The Bootstrap Hypothesis

To explain the symmetries in the particle world in terms of a dynamic model, that is, one describing the interactions between the particles, is one of the major challenges of present-day physics. The problem, ultimately, is how to take into account simultaneously quantum theory and relativity theory.

The particle patterns seem to reflect the ‘quantum nature’ of the particles, since similar patterns occur in the world of atoms.

In particle physics, however, they cannot be explained as wave patterns in the framework of quantum theory, because the energies involved are so high that relativity theory has to be applied. Only a ‘quantum-relativistic’ theory of particles, therefore, can be expected to account for the observed symmetries.

Quantum field theory was the ‘first model of that kind. It gave an excellent description of the electromagnetic interactions between electrons and photons, but it is much less appropriate for the description of strongly interacting particles.

As more and more of these particles were discovered, physicists soon realized that it was highly unsatisfactory to associate each of them with a fundamental field, and when the particle world revealed itself as an increasingly complex tissue of inter- connected processes, they had to look for other models to represent this dynamic and ever-changing reality. What was needed was a mathematical formalism which would be able to describe in a dynamic way &great variety of hadron oatterns: their continual transformation into one another, their mutual interaction through the exchange of other particles, the formation of ‘bound states’ of two or more hadrons, and their decay into various particle combinations. All these processes, which are often given the general name

‘particle reactions’, are essential features of the strong inter- actions and have to be accounted for in a quantum-relativistic model of hadrons.

The framework which seems to be most appropriate for the description of hadrons and their interactions is called ‘S-matrix theory’. Its key concept, the ‘S matrix’, was originally proposed by Heisenberg in 1943 and has been developed, over the past two decades, into a complex mathematical structure which seems to be ideally suited to describe the strong interactions.

The S matrix is a collection of probabilities for all possible reactions involving hadrons. It derives its name from the fact that one can imagine the whole assemblage of possible hadron reactions arranged in an infinite array of the kind mathe- maticians call a matrix. The letter S is a remainder of the original name ‘scattering matrix’ which refers to collision-or ‘scattering’-processes, the majority of particle reactions. In practice, of course, one is never interested in the entire collection of hadron processes, but always in a few specific reactions. Therefore, one never deals with the whole S matrix, but only with those of its parts, or ‘elements’, which refer to the processes under consideration.

These are represented symbolically by diagrams like the one above which pictures one of the simplest and most general particle reactions: two particles, A and B, undergo a collision to emerge as two different particles, C and D. More complicated processes involve a greater number of particles and are represented by diagrams like the following.

It must be emphasized that these S-matrix diagrams are very different from the Feynman diagrams of field theory. They do not picture the detailed mechanism of the reaction, but merely specify the initial and final particles. The standard process A+B+C+D, for example, might be pictured in field theory as the exchange of a virtual particle V, whereas in S-matrix theory, one simply draws a circle without specifying what goes on inside it. Furthermore, the S-matrix diagrams are not space-time diagrams, but more general symbolic representations of particle reactions. These reactions are not assumed to take place at definite points in space and time, but are described in terms of the velocities (or, more precisely, in terms of the momenta) of the incoming and outgoing particles.

This means, of course, that an S-matrix diagram contains much less information than a Feynman diagram. On the other hand, S-matrix theory avoids a difficulty which is characteristic of field theory. The combined effects of quantum and relativity theory make it impossible to localize an interaction between definite particles precisely. Due to the uncertainty principle, the uncertainty of a particle’s velocity will increase as its region of interaction is localized more sharply,* and consequently, the amount of its kinetic energy will be increasingly uncertain. Eventually, this energy will become large enough for new particles to be created, in accordance with relativity theory, and then one can no longer be certain of dealing with the original reaction. Therefore, in a theory which combines both quantum and relativity theories, it is not possible to specify the position of individual particles precisely. If this is done, as in field theory, one has to put up with mathematical incon- sistencies which are, indeed, the main problem in all quantum field theories. S-matrix theory bypasses this problem by

specifying the momenta of the particles and remaining suffi- ciently vague about the region in which the reaction occurs. The important new concept in S-matrix theory is the shift of emphasis from objects to events; its basic concern is not with the particles, but with their reactions. Such a shift from objects to events is required both by quantum theory and by relativity theory. On the one hand, quantum theory has made it clear that a subatomic particle can only be understood as a mani- festation of the interaction between various processes of measurement. It is not an isolated object but rather an occur- rence, or event, which interconnects other events in a particular way. In the words of Heisenberg:

[In modern physicsl, one has now divided the world not into different groups of objects but into different groups of connections . . . What can be distinguished is the kind of connection which is primarily important in a certain phenomenon . . . The world thus appears as a complicated tissue of events, in which connections of different kinds alternate or overlap or combine and thereby determine the texture of the whole.’

Relativity theory, on the other hand, has forced us to conceive of particles in terms of space-time: as four-dimensional patterns, as processes rather than objects. The S-matrix approach com- bines both of these viewpoints. Using the four-dimensional mathematical formalism of relativity theory, it describes all properties of hadrons in terms of reactions (or, more precisely, in terms of reaction probabilities), and thus establishes an intimate link between particles and processes. Each reaction involves particles which link it to other reactions and thus build up a whole network of processes.

A neutron, for example, may participate in two successive reactions involving different particles; the first, say, a proton and a z-, the second a Z- and a K+.The neutron thus inter- connects these two reactions and integrates them into a larger process (see diagram (a) opposite). Each of the initial and final particles in this process will be involved in other reactions; the proton, for example, may emerge from an interaction between a K+ and a A (see diagram (b) above) ; the K+ in the original reaction may be linked to a K-and a n0; then- to three more pions. The original neutron is thus seen to be part of a whole network of interactions; of a ‘tissue of events’,-all described by the S matrix. The interconnections in such a network cannot be determined with certainty, but are associated with probabilities.

Each reaction occurs with some probability, which depends on the available energy and on the characteristics of the reaction, and these probabilities are given by the various elements of the S matrix.

This approach allows one to define the structure of a hadron in a thoroughly dynamic way. The neutron in our network, for example, can be seen as a ‘bound state’ of the proton and the n- from which it arises, and also as a bound state of the C- and the K+ into which it disintegrates. Either of these hadron combinations, and many others, may form a neutron, and consequently they can be said to be components of the neutron’s ‘structure’. The structure of a hadron, therefore, is not understood as a definite arrangement of constituent parts, but is given by all sets of particles which may interact with one another to form the hadron under consideration.

Thus a proton exists potentially as a neutron-pion pair, a kaon-lambda pair, and so on. The proton also has the potential of disintegrating into any of these particle combinations if enough energy is available. The tendencies of a hadron to exist in various mani- festations are expressed by the probabilities for the corres- ponding reactions, all of which may be regarded as aspects of the hadron’s internal structure.