Aside from the group-theoretical, the spinor-field, and the model (Sakata) approaches to the systematization of particles, a new method was developed in the last few years, based on the dispersion relations and the analytical properties of scattering amplitudes, i.e. , probabilities of processes.
A number of promising results have been worked out by Goldberger, Nambu, Cini, N. N. Bogolyubov and colleagues, Chew, Frautschi, Yu. M. Lomsadze, A. M. Brodskii, Domokos, and Gribov, and particularly by S. Mandel'shtam, M. Regge and others, by means of the dispersion relations and calculations based on complex variables of the energy and the linear and angular moments. Briefly, dispersion theory excludes the possibility or the need of describing in detail the interaction and behavior of particles by means of a Lagrangian or a Hamiltonian; it proceeds to treat the scattering probability of particles, for instance, on the basis of the formalism of the scattering matrix S (Wheeler-Heisenberg). The definition of the scattering matrix rests on the most general requirements of invariance, causality, and unitarity. By means of the S matrix the asymptotic scattering of a wave is immediately derived from the incident wave, omitting a detailed description of the interaction. A program of this kind was first formulated by Heisenberg in the forties and was later revived in the form of a dispersion-relation theory. The decisive move was made when it was realized that the S matrix and the scattering amplitudes must be treated as analytical functions of the energy of angular momentum.
The dispersion relations provided the means of drawing up equations for the scattering probabilities of pions and nucleons in various directions, and some important formulas for the scattering probabilities at extremely high energies. The hope has arisen, most clearly expressed by Chew, of linking together the ordinary elementary particles and the resonons, which are all equally treated as poles in terms of the theory of analytic S-matrix functions. This theory gives rise again, though on a different basis, to the concept of excited states of particles. Thus, for instance, the particles in the family formed by the nucleon and the nucleon resonons, N*, N** N***, N**** , all lie on a single trajectory of Regge poles. The masses of
the resonons are obtained given the appropriate spin values. Chew's rather extreme program stands in sharp opposition to the old Lagrangian and all ordinary field-theoretical methods. But in spite of the fact that some of the notable successes of particle theory in recent times have been associated with dispersion theory, analytic properties, and Regge poles, we still have doubts about the possibility of constructing a particle theory divorced from field-dynamic conceptions, as do Schwinger, Heisenberg, Feynman, Sakurai, Nambu, and others, as well. At any rate, this field of inquiry is evolving quite rapidly, mainly due to the development produced by the discovery of the resonons, of new particles, and of many reactions.
In concluding, we must return to gravitation once more. Within nonlinear spinor theory it is in principle possible to derive the gravitons as quanta of a weak transverse gravitational field, though it is doubtful that the total field 4, can be obtained. On the other hand, by proceeding from various geometrical considerations (particularly, Wheeler's geometrodynamics) it might be possible to derive the electromagnetic field and perhaps other boson fields as well (the K and K mesons), but for the time being there are no chances that spinors will be obtained.
It thus seems that our representation of physical reality must, for the time being, remain dualistic—on the one hand there is space (curved or perhaps twisted) and time, and on the other, intimately connected with them, "ordinary" matter in the form of various elementary particles and "resonons"; the latter might perhaps be excited or compound states of a few basic particles or of some unique (spinor-type) "protomatter".
The graviton (the transverse part or waves of the gravitational field) would occupy an intermediate position, forming some sort of "ripples" on the fundamental space-time network and being at the same time capable of transforming into ordinary matter.
Modern physics has now at its disposal giant accelerators, recording devices on Earth and in space, and highly sophisticated computers. It is bound to uncover new horizons in our knowledge of matter and the universe, and to contribute to the technical achievements of civilization.