument why certain Standard Model parameters are realized in Nature and others are not. In fact the “chaotic strings” spectrum seem to reproduce the numerical values of the electroweak and strong couplings with a precision of (4-5) digits, and the (free) masses of the known quarks and leptons with a precision of (3-4) digits.

The W-boson mass comes out correctly and the neutrino-mass predictions are consistent with the present experimental findings. What C. Beck has found is a class 1+1 dimensional strongly self-interacting discrete field theories whose vacuum energy is minimized for string couplings that numerically coincide with running Standard Model couplings [2].

“Chaotic strings” can thus be used to provide theoretical arguments why the Standard Model parameters, including the unexpected flavour mixings, have the value we measure in our Laboratories.

Needless to say, this approach is at its first steps and no-one knows where it will go in the years to come.

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APPENDIX 10 - REFERENCES

[1] C. Beck, Phys. Rev. D. 69, 123515 (2004);

C. Beck, Physica D. 171, 72-106 (2002);

Spatio-Temporal Chaos and Vacuum Fluctuations of Quantized FieldsC. Beck (World Scientific, 2002).

[2] The Evolution of Gaugino Masses and the SUSY ThresholdF. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, Nuovo Cimento 105A, 581 (1992);

The Effective Experimental Constraints on MSUSY and MGUT F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, Nuovo Cimento 104A, 1817 (1991);

The Convergence of the Gauge Couplings at EGUT and above: Consequences for 3(MZ) and SUSY Breaking F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, Nuovo Cimento 105A, 1025 (1992);

The Simultaneous Evolution of Masses and Couplings: Consequences on Supersymmetry Spectra and Thresholds F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, Nuovo Cimento 105A, 1179 (1992);

Analytic Study of the Supersymmetry-Breaking Scale at Two Loops F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, Nuovo Cimento 105A, 1201 (1992);

A Study of the Various Approaches to MGUT and GUT F. Anselmo, L. Cifarelli and A. Zichichi, Nuovo Cimento 105A, 1335 (1992);

A 2-Test to Study the 1, 2, 3 Convergence for High-Precision LEP Data, Having in Mind the SUSY Threshold F. Anselmo, L. Cifarelli and A. Zichichi, Nuovo Cimento 105A, 1357 (1992).

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APPENDIX 11THE STATUS OF REDUCTIONISTIC ACHIEVEMENTS

AT THE TIME OF MAJORANA

211

APPENDIX 11THE STATUS OF REDUCTIONISTIC ACHIEVEMENTS

AT THE TIME OF MAJORANA

212

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REFERENCES

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REFERENCES

[1] Language, Logic and ScienceA. Zichichi, proceedings of the 26th Session of the International Seminar on Nuclear War and Planetary Emergencies, Erice 18-26 August 2002 (World Scientific, 2003).

[2] P.A.M. Dirac, “The Quantum Theory of the Electron”, Proc. Roy. Soc. (London) A117, 610 (1928); “The Quantum Theory of the Electron, Part II”, Proc. Roy. Soc. (London) A118, 351 (1928).

[3] The Positive ElectronC.D. Anderson, Phys. Rev. 43, 491 (1933);

Some Photographs of the Tracks of Penetrating RadiationP.M.S. Blackett and G.P.S. Occhialini, Proc. Roy. Soc. A139, 699 (1933).

[4] H. Weyl, “Gruppentheorie und Quantenmechanik”, 2nd ed., 234 (1931).

[5] E.P. Wigner, “Unitary Representations of the Inhomogeneous Lorentz Group”, Ann. Math., 40, 149 (1939).

[6] G.C. Wick, E.P. Wigner, and A.S. Wightman, “Intrinsic Parity of Elementary Particles”, Phys. Rev. 88, 101 (1952).

[7] E.P. Wigner, “Über die Operation der Zeitumkehr in der Quanten-mechanik”, Gött. Nach. 546-559 (1931). Here for the first time an anti-unitary symmetry appears.

[8] E.P. Wigner, Ann. Math. 40, 149 (1939).

[9] J. Schwinger, Phys. Rev. 82, 914 (1951).

[10] J.S. Bell, “Time Reversal in Field Theory”, Proc. Roy. Soc. (London) A231, 479-495 (1955).

[11] To the best of my knowledge, the CPT theorem was first proved by W. Pauli in his article “Exclusion Principle, Lorentz Group and Reflection of Space-Time and Charge”, in “Niels Bohr and the Development of Physics” [Pergamon Press, London, page 30 (1955)], which in turn is an extension of the work of J. Schwinger [Phys. Rev. 82, 914 (1951); “The Theory of Quantized Fields. II.”, Phys. Rev. 91, 713 (1953); “The Theory of Quantized Fields. III.”, Phys. Rev. 91, 728 (1953); “The Theory of Quantized Fields. VI.”, Phys. Rev. 94, 1362 (1954)] and G. Lüders, “On the Equivalence of Invariance under Time Reversal and under Particle-Anti-particle Conjugation for Relativistic Field Theories” [Dansk. Mat. Fys. Medd. 28, 5 (1954)], which referred to an unpublished remark by B. Zumino. The final contribution to the CPT theorem was given by R. Jost, in “Eine Bemerkung zum CPT Theorem” [Helv. Phys. Acta 30, 409 (1957)], who

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showed that a weaker condition, called “weak local commutativity” was sufficient for the validity of the CPT theorem.

[12] Observation of AntiprotonsO. Chamberlain, E. Segrè, C. Wiegand, and T. Ypsilantis, Physical Review 100, 947 (1955).

[13] Anti-Neutrons Produced from Anti-Protons in Charge Exchange CollisionsB. Cork, G.R. Lambertson, O. Piccioni, W.A. Wenzel, Physical Review 104, 1193 (1957).

[14] Observation of Long-Lived Neutral V ParticlesK. Lande, E.T. Booth, J. Impeduglia, L.M. Lederman, and W. Chinowski, Physical Review 103, 1901 (1956).

[15] Remarks on Possible Noninvariance under Time Reversal and Charge ConjugationT.D. Lee, R. Oehme, and C.N. Yang, Physical Review 106, 340 (1957).

[16] Question of Parity Conservation in Weak InteractionsT.D. Lee and C.N. Yang, Phys. Rev. 104, 254 (1956).

[17] Experimental Test of Parity Conservation in Beta DecayC.S. Wu, E. Ambler, R.W. Hayward, D.D. Hoppes, Phys. Rev. 105, 1413 (1957);

Observation of the Failure of Conservation of Parity and Charge Conjugation in Meson Decays: The Magnetic Moment of the Free MuonR. Garwin, L. Lederman, and M. Weinrich, Phys. Rev. 105, 1415 (1957);

Nuclear Emulsion Evidence for Parity Non-Conservation in the Decay Chain eJ.I. Friedman and V.L. Telegdi, Phys. Rev. 105, 1681 (1957).

[18] On the Conservation Laws for Weak InteractionsL.D. Landau, Zh. Éksp. Teor. Fiz. 32, 405 (1957).

[19] Evidence for the 2 Decay of the K20

MesonJ. Christenson, J.W. Cronin, V.L. Fitch, and R. Turlay, Physical Review Letters 113, 138 (1964).

[20] Experimental Observation of Antideuteron ProductionT. Massam, Th. Muller, B. Righini, M. Schneegans, and A. Zichichi, Nuovo Cimento 39, 10 (1965).

[21] The Discovery of Nuclear AntimatterL. Maiani and R.A. Ricci (eds), Conference Proceedings 53, Italian Physical Society, Bologna, Italy (1995); see also A. Zichichi in “Subnuclear Physics - The first fifty years”, O. Barnabei, P. Pupillo and F. Roversi Monaco (eds), a joint publication by University and Academy of Sciences of Bologna, Italy (1998); World Scientific Series in 20th Century Physics, Vol. 24 (2000).

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[22] The first report on “scaling” was presented by J.I. Friedman at the 14th International Conference on High Energy Physics in Vienna, 28 August-5 September 1968. The report was presented as paper n. 563 but not published in the Conference Proceedings. It was published as a SLAC preprint. The SLAC data on scaling were included in the Panofsky general report to the Conference where he says «... the apparent success of the parametrization of the cross-sections in the variable / q2 in addition to the large cross-section itself is at least indicative that point-like interactions are becoming involved». “Low q2

Electrodynamics, Elastic and Inelastic Electron (and Muon) Scattering”, W.K.H. Panofsky in Proceedings of 14th International Conference on High Energy Physics in Vienna 1968, J. Prentki and J. Steinberger (eds), page 23, published by CERN (1968). The following physicists participated in the inelastic electron scattering experiments: W.B. Atwood, E. Bloom, A. Bodek, M. Breidenbach, G. Buschhorn, R. Cottrell, D. Coward, H. DeStaebler, R. Ditzler, J. Drees, J. Elias, G. Hartmann, C. Jordan, M. Mestayer, G. Miller, L. Mo, H. Piel, J. Poucher, C. Prescott, M. Riordan, L. Rochester, D. Sherden, M. Sogard, S. Stein, D. Trines, and R. Verdier. For additional acknowledgements see J.I. Friedman, H.W. Kendall and R.E. Taylor, “Deep Inelastic Scattering: Acknowledgements”, Les Prix Nobel 1990, (Almqvist and Wiksell, Stockholm/Uppsala 1991), also Rev. Mod. Phys. 63, 629 (1991). For a detailed reconstruction of the events see J.I. Friedman “Deep Inelastic Scattering Evidence for the Reality of Quarks” in “History of Original Ideas and Basic Discoveries in Particle Physics”, H.B. Newman and T. Ypsilantis (eds), Plenum Press, New York and London, 725 (1994).

[23] Quark Search at the ISR T. Massam and A. Zichichi, CERN preprint, June 1968;

Search for Fractionally Charged Particles Produced in Proton-Proton Collisions at the Highest ISR EnergyM. Basile, G. Cara Romeo, L. Cifarelli, P. Giusti, T. Massam, F. Palmonari, G. Valenti and A. Zichichi, Nuovo Cimento 40A, 41 (1977); and

A Search for quarks in the CERN SPS Neutrino Beam M. Basile, G. Cara Romeo, L. Cifarelli, A. Contin, G. D'Alì, P. Giusti, T. Massam, F. Palmonari, G. Sartorelli, G. Valenti and A. Zichichi, Nuovo Cimento 45A, 281 (1978).

[24] A. Zichichi in “Subnuclear Physics - The first fifty years”, O. Barnabei, P. Pupillo and F. Roversi Monaco (eds), a joint publication by University and Academy of Sciences of Bologna, Italy (1998); World Scientific Series in 20th Century Physics, Vol. 24 (2000).

[25] New Developments in Elementary Particle PhysicsA. Zichichi, Rivista del Nuovo Cimento 2, n. 14, 1 (1979). The statement on page 2 of this paper, «Unification of all forces needs first a Supersymmetry. This can be broken later, thus generating the sequence of the various forces of nature as we observe them», was based on a work by A. Petermann and A. Zichichi in which the renormalization group running of the couplings using supersymmetry

217

was studied with the result that the convergence of the three couplings improved. This work was not published, but perhaps known to a few. The statement quoted is the first instance in which it was pointed out that supersymmetry might play an important role in the convergence of the gauge couplings. In fact, the convergence of three straight lines (α 1

−1 α 2−1 α 3

−1) with a change in slope is

guaranteed by the Euclidean geometry, as long as the point where the slope changes is tuned appropriately. What is incorrect about the convergence of the couplings is that, with the initial conditions given by the LEP results, the change in slope needs to be at MSUSY ~ 1 TeV as claimed by some authors not aware in 1991 of what was known in 1979 to A. Petermann and A. Zichichi.

[26] V.N. Gribov, G. ’t Hooft, G. Veneziano and V.F. Weisskopf “The Creation of Quantum ChromoDynamics and the Effective Energy”, L.N. Lipatov (ed), a joint publication by the University and the Academy of Sciences of Bologna, Italy (1998); World Scientific Series in 20th Century Physics, Vol. 25 (2000).

[27] The Effective Experimental Constraints on MSUSY and MGUTF. Anselmo, L. Cifarelli, A. Petermann and A. Zichichi, Nuovo Cimento 104A, 1817 (1991).

[28] The Simultaneous Evolution of Masses and Couplings: Consequences on Supersymmetry Spectra and ThresholdsF. Anselmo, L. Cifarelli, A. Petermann and A. Zichichi, Nuovo Cimento 105A, 1179 (1992).

[29] A Study of the Various Approaches to MGUT and GUT F. Anselmo, L. Cifarelli and A. Zichichi, Nuovo Cimento 105A, 1335 (1992).

[30] Are Matter and Antimatter Symmetric?T.D. Lee, in Proceedings of the “Symposium to celebrate the 30th anniversary of the Discovery of Nuclear Antimatter”, L. Maiani and R.A. Ricci (eds), Conference Proceedings 53, page 1, Italian Physical Society, Bologna, Italy (1995).

[31] String Theory: the Basic IdeasB. Greene, Erice Lectures - Discussion 1999 in “Basics and Highlights in Fundamental Physics”, A. Zichichi (ed), World Scientific (to be published).

[32] The Evolution of Gaugino Masses and the SUSY ThresholdF. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, Nuovo Cimento 105A, 581 (1992).

[33] Search for Supersymmetric Particles using Acoplanar Charged Particle Pairs from Z0 decaysALEPH Collab., D. Decamp et al., Phys. Lett. B236, 86 (1990).

[34] Search for Neutral Higgs Bosons from Supersymmetry in Z decays ALEPH Collab., D. Decamp et al., Phys. Lett. B237, 291 (1990).

[35] Search for Neutralino Production in Z decaysALEPH Collab., D. Decamp et al., Phys. Lett. B244, 541 (1990).

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[36] Search for the Neutral Higgs Bosons of the MSSM and other two Doublet Models ALEPH Collab., D. Decamp et al., Phys. Lett. B265, 475 (1991).

[37] Search for Heavy Charged Scalars in Z0 decays DELPHI Collab., P. Abreu et al., Phys. Lett. B241, 449 (1990).

[38] Search for Pair Production of Neutral Higgs Bosons in Z0 decays DELPHI Collab., P. Abreu et al., Phys. Lett. B245, 276 (1990).

[39] Search for Scalar Quarks in Z0 decays DELPHI Collab., P. Abreu et al., Phys. Lett. B247, 148 (1990).

[40] A Search for Sleptons and Gauginos in Z0 Decays DELPHI Collab., P. Abreu et al., Phys. Lett. B247, 157 (1990).

[41] Mass Limits for Scalar Muons, Scalar Electrons and Winos from e+e Collisions near S**(1/2) = 91GeV L3 Collab., B. Adeva et al., Phys. Lett. B233, 530 (1989).

[42] Search for the Neutral Higgs Bosons of the Minimal Supersymmetric Standard Model from Z0 Decays L3 Collab., B. Adeva et al., Phys. Lett. B251, 311 (1990).

[43] Search for the Charged Higgs Boson in Z0 decay L3 Collab., B. Adeva et al., Phys. Lett. B252, 511 (1990).

[44] A Search for Acoplanar Pairs of Leptons or Jets in Z0 decays: Mass Limits on Supersymmetric Particles OPAL Collab., M.Z. Akrawy et al., Phys. Lett. B240, 261 (1990).

[45] A Search for Technipions and Charged Higgs Bosons at LEP OPAL Collab., M.Z. Akrawy et al., Phys. Lett. B242, 299 (1990).

[46] A Direct Search for Neutralino Production at LEP OPAL Collab., M.Z. Akrawy et al., Phys. Lett. B248, 211 (1990); P.D. Acton et al., preprint CERN-PPE/91-115, 22 July 1991.

[47] Searches for Supersymmetric Particles Produced in Z Boson decay MARK II Collab., T. Barklow et al., Phys. Rev. Lett. 64, 2984 (1990).

[48] Searches for New Particles at LEPM. Davier, LP-HEP 91 Conference, Geneva, CH, Preprint LAL 91-48, December 1991.

[49] A Detailed Comparison of LEP Data with the Predictions of the Minimal Supersymmetric SU(5) GUTJ.R. Ellis, S. Kelley, D.V. Nanopoulos, preprint CERN-TH/6140-91, Nucl. Phys. B373, 55 (1992).

219

[50] The Effective Experimental Constraints on MSUSY and MGUT F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, Nuovo Cimento 104A, 1817 (1991).

[51] The Convergence of the Gauge Couplings at EGUT and above: Consequences for 3(MZ) and SUSY Breaking F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, Nuovo Cimento 105A, 1025 (1992).

[52] The Simultaneous Evolution of Masses and Couplings: Consequences on Supersymmetry Spectra and Thresholds F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, Nuovo Cimento 105A, 1179 (1992).

[53] Analytic Study of the Supersymmetry-Breaking Scale at Two Loops F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, Nuovo Cimento 105A, 1201 (1992).

[54] A Study of the Various Approaches to MGUT and GUT F. Anselmo, L. Cifarelli and A. Zichichi, Nuovo Cimento 105A, 1335 (1992).

[55] A 2-Test to Study the 1, 2, 3 Convergence for High-Precision LEP Data, Having in Mind the SUSY Threshold F. Anselmo, L. Cifarelli and A. Zichichi, Nuovo Cimento 105A, 1357 (1992).

[56] U. Amaldi, W. de Boer and H. Furstenau, Phys. Lett. B260, 447 (1991).

[57] John Bell and the Ten Challenges of Subnuclear PhysicsA. Zichichi, invited paper presented at the International Conference on “Quantum [un]speakables” in Commemoration of John S. Bell, International Erwin Schrödinger Institut (ESI), Universität Wien, 10-14 November 2000.