S-Matrix Advantages

The S-matrix formalism can describe the ‘exchange’ of a whole family of hadrons.

All hadrons fall into sequences whose members have identical properties except for their masses and spins. A formalism proposed originally by Tullio Regge makes it possible to treat each of these sequences as a single hadron existing in various excited states.

In recent years, it has been possible to incorporate the Regge formalism into the S-matrix framework where it has been used very successfully for the description of hadron reactions. This has been one of the most important developments in S-matrix theory and can be seen as a first step towards a dynamic explanation of particle patterns.

The framework of the S matrix, then, is able to describe the structure of hadrons, the forces through which they mutually interact, and some of the patterns they form, in a thoroughly dynamic way in which each hadron is understood as an integral part of an inseparable network of reactions. The main challenge, and so far unsolved problem, in S-matrix theory is to use this dynamic description to account for the symmetries which give rise to the hadron patterns and conservation laws discussed in the previous chapter. In such a theory, the hadron symmetries would be reflected in the mathematical structure of the S matrix in such a way that it contains only elements which correspond to reactions allowed by the conservation laws. These laws would then no longer have the status of empirical regularities but would be a consequence of the S-matrix structure, and thus a consequence of the dynamic nature of hadrons.

At present, physicists are trying to achieve this ambitious aim by postulating several general principles which restrict the mathematical possibilities of constructing S-matrix elements and thus give the S matrix a definite structure. So far, three of these general principles have been established.

The first is suggested by relativity theory and by our macroscopic experience of space and time. It says that the reaction probabilities (and thus the S-matrix elements) must be independent of displacements of the experimental apparatus in space and time independent of its orientation in space, and independent of the state of motion of the observer. As discussed in the previous chapter, the independence of a particle reaction with regard to changes of orientation and displacements in space and time implies the conservation of the total amount of rotation, momentum and energy involved in the reaction.

These ‘symmetries’ are essential for our scientific work. If the results of an experiment changed according to where and when it was performed, science in its present form would be impossible. The last requirement, finally-that the experimental results must not depend on the observer’s motion-is the principle of relativity which is the basis of relativity theory.* The second general principle is suggested by quantum theory. It asserts that the outcome of a particular reaction can only be predicted in terms of probabilities and, furthermore, that the sum of the probabilities for all possible outcomes- including the case of no interaction between the particles- must be equal to one. In other words, we can be certain that the particles will either interact with one another, or not. This seemingly trivial statement turns out to be, in fact, a very powerful principle, known under the name of ‘unitarity’, which severely restricts the possibilities of constructing S-matrix elements. The third and final principle is related to our notions of cause and effect and is known as the principle of causality. It states that energy and momentum are transferred over spatial distances only. by particles, and that this transfer occurs in such a way that a particle can be created in one reaction and destroyed in another only if the latter reaction occurs after the former. The mathematical formulation of the causality principle implies that the S matrix depends in a smooth way on the energies and momenta of the particles involved in a reaction, except for those values at which the creation of new

particles becomes possible. At those values, the mathematical structure of the S matrix changes abruptly; it encounters what mathematicians call a ‘singularity’. Each reaction channel contains several of these singularities, that is, there are several values of energy and momentum in each channel at which new particles can be created. The ‘resonance energies’ men- tioned before are examples of such values. The fact that the S matrix exhibits singularities is a con- sequence of the causality principle, but the location of the singularities is not determined by it. The values of energy and momentum at which particles can be created are different for different reaction channels and depend on the masses and other properties of the created particles. The locations of the singularities thus reflect the properties of these particles, and since all hadrons can be created in particle reactions, the singularities of the S matrix mirror all the patterns and sym- metries of hadrons. The central aim of S-matrix theory is, therefore, to derive the singularity structure of the S matrix from the general principles. Up to now, it has not been possible to construct a mathematical model which satisfies all three principles, and it may well be that they are sufficient to determine all the properties of the S matrix-and thus all the properties of hadrons-uniquely.* If this turns out to be the case, the philosophical implications of such a theory would be very profound. All three of the general principles are related to our methods of observation and measurement, that is, to the scientific framework. If they are sufficient to determine the structure of hadrons, this would mean that ‘the basic structures of the physical world are determined, ultimately, by the way in which we look at this world. Any fundamental change in our observational methods would imply a modification of the general principles which would lead to a different structure of the S matrix, and would thus imply a different structure of hadrons.

Such a theory of subatomic particles reflects the impossibility of separating the scientific observer from the observed pheno-

*This conjecture, known as the ‘bootstrap’ hypothesis, will be discussed in more detail in the subsequent chapter.

mena, which has already been discussed in connection with quantum theory,* in its most extreme form. It implies, ultimately, that the structures and phenomena we observe in nature are nothing but creations of our measuring and categorizing mind. That this is so is one of the fundamental tenets of Eastern philosophy. The Eastern mystics tell us again and again that all things and events we perceive are creations of the mind, arising from a particular state of consciousness and dissolving again if this state is transcended. Hinduism holds that all shapes and structures around us are created by a mind under the spell of maya, and it regards our tendency to attach deep significance to them as the basic human illusion. Buddhists call this illusion avidya, or ignorance, and see it as the state of a ‘defiled’ mind. In the words of Ashvaghosha, When the oneness of the totality of things is not recognised, then ignorance as well as particularisation arises, and all phases of the defiled mind are thus developed . . . All phenomena in the world are nothing but the illusory manifestation of the mind and have no reality on their own.3 This is also the recurring theme of the Buddhist Yogacara school which holds that all forms we perceive are ‘mind only’; projections, or ‘shadows’, of the mind: Out of mind spring innumerable things, conditioned by discrimination . . . These things people accept as an external world . . . What appears to be external does not exist in reality; it is indeed mind that is seen as multiplicity; the body, property, and above-all these, I say, are nothing but mind.4 In particle physics, the derivation of the hadron patterns from the general principles of S-matrix theory is a long and arduous task, and so far only a few small steps have been taken towards achieving it. Furthermore, the theory in its present form cannot be applied to the electromagnetic interactions that

give rise to the atomic structures and dominate the world of chemistry and biology. Nevertheless, the possibility that the hadron patterns will some day be derived from the general principles, and thus be seen to depend on our scientific framework, must be taken seriously. It is an exciting con- jecture that this may be a general feature of particle physics which will also appear in future theories of electromagnetic, weak, and gravitational interactions. If this turns out to be true, modern physics will have come a long way towards agreeing with the Eastern sages that the structures of the physical world are maya, or ‘mind only’.

S-matrix theory comes very close to Eastern thought not only in its ultimate conclusion, but also in its general view of matter. It describes the world of subatomic particles as a dynamic network of events and emphasizes change and trans- formation rather than fundamental structures or entities. In the East, such an emphasis is particularly strong in Buddhist thought where all things are seen as dynamic, impermanent and illusory. Thus S. Radhakrishnan writes:

How do we come to think of things, rather than of pro- cesses in this absolute flux? By shutting our eyes to the successive events. It is an artificial attitude that makes sections in the stream of change, and calls them things . . . When we shall know the truth of things, we shall realise how absurd it is for us to worship isolated products of the incessant series of transformations as though they were eternal and real. Life is no thing or state of a thing, bu It a continuous movement or change.5

Both the modern physicist and the Eastern mystic thave realized that all phenomena in this world of change and transformation are dynamically interrelated. Hindus Buddhists see this interrelation as a cosmic law, the law of karma, but they are generally not concerned with any specific patterns in the universal network of events. Chinese philosophy, on the other hand, which also emphasizes movement and change, has developed the notion of dynamic patterns which are continually formed and dissolved again in the cosmic flow of the Tao. In the I Ching, or Book of Changes, these patterns have been elaborated into a system of archetypal symbols, the so-called hexagrams.

The basic ordering principle of the patterns in the /Ching* is the interplay of the polar opposites yin and yang. The yang , the yin by a broken line is represented by a solid line (I I-1, and the whole system of hexagrams is built up naturally from these two lines. By combining them in pairs, four configurations are obtained,

and by adding a third line to each of these, eight ‘trigrams’ are generated : In ancient China, the trigrams were considered to represent all possible cosmic and human situations. They were given names reflecting their basic characteristics-such as The Creative’, The Receptive’, ‘The Arousing’, etc.-and they were associated with many images taken from nature and from social life. They represented, for example, heaven, earth, thunder, water, etc., as well as a family consisting of father, mother, three sons and three daughters. They were, furthermore, associated with the cardinal points and with the seasons of the year, and were often arranged as follows:

In this arrangement, the eight trigrams are grouped around a circle in the ‘natural order’ in which they were generated, starting from the top (where the Chinese always place the south) and placing the first four trigrams on the left side of the circle, the second four on the right side. This arrangement shows a high degree of symmetry, opposite trigrams having yin and yang lines interchanged.

In order to increase the number of possible combinations further, the eight trigrams were combined in pairs by placing one above the other. In this way, sixty-four hexagrams were obtained, each consisting of six solid or broken lines. The hexagrams were arranged in several regular patterns, among which the two illustrated on the opposite page were the most common; a square of eight times eight hexagrams, and a circular sequence showing the same symmetry as the circular arrangement of the trigrams. The sixty-four hexagrams are the cosmic archetypes on which the use of the I Ching as an oracle book is based. For the interpretation of any hexagram, the various meanings

of its two trigrams have to be taken into account. For example, when the trigram The Arousing’ is situated above the trigram The Receptive’ the hexagram is interpreted as movement meeting with devotion and thus inspiring enthusiasm, which is the name given to it. the Arousing the Receptive Enthusiasm The hexagram for Progress, to give another example, represents ‘The Clinging’ above ‘The Receptive’ which is interpreted as the sun rising over the earth and thus as a symbol of rapid, easy progress. the Clinging ~~ the Receptive Progress In the I Ching, the trigrams and hexagrams represent the patterns of the Tao which are generated by the dynamic inter- play of the yin and the yang, and are reflected in all cosmic and human situations. These situations, therefore, are not seen as static, but rather as stages in a continuous flow and change. This is the basic idea of the Book of Changes which is expressed in its very title. All things and situations in the world are subject to change and transformation, and so are their images, the trigrams and hexagrams. They are in a state of continual transition; one changing into another, solid lines pushing outwards and breaking in two, broken lines pushing inwards and growing together. Because of its notion of dynamic patterns, generated by change and transformation, the I Ching is perhaps the closest analogy to S-matrix theory in Eastern thought. In both systems, the emphasis is on processes rather than objects. In S-matrix theory, these processes are the particle reactions that give rise to all the phenomena in the world of hadrons. In the I Ching, the basic processes are called ‘the changes’ and are seen as essential for an understanding of all natural phenomena:

The changes are what has enabled the holy sages to reach all depths and to grasp the seeds of all things6 These changes are not regarded as fundamental laws imposed on the physical world, but rather-in the words of Hellmut Wilhelm-as ‘an inner tendency according to which develop- ment takes place naturally and spontaneously’.7 The same can be said of the ‘changes’ in the particle world. They, too, reflect the inner tendencies of the particles which are expressed, in S-matrix theory, in terms of reaction probabilities. The changes in the world of hadrons give rise to structures and symmetric patterns which are represented symbolically by the reaction channels. Neither the structures nor the symmetries are regarded as fundamental features of the hadron world, but are seen as consequences of the particles’ dynamic nature, that is, of their tendencies for change and transformation.

In the I Ching, too, the changes give rise to structures-the trigrams and hexagrams. Like the channels of particle reactions, these are symbolic representations of patterns of change. As the energy flows through the reaction channels, the ‘changes’ flow through the lines of the hexagrams: Alteration, movement without rest, Flowing through the six empty places, Rising and sinking without fixed law, It is only change that is at work here!

In the Chinese view, all things and phenomena around us arise out of the patterns of change and are represented by the various lines of the trigrams and hexagrams. Thus the things in the physical world are not seen as static, independent objects, but merely as transitional stages in the cosmic process which is the Tao:

The Tao has changes and movements. Therefore the lines are called changing lines. The lines have gradations, there- fore they represent things.9

As in the world of particles, the structures generated by the changes can be arranged in various symmetric patterns, such as the octagonal pattern formed by the eight trigrams, in which opposite trigrams have yin and yang lines interchanged.

This pattern is even vaguely similar to the meson octet discussed in the previous chapter, in which particles and anti- particles occupy opposite places. The important point, how:

ever, is not this accidental similarity, but the fact that both modern physics and ancient Chinese thought consider change and transformation as the primary aspect of nature, and see the structures and symmetries generated by the changes as secondary. As he explains in the introduction to his translation’ of the I Ching, Richard Wilhelm regards this idea as the funda- mental concept of the Book of Changes:

The eight trigrams . . . were held to be in a state of continual transition, one changing into another, just as transition from one phenomenon to another is continually taking place in the physical world. Here we have the fundamental concept of the Book of Changes.

The eight trigrams are symbols standing for changing transitional states; they are images that are constantly undergoing change. Attention centers not on things in their state of being-as is chiefly the case in the Occident-but upon their movements in change. The eight trigrams therefore are not representations of things as such but of their tendencies in movement.lO

In modern physics, we have come to see the ‘things’ of the subatomic world in very much the same way, laying stress upon movement, change and transformation and regarding the particles as transient stages in an ongoing cosmic process.