Partial Models

When the bootstrap idea is formulated in a scientific context, it has to be limited and approximate, and its main approximation consists in neglecting all but the strong interactions. Since these interaction forces are about a hundred times stronger than the electromagnetic ones, and many more orders of magnitude stronger than weak and gravitational interactions, such an approximation seems reasonable. The scientific bootstrap, then, deals exclusively with strongly interacting particles, or hadrons, and is therefore often called the ‘hadron bootstrap’.

It is formulated in the framework of S-matrix theory and its aim is to derive all properties of hadrons and their interactions uniquely from the requirement of self-consistency. The only ‘fundamental laws’ accepted are the general S-matrix principles discussed in the previous chapter, which are required by our methods of observation and measurement and thus constitute the unquestioned framework necessary for all science. Other properties of the S matrix may have to be postulated temporarily as ‘fundamental principles’, but will be expected to emerge as a necessary consequence of self-consistency in the complete theory. The postulate that all hadrons form sequences described by the Regge formalism* may he of that kind.

In the language of S-matrix theory, then, the bootstrap hypothesis suggests that the full S matrix, and thus all the properties of hadrons, can be determined uniquely from the general principles because there is only one possible S matrix consistent with all three of them. This conjecture receives support from the fact that physicists have never come close to constructing a mathematical model which satisfies the three general principles. If the only consistent S matrix is the one describing a// properties and interactions of hadrons, as the bootstrap hypothesis assumes, the physicists’ failure to construct a-consistent partial S matrix becomes understandable.

The phenomena involving hadrons are so complex that it is by no means certain whether the complete self-consistent S matrix will ever be constructed, but one can envisage a series of partially successful models of smaller scope. Each of them would be intended to cover only a part of hadron physics and would therefore contain some unexplained parameters representing its limitations, but the parameters of one model may be explained by another. Thus more and more hadron phenomena may gradually be covered with ever-increasing accuracy by a mosaic of interlocking models whose net number of unexplained parameters will keep decreasing. The adjective ‘bootstrap’ is thus never appropriate for any individual model, but can only be applied to a combination of mutually consistent models, none of which is any more fundamental than the others. As Chew has put it, ‘A physicist who is able to view any number of different partially successful models without favoritism is automatically a bootstrapper.’

A number of partial models of that kind already exist, formulated in S-matrix language and describing certain aspects of hadron phenomena. The most successful of them are the so-called ‘dual models’ which make extensive use of the dual description of hadron reactions in terms of direct and cross channels.* These models incorporate, for the first time, two of the three general principles** plus the crossing property of the S matrix and the Regge formalism. They constitute so far the most promising attempts at carrying out the bootstrap programme.

The picture of hadrons which emerges from these bootstrap models is often summed up in the provocative phrase, ‘every particle consists of all other particles’. It must not be imagined, however, that each hadron contains all the others in a classical, static sense. Rather than ‘containing’ one another, hadrons ‘involve’ one another in the dynamic and probabilistic sense of S-matrix theory, each hadron being a potential ‘bound state’ of all sets of particles which may interact with one another to form the hadron under consideration.***

In that sense, all hadrons are composite structures whose components are again hadrons, and none of them is any more elementary than the others. The binding forces holding the structures together manifest themselves through the exchange of particles, and these exchanged particles are again hadrons. Each hadron, therefore, plays three roles: it is a composite structure, it may be a constituent of another hadron, and it may be exchanged between constituents and thus constitute part of the forces holding a structure together. The concept of ‘crossing’ is crucial for this picture. Each hadron is held together by forces associated with the exchange of other hadrons in the cross channel, each of which is, in turn, held together by forces to which the first hadron makes a contribution. Thus, ‘each particle helps to generate other particles, which in turn generate it.“’ The whole set of hadrons generates itself in this way or pulls itself up, so to say, by its ‘bootstraps’. The idea, then, is that this extremely complex bootstrap mechanism is self-determining, that is, that there is only one way in which it can be achieved. In other words, there is only one possible self-consistent set of hadrons- the one found in nature.

In the hadron bootstrap, all particles are dynamically composed of one another in a self-consistent way, and in that sense can be said to ‘contain’ one another. In Mahayana Buddhism, a very similar notion is applied to the whole universe. This cosmic network of interpenetrating things and events is illustrated in the Avatamsaka Sutra by the metaphor of Indra’s net, a vast network of precious gems hanging over the palace of the god Indra. In the words of Sir Charles Eliot: In the heaven of Indra, there is said to be a network of pearls, so arranged that if you look at one you see all the others reflected in it. In the same way each object in the world is not merely itself but involves every other object and in fact is everything else. ‘In every particle of dust, there are present Buddhas without number.‘18 The similarity of this image with that of the hadron bootstrap is indeed striking. The metaphor of Indra’s net may justly be called the first bootstrap model, created by the Eastern sages some 2,500 years before the beginning of particle physics.

Buddhists insist that the concept of interpenetration is not comprehensible intellectually, but is to be experienced by an enlightened mind in the state of meditation. Thus D. T. Suzuki writes : The Buddha [in the Gandavyuhal is no more the one who is living in the world conceivable in space and time. His consciousness is not that of an ordinary mind which must be regulated according to the senses and logic . . . The Buddha of the Gandavyuha lives in a spiritual world which ’ has its own rules.19

In modern physics, the situation is quite similar. The idea of every particle containing all the others is inconceivable in ordinary space and time. It describes a reality which, like the one of the Buddha, has its own rules. In the case of the hadron bootstrap, they are the rules of quantum theory and relativity theory, the key concept being that the forces holding particles together are themselves particles exchanged in the cross channels. This concept can be given a precise mathematical meaning, but is almost impossible to visualize. It is a specifically relativistic feature of the bootstrap, and since we have no direct experience of the four-dimensional world of space-time, it is extremely difficult to imagine how a single particle can contain all other particles and at the same time be part of each of them. This, however, is exactly the view of the Mahayana: When the one is set against all the others, the one is seen as pervading them all and at the same time embracing them all in itself.” The idea of each particle containing all the others has not only arisen in Eastern mysticism, but also in Western mystical thought. It is implicit, for example, in William Blake’s famous lines : To see a world in a grain of sand And a heaven in a wild flower, Hold infinity in the palm of your hand, And eternity in an hour.

Here again, a mystical vision has led to an image of the boot- strap type; if the poet sees the world in a grain of sand, the modern physicist sees the world in a hadron. A similar image appears in the philosophy of Leibniz who considered the world as being made of fundamental substances called ‘monads’, each of which mirrored the whole universe. This led him to a view of matter which shows similarities to that of Mahayana Buddhism and to the hadron bootstrap.*

In his Monadology, Leibniz writes :

Each portion of matter may be conceived of as a garden full of plants, and as a pond full of fishes. But each branch of the plant, each member of the animal, each drop of its- humors, is also such a garden or such a pond.21 It is interesting that the similarity of these lines to the passages of the Avatamsaka Sutra mentioned before may stem from an actual Buddhist influence on Leibniz. Joseph Needham has argued 22 that Leibniz was well acquainted with Chinese thought and culture through translations he received from Jesuit monks, and that his philosophy might very well have been inspired by the Neo-Confucian school of Chu Hsi with which he was familiar. This school, however, has one of its roots in Mahayana Buddhism, and in particular in the Avatamsaka (Chinese: ha-yen) school of the Mahayana branch. Needham, in fact, mentions the parable of Indra’s net of pearls explicitly in connection with the Leibnizian monads.

A more detailed comparison of Leibniz’ notion of ‘mirroring relations’ between monads with the idea of interpenetration in the Mahayana seems to show,. however, that the two are rather different, and that the Buddhist conception of matter comes much closer to the spirit of modern physics than that of Leibniz. The principal difference between the Monadology and the Buddhist view seems to be that the Leibnizian monads are fundamental substances which are seen as the ultimate constituents of matter. Leibniz begins the Monadology with the words, The monad of which we shall here speak is merely

*The parallels between Leibniz’ view of matter and the hadron bootstrap have recently been discussed; see C. Gale, ‘Chew’s Monadology’, journal of History of Ideas, vol. 35 (April-June 1974), pp. 339-48.

a simple substance, which enters into composites; simple, that is to say, without parts.’ He goes on to say, ‘And these monads are the true atoms of nature, and, in a word, the elements of all things.‘23 Such a ‘fundamentalist’ view is in striking contrast to the bootstrap philosophy, and is also totally different from the view of Mahayana Buddhism which rejects all fundamental entities or substances. Leibniz’ fundamentalist way of thinking is also reflected in his view of forces which he regards as laws ‘imprinted by divine decree’ and essentially different from matter. ‘Forces and activity’, he writes, ‘cannot be states of a merely passive thing like matter.‘” Again, this is contrary to the views of modern physics and of Eastern mysticism. As far as the actual interrelation between the monads is concerned, the main difference to the hadron bootstrap seems to be that monads do not interact with each other; they ‘have no windows’, as Leibniz says, and merely mirror one another. In the hadron bootstrap, on the other hand, as in the Mahayana, the emphasis is on the interaction, or ‘interpenetration’, of all particles. Furthermore, the bootstrap and the Mahayana views of matter are both ‘space-time’ views which see objects as events whose mutual interpenetration can only be understood if one realizes that space and time, too, are interpenetrating. The bootstrap hypothesis is not yet firmly established and the technical difficulties involved in its implementation are con- siderable. Nevertheless, physicists already speculate about extending the self-consistent approach beyond the description of hadrons. In the present context of S-matrix theory, such an extension is not possible. The framework of the S matrix has been developed specifically to describe the strong interactions and cannot beapplied to the rest of particle physics, the principal reason being that it cannot accommodate the massless particles which are characteristic of all the other interactions.

To enlarge the hadron bootstrap, therefore, a more general framework will have to be found, and in this new framework some of the concepts which are at present accepted without explanation will have to be ‘bootstrapped’; they will have to be derived, that is, from the overall self-consistency. According to Geoffrey Chew, these might include our conception of macroscopic

space-time and, perhaps, even that of human consciousness: Carried to its logical extreme, the bootstrap conjecture implies that the existence of consciousness, along with all other aspects of nature, is necessary for self-consistency of the whole.25

This view is in perfect harmony with the views of the Eastern mystical traditions which have always regarded consciousness as an integral part of the universe. In the Eastern view, human beings, like all other life forms, are parts of an inseparable organic whole. Their intelligence, therefore, implies that the whole, too, is intelligent.

Man is seen as the living proof of cosmic intelligence; in us, the universe repeats over and over again its ability to produce forms through which it becomes consciously aware of itself.

In modern physics, the question of consciousness has arisen in connection with the observation of atomic phenomena. Quantum theory has made it clear that these phenomena can only be understood as links in a chain of processes, the end of which lies in the consciousness of the human observer.* In the words of Eugene Wigner, ‘It was not possible to formulate the laws of [quantum theory1 in a fully consistent way without reference to consciousness.‘26 The pragmatic formulation of quantum theory used by the scientists in their work does not refer to their consciousness explicitly. Wigner and other physicists have argued, however, that the explicit inclusion of human consciousness may be an essential aspect of future theories of matter.

Such a development would open exciting possibilities for a direct interaction between physics and Eastern mysticism. The understanding of one’s consciousness and of its relation to the rest of the universe is the starting point of all mystical experience. The Eastern mystics have explored various modes of consciousness throughout centuries, and the conclusions they have reached are often radically different from the ideas held in the West. If physicists really want to include the nature of human consciousness in their realm of research, a study of Eastern ideas may well provide them with stimulating new viewpoints.

Thus the future enlargement of the hadron bootstrap, with the ‘bootstrapping’ of space-time and of human consciousness it may require, opens up unprecedented possibilities which may well go beyond the conventional framework of science: Such a future step would be immensely more profound than anything comprising the hadron bootstrap; we would be obliged to confront the elusive concept of observation and, possibly, even that of consciousness. Our current struggle with the hadron bootstrap may thus be only a foretaste of a completely new form of human intellectual endeavor, one that will not only lie outside of physics but will not even be describable as ‘scientific’.27

Where, then, does the bootstrap idea lead us?This, of course, nobody knows, but it is fascinating to speculate about its ultimate fate. One can imagine a network of future theo,ries covering an ever-increasing range of natural phenomena with ever-increasing accuracy; a network which will contain fewer and fewer unexplained features, deriving more and more of its structure from the mutual consistency of its parts. Some day, then, a point will be reached where the only unexplained features of this network of theories will be the elements of the scientific framework. Beyond that point, the theory will no longer be able to express its results in words, or in rational concepts, and will thus go beyond science.

Instead of a bootstrap theory of nature, it will become a bootstrap vision of nature, transcending the realms of thought and language; leading out of science and into the world of acintya, the un- thinkable. The knowledge contained in such a vision will be complete, but cannot be communicated in words. It will be the knowledge which Lao Tzu had in mind, more than two thousand years ago, when he said:

He who knows does not speak, He who speaks does not know.*8