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Philosophy of Science

Ontological Levels and Symmetry Breaking

Gyorgy Darvas
The Institute for Advanced Symmetry Studies, Budapest, Hungary
h492dar@ella.hu, symmetry@mailhost.net

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ABSTRACT: I discuss the role of symmetry breaking in a philosophical context, and formulate laws of symmetry breaking. I deal with their conceptual and ontological background, limits of validity, their relation to the theories of evolution and reductionism and to level theories. Level theories are used to make a sequential arrangement of the forms of appearance of moving matter. Aspects of symmetry or symmetry breaking have never been involved in the treatment of these theories. Here, I first attempt to bring knowledges of different origins together. There are two types of level theories: a general one (in philosophy) and particular ones (in the inanimate, the organic nature and in the human society). Particular level theories differ from each other in the three fundamental ontological spheres, and in their description and contents . At the same time they may have common features, e.g., all are particular theories concerning their width of validity, and all are based on an arrangement by a common concept, namely the forms of interaction. The clarification of these conceptual problems was necessary to understand the laws of symmetry breaking. The law of correspondence between the ontological levels and their potential symmetry properties is formulated in four constituent statements and two concluding laws are also presented. The new features of this treatment will link level theories with (dis)symmetry principles, and formulate the laws of symmetry breaking.

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Introduction

The roots of symmetry principles are partly in ancient philosophical thinking, partly in ancient mathematics, when science and philosophy, arts and crafts, crafts and science had not yet been separated. Symmetry formed a part of the picture of the known world of the Greek, Indian, and Chinese philosophers (cf., the symbols: Platonic perfect solids, mandala, yin-yang). Most of the ancient philosophers tried to find as perfect a model of the world as could be fitted to their experience of reality. They were convinced that our world must be perfect. The perfection of the world could be treated as an axiom; deviations from this state of perfection were what had to be explained. Their efforts to describe perfection culminated in the search for perfection in forms. Form meant different things in different (later so called) 'disciplines': (a) the perfectness of the statements in logic, (b) the perfectness of the physical appearance in geometry, and (c) the perfectness of works of art. In the Greek culture these different meanings were not separated. Soon after, the first became associated with rationality (science), the third with emotionality, impression (art), while the middle one more or less with both. This explains why this second could maintain its bridging role (proportion, harmony, golden section, etc., i.e., symmetry; cf., the synonymes of symmetry before the Renaissance, Nagy, 1995) between sciences and arts in the later centuries when a separation of these activities had taken place. Their search for the most perfect forms reached its climax in finding the five so-called perfect Platonic solids (not mentioning now the evidently most perfect body: the sphere), namely the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. The perfect Greek 'forms' run an unprecedented career in the role they have played in the sciences (a) and the arts (c) since then (Darvas, 1997).

Nevertheless, nature proved to be not always symmetric. Although basic laws could be formulated by the application of symmetry principles, most new phenomena appeared by certain distortion of symmetry. Therefore, it was symmetry breaking what led scientists to new discoveries. This is why P. Curie stated ''dissymmetry makes the phenomenon''. (Per definitionem something is dissymmetric, if it shows symmetry in its main features, but this symmetry is partially distorted in a single property.)

Symmetry plays several different roles in philosophical thought

In epistemology it plays a heuristic role, since the mind often prefers symmetric solutions of problems from among alternatives. Not only prefers, even seeks for such solutions if there are available any such. Thus, symmetry performs a methodological function in the formulation of scientific knowledge. Many examples can be quoted when great discoverers' minds were led by symmetry principles. (Cf. e.g., Plato's influence on Heisenberg when he applied symmetry groups into particle physics.)

The ontological basis of the importance of (dis)symmetry is that material reality indeed has both symmetry properties and symmetry breakings. It has been less realised that symmetry breaking plays important role in the construction of the material world. There is an order (of symmetry breakings), what can be traced along the evolution. These material properties and order can be reflected in the laws of the nature. These are the laws of symmetry and of symmetry breaking.

Science and mathematical description looked for order and for the linear phenomena for many centuries, because those were so perfect and beautiful. Less attention was paid to chaos, disordered structures and nonlinear phenomena. The recent two decades turned the attention of scientists to the systematic description (and discovery of laws) of the latter.

Similar is the situation with the phenomena of symmetry breaking. We knew and phenomenologically described them. We knew, that they were present in any phenomenon of nature, and most new promising areas for a scientist could be identified by the study of dissymmetric phenomena. However, most works dealing with symmetry itself discussed symmetry and treated the laws of symmetry (e.g., Rosen, 1995; van Fraassen, 1989; de Gortari, 1970). Many new transdisciplinary discoveries were made on the basis of the application of symmetry (cf., the discovery of quasicrystals, fullerenes, in the recent two decades). Dissymmetry, and so symmetry breaking was left for the so called ''puzzle solving'' research (using this term after T. Kuhn, 1963). Making an order in symmetry breaking was a subject only within separated disciplines, like in particle physics and cosmology, as well as in some biological subdisciplines.

Now, we make an attempt to discuss the role of symmetry breaking at a philosophical level. Laws of symmetry breaking in a wide context will be formulated first. We will study their conceptual and ontological background, limits of validity, their relation to the theories of evolution and reductionism and to the so called level theories.

Level theories

We must start the discussion with some preliminary remarks about the level theories, since most of the conceptual problems of the laws of symmetry breaking originate in them.

The conception formed about the structure of matter has in our world view traditionally been based on the forms of motion. Various theories, both philosophical and scientific, tended to carry out several systematizations within the great variety of the forms of appearance of matter and motion. These were in main lines summed up in what are called level theories.

The fundamental concepts used in the formulation of the laws of symmetry breaking are based in the level theories. However, there has never existed a consensus on the use of the concepts of the level theories, including the concept of the level itself. The basic problems occur in two respects: (i) there was no agreement what is meant under a level, or in other words, on what criteria one distinguishes levels; as well as (ii) what are the limits of a given level, and what follows from this: how many qualitatively different levels and consequently different level theories exist?

Level theories used to make a sequential arrangement of the forms of appearance of moving matter. Aspects of symmetry or symmetry breaking have never been involved in the treatment of these theories. In the present paper we make an attempt to bring knowledges of different origin together for the first time; in this course (a) sketch a critical picture of the existing level theories, (b) discuss the role of symmetry-breaking in the constitution of qualitatively higher level material structures, and (c) formulate the laws of the process.

The origins of the discussion

Levels had been mentioned in both philosophical and scientific publications, however, the use of the concept had been equivocal and no one had tried to formulate a clear-cut theory before Vigier and Bohm in the late fifties. In parallel, another discussion developed among philosophers on updating the classification of different forms of motion. The two debates started on separate lines and united, as a matter of course, only later. However, the critique of the level theory theses presented by Vigier gives the key to understand, why the later unified ontological level theory (founded both on the forms of motion and types of interaction) was predestined to fail; viz., because two differently based level theories were mixed in it. These limits led to inconsistencies and were only gradually recognised in the literature. These inconsistencies involve the necessity of the idea of distinguishing separate level theories. Later one could notice in the discussion papers the statement (which had never been expressed explicitly), that there were two kinds of level theories: a general, philosophical one and the other kind in the sphere of particularity. The foundations of the particular level theories have not been elaborated in details. This concerns mainly the 'higher' fundamental levels (biological, social); but there can be proposed, even at the level of inanimate nature, at least two basic approaches: a structural and a genetic (among others). Therefore, many authors enforced a unified treatment in a common framework.

Our critique leads to the conclusion that philosophy, with its total concepts of matter and motion, cannot do more than to identify the three fundamental levels corresponding to the three fundamental forms of motion. The treatment of all further levels belongs to the sphere of particularity and can be discussed (although at a higher stage than the individual sciences) in meta-science frameworks. This meta-science framework can operate with the terminology and methodology of philosophy, but without laying claim to philosophical totality. This provides the opportunity to avoid the analogy with the three fundamental philosophical levels when searching for the differentia specifica of the further, particular levels in the study of the fine structure of the material organisation within an individual fundamental level.

These new foundations made possible to get a better understanding of the laws of the structure of matter. In formulating the laws we base the levels on the sequence of the characteristic types of interactions assigned to any given level. (Justification of this choice, which is one of the crucial points of the theory, will be given later.)

The laws of ontological levels and symmetry breaking

(1) The law on the determining role of the lower levels.

(1a) Among two consecutive (a lower and an upper) levels, the lower level potentially (but only potentially) possesses the characteristic type of interaction of the consecutive upper level; i.e., that the preceding lower level's types of interaction play the determining role in the development and existence of any level's characteristic interaction. However,

(1b) In the interrelation of two different (an upper and a lower) levels, generally the upper level's structure affects actively the other, since

(1c) Any lower level material structure can reflect its environment only on its own (lower) quality and own level. Within that, the material structure of a lower level can reflect the material structures corresponding to the upper level's forms of material motion also only on its own (lower) level.

(The two statements in (1c) are not-certainly equivalent, because the given levels are determined per definitionem by their characteristic interaction and not by the corresponding form of material motion.) For example, any inanimate being can reflect an animal only as a physical object, and cannot reflect its biological properties; no animal can discern the social differences between human beings.

Since the relation of the two, lower and upper, levels are not symmetric, this law does not open the door to any reductionism. A reductionist approach would allow only the following kind of statement, viz., ''among two consecutive levels, the lower level possesses the characteristic type of interaction of the consecutive upper level.'' But, according to our laws, (1a) limits the existence of the upper level's characteristic interaction at the lower levels to potentiality, while (1b) and (1c) together contradict any statement which denies the appearance of new qualities at the upper levels.

(2) The law of correspondence between the ontological levels and their potential symmetry properties.

(2a) Each qualitatively higher organisational form in the evolution of matter is marked by the loss of a certain symmetry property, and

(2b) Each loss of a potential symmetry property of matter traces a new material quality.

Consequently, the precondition of the development (in its relative totality) of a qualitatively new (material) level is the breaking of a certain symmetry (property), and at the same time, the condition of the continuance (existence) of the new level is to possess (new?) conserved properties. Therefore

(2c) Parallel with the appearance of new material qualities and new (higher) ontological levels, there appear also new symmetries.

(2d) These new symmetries qualitatively differ from those what existed at the previous (lower) levels and what have been broken at the given level. These new symmetries involve new conserved properties.

As a conclusion, the lower and higher ontological levels can be traced by a sequence of symmetry breakings, thus it can be formulated, that

(3) Each symmetry breaking leads to a higher organisational level of matter.

(4) Each higher organisational level of matter is- in a certain sense- less stable than the former one.

The latter statement needs some further explanation. This will be given in the detailed treatise by encountering examples for all the above four laws. Let's now mention only the decreasing self-reproducibility of the living organisms along phylogeny, or the decreasing forces keeping together the inanimate structures from the subatomic particles to the large molecules.

Some unsolved conceptual problems

What is an individual level? What are the criteria of speaking about a new, higher level? What is the differentia specifica of a given level? The question whether nothing but the forms of motion may serve as a basis for a systematization of the hierarchy of levels of material structures still remains to be answered (Darvas, 1987). The same applies to the question of what else may underlie an arrangement and which of them, if any, might be given the primacy?

Traditionally the forms of motion were seen as playing the typifying role, at least in philosophy. Based on them were the original level theories as they are called. Some of the questions posed by the criticism of level theories relate to the quantity of levels. (Later we will also return to this question.) Are their ranges broken by qualitative bounds; are there some of them that may play key-role in our world view while others may not, etc.? We also try below to explore how are they closely related to symmetry breaking.

Philosophical level theory may work successfully within certain limits only: it gives a sound description of the three fundamental levels of the material world; namely, inanimate nature, the organic world, and human society and thought. From the aspect of understanding the structure of matter, however, this depth is not satisfactory. Thus we have to find possibilities for its 'working' extension to the specific spheres of material reality, to the understanding of the finer structure of matter- far beyond the proficiency of the philosophy, deep in the competence of the sciences- and to its interpretation in context of the whole world view.

The philosophical level theory sought to attach the three levels to the respective three fundamental forms of motion. The individual fundamental forms of motion can be found with the corresponding fundamental interactions. With the genesis of the three fundamental forms of motion an unambiguous evolutionary line became clearly marked, along which any new fundamental form of motion would appear as a new quality compared to the former. Their being built upon each other, along with the appearance of the new qualities, takes place so that the new ones will retain the former as less developed, subordinate qualities. (E.g., a living organism is characterised by its biological properties, but it also has physical properties, like mass, volume, temperature, etc.) The evolution of a new fundamental form of motion is conditional upon the pre-existence of the developed forms of motion pertaining to the previous level in the totality of their own level (cf. law (1a)). Both presently known higher fundamental levels (the organic world and human society) imply the pre-existence of the previous ones (inanimate and living nature, respectively). Fundamental levels representing more developed qualities will, as a rule, produce more sophisticated material structures: they are capable of reflecting their environment in a more differentiated manner (cf. law (1c)). E.g., an inanimate material structure can reflect the objects of its environment only in their physical/chemical quality, and cannot reflect (get in interaction with) any biological, social properties of the surrounding objects.

The main arranging aspects (forms of motion, interactions, genetic sequences, their nest of tables) used so far largely coincide with each other at the three fundamental levels: it is thus possible to bring them into an unambiguous correspondence with each other. Their respective role is equal in rank.

Now the question is whether the same holds true in the case of an arrangement carried through within the individual fundamental levels, (particularly within the level of inanimate nature), so that its finer structure of matter could be understood within the framework of a consistent theory? At this particular level, however, the role of the above concepts as arranging aspects do not coincide with each other, not even in the sense that they could be brought unambiguously into proper correspondence. Which of them, then, may be given primacy, on the basis of which, one might form an image of the structure of the inanimate matter, or in other words: on the basis of which a particular physical level theory could be developed?

Let's see now how the concepts, more often applied to characterise 'level formation', are suitable for a generalisation applicable at all (both fundamental and particular) levels.

The concept of the forms of motion has proven to be feasible mainly in the classification of scientific disciplines, being less useful in reaching a better understanding of the structure of physical matter. (Scientific disciplines are formed in many, but not in all, cases to explain phenomena at an individual level. There is no one-to-one correspondence among them. This statement is more valid in the biological sciences, where disciplines are organised along a perpendicular axis compared to several other sciences; i.e., the biological sciences can be arranged not only on a structural basis: cell, organ, organism, but also along phylogeny.)

To understand the role of interactions, a clear conceptual distinction should be made between them and the forms of motion. While the two can be clearly discerned in physics, not infrequently are they mixed up in philosophy. In a precise treatment, the concept of interaction- in a philosophical sense- has certain additional substance as compared to that of the forms of motion, namely, it contains the time and space conditions of motion and transition, their dynamics, as well as the formal- adding to the mentioned content- elements of motion.

It is usual to speak of the types of matter as they vary from level to level, and even of its particular space-time forms characteristic of the individual levels. Serving as arranging aspects are also the order of magnitude relations (the principle of 'nest of tables'), the sequence of genetic evolution, and the degree of complexity.

The order of magnitude relations can be interpreted as properties within certain limits only: basically, the metric properties are emphasised, while in certain extreme spheres (small and large scales) the topological properties are predominant. (In topological terms, it is meaningless to say that a quark is smaller or larger than a neutron or an electron. Similarly, there is no meaning to say that the extension of a neutron-star or a black hole characterises the level of the deterministic physical processes in the given material structure.)

Space-time forms, while having certain content elements too, seem less suitable to furnish a basis for systematisation than the categories involving substantiality (motion, interaction).

Essentially, the concept of the types of matter only proves feasible to describe either the forms of motion or structures belonging to spheres of various orders of magnitude, but does not have additional substantial elements.

Sequence of genetic evolution can be hardly interpreted amidst the fundamental physical interactions. No genetic order can be defined among the strong, electromagnetic, or gravitational interactions. Note, that while physics uses the concept of interaction in a way similar to that of philosophy, the concepts of fundamental interactions within the framework of physics are meant quite different from the characteristic interactions attributed in philosophy to the three fundamental philosophical levels.

The degree of complexity becomes subject to its ability to participate in various interactions. Nevertheless, it determines, among others, the degree of ability to reflect the environment of the given object.

Also to be met with is the arrangement of levels by structures which, in the process of understanding the structure of matter, would lead to tautology; but it reminds us, by all means, to update our traditional concept of structure (Darvas and Nagy, 1989).

General and particular theories

Since in the case of arrangement within a particular fundamental level the contents of the listed arranging concepts will not coincide (in contrast to the philosophical level theory, which serves for the arrangement of the fundamental levels), one must suppose a justification for the existence of particular level theories too. These serve for the arrangement of levels considered (in philosophical terms) not fundamental. The experience allows us to suppose that a level theory of philosophy cannot be transferred analogically to-e.g.,-a physical level theory. The provable collapse of any analogy with the level theory of philosophy makes it clear that the theory of levels within the inanimate nature (and also within the organic nature) may by no means be an extension of the former. Thus there arises the demand on new- let us call them physical (biological)- level theories laid on other foundations. To organise its conceptual basis, an arrangement by (physical) interactions seems to be the most practicable that joins with the concept of structure including both actual and potential elements.

With all this accepted, we should draw the conclusion, retrospectively, also for the level theory of philosophy. Namely,-to retain, at least partially, those properties of the theories that are complementary to each other- that among concepts which play equivalent roles, and which appear also in the level theory of philosophy, primacy should be given to the concept of interactions rather than to the forms of motion. (Mention should be made that several attempts have being been made at developing a largely axiomatic physical theory built on the fundamental physical interactions. Relying on our present-day knowledge of physics, as well as on the empirical results of recent years (cross-interactions, their carriers, unification theories), such a description of nature seems to be possible.)

To be sure, beyond a certain point within physics one should always reckon with the evolution of matter, as well as with the appearance of new qualities. These new qualities, however, will not transgress the bounds of the fundamental level, but-within this-they should be interpreted as new qualities having come into being on the grounds of interactions at the preceding levels, yet they will act as carriers of abilities to take part in such, actual and potential, interactions to which their components were not attributed; (cf., the law (1a)).

It may be stated, then, that level theories of philosophy have clear-cut limitations, and that the- particular- level theories existing within one fundamental level ought to be quite different (and more differentiated) in order to be more instrumental in our reaching a better understanding and interpretation of the structure (in general and of all appearances) of matter.

Therefore one can speak about two types of level theories: a general one (in philosophy) and particular ones (in the inanimate, the organic nature, and in the human society). Particular level theories differ from each other in the three fundamental ontological spheres, nevertheless in their description and contents. At the same time they may have common features, e.g., all are particular theories concerning their width of validity, and all are based on an arrangement by a common concept, namely the forms of interaction. The clarification of these conceptual problems was necessary to understand the laws of symmetry breaking.

Is unification of the different types of level theories possible? With certain limits, yes. For this reason, one should accept that all levels can be characterised by a given type of interaction, and they are submitted to the laws (1a-c). Any further detail belongs to the competency of the discipline studying the phenomena of the given level.

Levels and symmetry breakings

Is there a one-to-one correspondence between the levels and symmetry properties broken at the given level? One cannot give a definite answer yet, since it has not been studied thoroughly in all disciplines. However, all the available examples affirm the presumption. E.g., the stronger a basic physical interaction is, the more quantities are conserved, and with weakening the type of interaction, the number of symmetry breaking increases. That means also, that the weaker an interaction is, the greater number of material structures (particles) are affected by it, and their interactions are limited by fewer conservation laws. Strong interaction conserves all elementary particle quantities. In electromagnetic interaction Isospin is not conserved, but all the others are; in weak interaction Parity, Charge conjugation and others are not conserved, (however the combination of them with Time reversal (CPT) is conserved). Parity conservation is also violated in the so called united electroweak interaction. In the electroweak interaction the antineutrinos play an important role. These particles exist only in a right-handed chiral form. Antineutrinos are produced during beta decay, where the majority of the electrons produced simultaneously with the antineutrinos have a left-handed chirality (spin) (Ne'eman, 1986). The participants of the electroweak interaction are the electrons of the atom on the one side and the protons and neutrons of the nucleus on the other. From the chirality of the participants follows the chirality of the atoms, and the molecules built of them. This leads to the existence of the enantiomers in the organic molecules (e.g., glucose and fructose), then the L- and D-aminoacids. Proteins are built up (almost) exclusively from L-aminoacids, and therefore it is not by chance that RNA and DNA form only right-handed helices. Is it surprising that living creations are chiral? All this follows from the electroweak interaction what distinguishes 'left' and 'right' by the charged weak currents and neutral weak currents (or in other words by W and Z forces) (Hegstrom and Kondepudi, 1990).

However, nature is not so simple. Nature reproduces the dominance of left- or right-handedness at any new level by new properties. While left-handed DNA helices are very rare, we find both left- and right-handed helices among bacteria, plants, snails, etc. Nature produces again both kinds, although, by a spontaneous symmetry breaking their numbers are different.

The dominance of morphological asymmetry is becoming prevailing in the morphology at the more evolved animals (e.g., circulation system). Another symmetry: irreversibility (e.g., reproducibility of the organs) weakens during the evolution too. Nevertheless, the brain remains symmetric, even at mammals. A new, qualitative change (mutation) takes place, when the lateralisation of the brain starts. This makes possible the real right- and left-handedness, differentiation of the kinetic and the speech centres in the brain, and the separation of the emotional and rational, etc. functions. The loss of the symmetry of the brain is also a typical example of the violation of a symmetry, which did not exist 'always', only since one can speak of 'brain' or 'neural system' as a quality, as an organ of living organisms (2c-d).

Level theory, evolution theory, reductionism, and symmetry principles

This treatment could not give a detailed introduction to the level theories. That was not the aim of this paper. Our goal was only to introduce the laws of symmetry breaking. We did not want to replace any evolution theory; these laws touch them only tangentially. It is important to note that none of the treated laws take stand on the debate of reductionism. We stressed, that they may be used as arguments by both parties-probably they can bring the parties closer to a decision- but in their presented form they do not fulfil a decisive function. This was also not the aim of this paper. The new features of this treatment were to link the level theories with the (dis)symmetry principles, and the laws of symmetry breaking. Both are subjects of the philosophy of science.

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Bibliography

B‚rczi, S., Luk cs, B., guest eds. [1993] Symmetry and Topology in Evolution, Special issues of Symmetry: Culture and Science, Vol. 4, 1-2, 1-224.

Darvas, G. [1987] Level theories in philosophy and physics- Exploring and interpreting the structure of matter, Vol. 5, Part 3, pp. 76-79, In: 8th International Congress of Logic, Methodology, and Philosophy of Science, Abstracts, Moscow: Nauka.

Darvas, G. [1995] ISIS-Symmetry: Looking for truth and beauty, pp. 9-18, In. Darvas, G., Nagy, D., Pardavi, M., eds., Symmetry: Natural and Artificial, Budapest: Symmetrion, 734 p.

Darvas, G. [1997] Mathematical symmetry principles in the scientific world view, pp. 319-334, In: Agazzi, E., Darvas, G., eds. Philosophy of Mathematics Today, Dordrecht-Boston-London: Kluwer Academic Publ., 361 p.

Darvas, G., Nagy, D., eds. [1989] Symmetry of Structure, Vols. 1-2, Budapest: ISIS-Symmetry, 656 p.

van Fraassen, B. C. [1989] Laws and Symmetry, Oxford: Clarendon Press, 395 p.

de Gortari, E. [1970] Symmetry as a scientific method, pp. 30-34, In: La symetrie. Comme principe heuristique dans les differentes sciences, Bruxelles: Office international de librairie, 135 pp.

Hahn, W. [1989] Symmetrie als Entwicklungsprinzip in Natur und Kunst, Künigstein: Langewiesche, 320 p.

Hegstrom, R. A., Kondepudi, D. K., [1990] Is the Universe left- or right-handed? Scientific American, 3.

Heisenberg, W. [1971] Physics and Beyond. Encounters and Conversations, New York: Harper and Row Publ. Inc.

Kuhn, T. [1963] The Structure of Scientific Revolutions, Chicago: University of Chicago Press.

Mainzer, K. [1990] Symmetry in philosophy and history of science, Symmetry: Culture and Science, 1, 3, 319-328.

Nagy, D. [1990] Manifesto on (dis)symmetry, with some preliminary symmetries, Symmetry: Culture and Science, 1, 1, 3-26.

Nagy, D. [1995] The 2,500-year old term symmetry in science and art and its 'missing link' between the antiquity and the modern age, Symmetry: Culture and Science, 6, 1, 18-28.

Ne'eman, Y., Kirsh, Y. [1986] The Particle Hunters, Cambridge: Cambridge University Press, p. 272.

Rosen, J. [1995] Symmetry in Science, New York: Springer-Verlag, 213 pp.

Wigner, E. P. [1964] Symmetry and conservation laws, Proceedings of the National Academy of Sciences, 51, 5, 956-965.

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