Does this mean that the geometrized picture of the world is to be committed to history, like its predecessors, the classical-mechanical and the electromagnetic world pictures? It appears that this question must be answered in the negative. At the present time, it would no longer occur to anyone to raise the possibility of constructing a purely electromagnetic unified picture of matter, it being manifestly too restrictive and altogether impracticable, whereas a number of leading scientists are still trying to evolve a geometrized unified field theory.
A particularly noteworthy attempt to arrive at a geometrized field theory, proceeding from the wealth of information available on the elementary particles and based on quantum field theory, was made in recent years by the American physicist J. A. Wheeler. His theory, which he regards as a continuation of the works of Riemann, Clifford and Einstein, Wheeler called "geometrodynamics". The field of geometrodynamics constitutes a serious attempt to construct a unified world picture. Work on it has still not gone beyond Princeton University, but it has already drawn the attention of scientific circles, for instance, at recent international conferences on gravitation (at Royaumont in 1959 and, to a lesser extent, at Jablonna Warsaw in 1962), and in the scientific literature of the last few years.
Geometrodynamics, like any other theory of matter, can be realistic only if it takes into account quantum phenomena. Quantum geometro dynamics, following behind the classical theory, is just making its first steps. It stresses the topological aspects of geometry and emphasizes the fundamental role which is bound to be played by quantum fluctuations (random deviations) of the metric.
Fluctuations of the vacuum necessarily take place in any kind of field. It is, for instance, well known that when the vacuum fluctuations of the photon field and electron-positron field "knock off" an electron, they shift slightly the energy levels of the electron, as Lamb has shown for the electron in the hydrogen atom (1947). The theory of the Lamb shift has been developed to a high degree of precision and has been found in excellent agreement with experiments. In the course of developing the theory some methods typical of modern quantum field theory were worked out; for instance, various singular functions were used, entirely relativistic calcu lations were made, and the infinite electromagnetic mass, and in general any field mass, of particles (and also the field portions of the charge and other coupling constants) were isolated by means of the "renormalization" procedure.
The necessity of taking into account the fluctuations of a weak gravita tional field was pointed out in some of our papers, and also recently by D. I. Blokhintsev. Wheeler goes even farther than that and makes the daring assumption that there may be fluctuations of all the gravitational potentials and of the metrical tensor, and points out that at very small distances of the order of the critical universal gravitational length r - n cm, where G is the gravitational constant, such fluctuations must attain con siderable values. This may cause the topology of space to change, with the formation of multiply connected regions, gaps, holes connected by "shafts", etc. The suggestion as to the necessity of fluctuations in the metric
or in the gravitational field and the need for taking into account topological factors, which have been disregarded before, seem to be the most significant aspects of geometrodynamics. The theory also puts forward the hypothesis that the "bare" electron, without its surrounding field, which is associated, according to quantum theory, with nonrenormalized mass and charge, may be identified with a hole; its antiparticle, the positron, is then identified with the hole at the other end of the "shaft". However, quite independently of the incompleteness of classical geometrodynamics, there is in quantum geometrodynamics a fundamental difficulty, stemming from the fact that, proceeding from a metric described by a second-rank tensor (in other words, from a Bose gravitational field, whose waves have a spin S=2, in fractions of it is very difficult to go over to fermions, in particular to electrons and neutrinos, represented by spinors (which maybe described as sem ivectors or tensors of and possessing half-integral spin, Let us remark at this point that, on the other hand, modern unified nonlinear spinor theory considers fermions as basic, although it is not yet capable of describ ing the total gravitational field. In any case, Wheeler's interesting and stimulating work on geometrodynamics has its positive side, in that it endeavors to link the theory of elementary particles with the processes occurring in stars, galaxies, and all the known universe. Wheeler also considered "geons" —hypothetical giant concentrations of the electromag netic field, or of neutrinos, or of the gravitational field, held together by their own gravity. We may note in this connection that Wheeler entertains the hypothesis, as we do, that it may be possible for electron-positron and other particle pairs to be transformed not only into photons but also into gravitons (and also for proton-antiproton pairs to change into a mesons); moreover, he believes, conversely, that it may be possible for the gravi tational field to be transformed into ordinary matter. Somewhat earlier we calculated, together with A. A. Sokolov and A. M. Brodskii, the probability of an electron-positron pair (or more precisely, of two scalar particles) transforming into two gravitons, as defined by the effective cross section: where the gravitational radius of the particle, E is the energy, v is the velocity of the electron, and c is the velocity of light. Afterward, I. Piir (at Tartu) computed the probability of transformation of photons into gravitons, and Wheeler and Brill estimated the probability of trans formation of a neutrino-antineutrino pair into gravitons. More precise calculations were recently carried out by Vladimirov and Feynman. Con siderable attention is devoted in these papers to various cases of emission of gravitational waves (in the motion of stars, or of whole galaxies) and their subsequent behavior is analyzed, specifically with respect to the expansion of the universe.