After our brief account of the group-theoretical approach, let us turn to the attempt to produce a unified field picture of particles and fields. The basic idea was originated by Louis de Broglie, who proceeded from a simple spinor wave function 'I), describing the particle of least nonvanishing angular momentum, i.e. , with spin If we combine these wave functions under given supplementary conditions, we obtain by such a "fusion" all the other possible wave functions of particles with the spins 0, I, 2... etc. Bycombiningthe two angular momenta +'/2 and we obtain 0, by combining the two angular momenta and we obtain 1 (since the spins may be oriented only parallel or antiparallel), etc. By means of the coalescence method it is possible to combine two Dirac equations, each describing a spinor-type particle (fermion) of spin thus obtaining the Klein-Gordon and the Proca equations, and in the special case of a vanishing rest mass, Maxwell's electrodynamic equations; in principle, therefore, photons could be constructed out of neutrino-anti neutrino parts. The ideas of de Broglie's neutrino theory of light have been developed by Kronig, Jordan, and A. A. Sokolov.
A weak point in the coalescence method is the absence of any forces responsible for the coalescence. It fails to explain, for instance, what would cause the neutrino functions to transform into the electromagnetic field. An answer to this has been attempted by a unified nonlinear spinor theory of matter. If one postulates for matter some unique spinor field, then this field would be able to interact only with itself. This leads to the appearance of the nonlinear terms in Dirac's equation, which we introduced for the first time in 1938 and later examined in more detail together with A. M. Brodskii. If it is required that not only relativistic, Lorentz invariance should hold, but also invariance with respect to isotopic rotations (Pauli-Gyurshi transformations) as well as transformations conserving the baryon number (Salam-Touschek transformations), then, as Heisenberg and Pauli have shown, it is necessary to choose the pseudo vector term from the set of possible corrections to Dirac's equations. Heisenberg further suggests discarding the mass term, since mass should directly follow from a unified theory. We then obtain the following basic nonlinear spinor equation of a unified theory of matter, which we simply give in symbolic form, without any special explanations: where are Dirac matrices, h is Planck's constant, c is the velocity of light, is a new constant denoting the minimal length, and 41 is the spinor wave function.
By formulating some new field-quantization rules, Heisenberg and his colleagues obtained approximate solutions to these equations and derived a basic particle, the nucleon, having a finite mass, and also some mesons with lower masses, which could be associated with a and K mesons. The theory also yields the magnitude of the electric charge and of the Fermi constant of weak interaction between four fermions (for example, the neutron, proton, electron, and antineutrino in beta-decay). Although the
theory does not yet provide precise values for the masses of particles or for the coupling constants, it is likely that quite a plausible unified picture of known matter is beginning to take shape. In summing up the proceedings of the International Conference on Elementary Particles of 1959 in Kiev, I.E. Tamm correctly pointed out that the nonlinear spinor theory looks like one of the most promising trends in particle theory. Some subsequent interesting work on nonlinear spinor theory was done by D. F. Kurdgelaidze, Ya. I. Granovskii, and by Marshak-Okuba and Nambu, who found some points of similarity between this theory and the Bardin-Bogolyubov super conductivity theory. As we mentioned in the first section of this article, according to V. I. Rodichev the nonlinear term in the Dirac equation follows in a straightforward manner if one considers spinor motion not in ordinary flat or curved space, but in a twisted space.
A second version of unified particle theory takes as its point of departure the proposal of Fermi and Yang to consider the it meson as formed from a nucleon and an antinucleon by means of some as yet unknown forces acting at very small ranges, i. e. : The enormous binding energy involved "swallows up" almost all the mass of the two nucleons, leaving only the mass of the pion. In this case, again, a boson is constructed from two fermions.
Goldhaber proceeded from the p, p, n, n, k, and k particles. The proposal that proved most successful was Sakata' s, further developed by M. A. Markov, B. L. Okun', and others, who took as point of departure the proton, neutron, and A particle and the three corresponding antiparticles. It is then possible, by combining these basic particles, to obtain all the pions, K mesons, and hyperons. For example, In this case also the nature of the binding or coalescence forces remains unex plained. A minimum of three basic particles is necessary to ensure the presence of the fundamental properties of charge, isospin, and strangeness. Obviously, it is necessary to start off with "rotating", spinor-type particles or fermions, because if "rotation" is not provided to begin with, there is no way of deriving it. These attempts, as the coalescence theory and the nonlinear spinor theory, can be seen to revive in their way the old ideas of Helmholtz and Kelvin, who tried in the middle of the 19th century to construct matter out of hypothetical ether vortices. An attempt was recently made by Sakata and his coworkers at the University of Nagoya to include the leptons in the above model too. They proceeded from the leptons e , v, p (in the last version of Katayama and others, and some "baryon" field B. By combining every lepton with the field B, one obtains the basic heavy particles of the old Sakata model. This also achieves the correspondence between the baryons (p, n, A) and leptons (v, e, µ), noted by Marshak, Gamba, and Okuba; the same symmetry is also realized in the nonlinear spinor theory of particles. We may remark that Sakata's scheme follows directly from group-theoretical considerations, within the SU3 group.