Meetings and Conferences

All-Union Conference on Quantum Electrodynamics and the Theory of Elementary Particles

In Moscow, from March 31 to April 7, 1955, the All-Union Conference on Quantum Electrodynamics and the Theory of Elementary Particles was held at the Academy of Sciences of the USSR. The conference, organized by the Division of Physico-Mathematical Sciences of the Academy of Sciences of the USSR, attracted wide attention among Soviet scientists. More than 500 people from Moscow and other cities of the country, as well as foreign guests—representatives of the Chinese People’s Republic, Hungary, Poland, Czechoslovakia, the German Democratic Republic, Bulgaria, and Sweden—took part in it.

The theory of elementary particles, to which the work of the conference was devoted, is the central problem of modern physics, determining its development as the science of the structure of matter. During the conference there were 11 plenary sessions, at which 54 scientific reports and communications devoted to various questions of quantum electrodynamics and the theory of elementary particles were heard and discussed.

Opening the conference, I. E. Tamm, in a brief introductory address, characterized the present state of the theory of quantum fields and pointed out the well-known difficulties encountered by existing theoretical constructions. In this connection he emphasized the circumstance that, alongside the further development of existing theories, it is necessary to devote great attention to the construction of new theories which, in his opinion, solve the difficulties mentioned and are built on fundamentally new foundations.

In a survey report devoted to the most important problems of quantum field theory, L. D. Landau noted a number of difficulties of modern field theory, connected in particular with the point-like character of the interaction and the presence of divergent expressions in the theory. In a consistent treatment, the point interaction in electrodynamics must be introduced as the limit of a “smeared” interaction as the radius of “smearing” decreases. Charge renormalization is then expressed in the fact that the physical charge \(e\) differs from the constant \(e_1\) that appears as the coefficient in the interaction. The quantity \(e_1\), therefore, proves to depend on the radius of the interaction. If the treatment is carried out by the method of perturbation theory, then the results obtained are valid under the condition \(e_1^2 \ln(\Lambda^3/m^2) \ll 1\), where \(m\) is the electron mass and \(\Lambda\) is a constant connected with the “smearing” of the interaction. However, as L. D. Landau, A. A. Abrikosov, and I. M. Khalatnikov showed, all series can be summed under the much weaker assumption: \(e_1^2 \ll 1\). From the formulas thereby obtained it follows that for sufficiently large \(\Lambda\), \(e\) tends to 0, however large \(e_1\) may be; this means that

point interaction in electrodynamics ultimately leads to the absence of any interaction.

A similar analysis may also be carried out for weak pseudoscalar meson coupling, and in this case analogous formulas for the renormalization of the coupling constant arise. From these formulas, and from the qualitative consideration of pseudovector coupling, it follows that one of the possibilities in the modern theory of meson interaction consists in the fact that the intensity of meson couplings grows without bound with increasing energy. In this case the principal task of the theory is to construct a theory of strong coupling at high energies. Another possibility is that point interaction, by its very nature and with an unlimited increase in its intensity, may lead to the absence of interaction. In this case the construction of meson theories is possible only as a result of abandoning point interaction, i.e. of the entire existing methodology.

After L. D. Landau’s report and its discussion, a series of communications was heard devoted to quantum electrodynamics and, in general, to quantum field theory. A. A. Abrikosov, in his report “On the Infrared Catastrophe in Quantum Electrodynamics,” touched on a number of questions arising in electrodynamics when photons of low energies are considered. The consideration was carried out with the aid of the general equations of quantum electrodynamics; under certain conditions Green’s functions and the vertex part $\Gamma_\mu(p;q;l)$ were calculated. In the consideration of the emission of quanta, the formalism of quantum electrodynamics leads to Poisson’s formula. The question of multiphoton annihilation of positronium was also considered. It turns out here that two of the annihilation quanta are the principal ones, i.e. they practically do not differ from those that arise in two-quantum annihilation. The emission of the remaining quanta has little effect on the basic process. Certain classical formulas were obtained, but with additional restrictions on the domain of applicability. It was pointed out that the formulas for multiphoton annihilation obtained by Fultz are erroneous.

In the report by L. P. Gor’kov and I. M. Khalatnikov, the question of finding the asymptotic form of the Green functions of a spin-zero particle interacting with the electromagnetic field was discussed. The asymptotics of the Green functions is found from the solution of integral equations in an approximation in which all terms of the perturbation-theory series are taken into account that give, for large impulses, a logarithmic contribution of degree equal to the order of perturbation theory. The consideration is carried out with the aid of first-order equations ($\beta$-formalism), and the problem is simplified by a special choice of the photon Green function, when its transverse and longitudinal parts are equal to one another. The results can then be written for an arbitrary longitudinal part of the photon Green function by means of the corresponding gradient transformation.

V. V. Sudakov described a method that makes it possible to carry out calculations of Feynman integrals in an arbitrary order of perturbation theory with logarithmic accuracy. Essential for this method is the introduction of certain auxiliary variables. As an example, this method was applied to obtaining the sum of the perturbation-theory series determining the vertex part $\Gamma_\mu(p;q;l)$ under the conditions $e^2\ln|l^2/p^2|\,|\ln|l^2/q^2||\gg1$, but $e^2\ln|l^2/m^2|\ll1$.

In B. G. Bersch’s report it was indicated that a stronger nonelectromagnetic interaction does not affect asymptotically the contribution of mesons to the electromagnetic polarization of the vacuum. Nonelectromagnetic interactions under these conditions do not lead to the formation in the particle of a form factor that changes its interaction with the electromagnetic field. This result is organically connected with the renormalizability

of the corresponding interactions. The requirement of renormalizability by itself already imposes substantial restrictions on the character of possible corrections to the Green’s function. The appearance of a form factor would at the same time signify a violation of the property of renormalizability of the theory.

I. M. Gelfand and R. A. Minlos presented a simple method for introducing continual integrals into the theory, starting from Fock’s representation for a field functional and using the Schrödinger equation. For simplicity the consideration is carried out for the Bose field, but it can also be generalized to the case of anticommuting Fermi fields. It is noted that continual integrals used recently in theoretical physics are well known in mathematics and are applied, in particular, in probability theory.

In the report by N. N. Bogolyubov, a method was presented for finding Green’s functions in the interaction representation. In this connection the relation between vacuum expectation values for various quantum fields and the Feynman integral was discussed. The formalism developed was applied to finding the Green’s function of a single fermion, and a generalization was made to the case of several particles. The formalism employed makes it possible to obtain concrete results rather simply, as was demonstrated with the example of the infrared catastrophe (the Bloch—Nordsieck formula).

The report by G. Tsel’na (Sweden), “The Mathematical Structure of T. D. Lee’s Model of a Renormalizable Field Theory,” was devoted to a discussion of the theory proposed by T. Lee, in which three interacting fields are considered—two fermion fields, treated nonrelativistically, and one boson field carrying the interaction. The equations of the theory admit an analytic solution, from which a formula is obtained for the renormalization of the interaction constant, leading to the appearance of imaginary quantities. In view of this, Lee’s theory was supplemented by the introduction of a certain cutoff parameter making the divergent integrals finite. The formulas obtained after this lead to the conclusion that the renormalized charge is equal to zero, which is analogous to the conclusion drawn by L. D. Landau. Some features of the theory considered were also discussed.

In the report by Yu. A. Gol’fand, the question of constructing propagation functions by the method of quasipoles was discussed. In some respects this method is simpler than existing methods, thanks to which it is possible to advance considerably in the study of the exact expression for propagation functions. The method is based on secondary quantization of “virtual” particles, i.e. particles whose 4-momentum is not connected by the relation \(p^2=m^2\). A technique is developed for “disentangling” the operators entering into the theory, thanks to which it is possible to represent the expression for the propagation function in the form of an infinite multiple integral.

In the report by E. S. Fradkin, a derivation was given of functional equations for the \(S\)-matrices of interacting Fermi and Bose fields. A solution of the functional equations was found in the form of infinite multiple integrals. An operator form of the solution of the functional equations under consideration was obtained in the form of a certain exponential operator depending on the functional derivatives with respect to the sources of both fields, acting on the solution of the system of the same functional equations in the absence of interaction.

The report by A. D. Galanin, B. L. Ioffe, and I. Ya. Pomeranchuk was devoted to the consideration of systems of covariant equations and to carrying out in them the renormalization of the proper mass and charge. The derivation of the Schwinger equations is made by considering the arbitrary diagrams in perturbation theory. By expanding these equations in a series in the external sources, an exact system of covariant equations is obtained, relating the Green’s functions for a system with several fermions and bosons. It is shown that in the system of equations obtained one can carry out the renormalization

of the proper mass and charge, so that the renormalized system, when solved, will not contain infinities of this type.

E. S. Fradkin obtained an infinitely coupled system of renormalized equations for Green’s functions in the presence of external sources of the Bose field. The method developed by him makes it possible, more simply than in comparison with other works, to obtain a number of results. A method is given for truncating the coupled equations in which the equations obtained remain completely renormalized. The asymptotic form of the equations obtained has been found, expressed in terms of the experimental charges and masses.

N. P. Klepikov obtained a system of equations in variational derivatives determining the vacuum functional (i.e. the vacuum average of the \(S\)-matrix) up to an arbitrary constant factor. All Green’s functions for systems consisting of electrons, positrons, and photons can be obtained from the vacuum functional by functional differentiation with respect to the sources of the electron-positron and electromagnetic fields. Solutions of the equations obtained are found in the form of functional integrals over the field sources.

D. V. Shirkov devoted his communication to questions of renormalization in quantum field theory. The construction of the scattering matrix is carried out starting not from the Hamiltonian formalism and the Schrödinger equation, but from explicitly formulated conditions of causality, covariance, unitarity, and the correspondence principle; moreover, a certain function is introduced into the formalism, equal to unity in the region where the interaction is included and equal to zero where the interaction is absent. The developed formalism makes it possible to investigate a number of questions connected with the introduction of renormalization terms into the Lagrangian.

The report by I. Ya. Pomeranchuk was devoted to a generalization of Ward’s theorem for particles with spin \(0\) to the case where the wavelength of the electromagnetic field is arbitrary, but the field itself is lightlike, i.e. \(k^2 = 0\).

Yu. V. Novozhilov considered the question of causal operators in quantum field theory. Causal operators are understood to mean operators that satisfy the condition of chronological ordering associated with the principle of causality. Causal operators also satisfy the usual conditions imposed on operators in quantum field theory. One can construct a field theory with causal operators and introduce nonlocal interaction without violating the condition of compatibility.

V. L. Bonch-Bruevich discussed the conditions of reality in quantum field theory. Owing to the inevitable “smearing” of a particle (because of its interaction with zero-point oscillations), it is natural to regard real particles as non-pointlike. For stable particles the condition of “reality” consists in requiring that all vacuum corrections already be contained in the “initial” propagation functions. The reality conditions make it possible to determine uniquely the values of arbitrary constants that appear in integrating singular functions.

The report by L. D. Landau and I. M. Khalatnikov was devoted to solving the problem of the gradient transformation of the Green’s functions and vertex parts for charged particles interacting with the electromagnetic field.

In the communication by A. I. Akhiezer and R. V. Polovin, results were presented for the calculation of the cross section for electron-electron scattering with allowance for radiation corrections caused by the interaction of the electrons with the zero-point oscillations of the electromagnetic field and by the polarization of the electron-positron vacuum. The radiation corrections were taken into account up to the fourth order of perturbation theory (inclusive). In the nonrelativistic region the radiation corrections tend to zero

together with the velocity of the electron. In the relativistic region the radiative corrections are, in order of magnitude, \(1/137\) of the main effect.

V. G. Bar’yakhtar, S. V. Peletminskii, and P. I. Fomin reported on calculations, carried out in fourth order of perturbation theory, of the Green function in the region of large impulses. The regularization was performed by Dyson’s method, with the initial integration over a finite invariant region. Overlapping divergences were eliminated by Salam’s method.

V. F. Aleksin and D. V. Volkov reported on an investigation of certain effects connected with the polarization of the vacuum by scalar mesons in an electromagnetic field. Corrections to the Coulomb and Lamb shifts of levels in atoms were found. The polarization tensor \(I_{\mu\nu\lambda\sigma}\), determining light-by-light scattering, was calculated. A correction to the Lagrange function for weak, slowly varying fields, allowing for nonlinear effects, was found. The cross section for coherent scattering of light by a Coulomb field at small angles was calculated.

The above were the reports heard at the first four sessions and devoted mainly to quantum electrodynamics and to questions of the quantum theory of the meson field related to it in their methods. At the next seven sessions a number of other directions and problems in the theory of elementary particles were discussed.

Thus, the fifth session was devoted to the so-called Tamm–Dancoff method. The corresponding survey report entitled “The Method of Cutting Off Equations by the Number of Particles in the Theory of Mesons,” after V. P. Silin, V. Ya. Fainberg, and himself, was given by I. E. Tamm. In order to clarify to what extent the modern theory of mesons can correctly describe phenomena occurring at not too large energies, it is necessary to use methods of calculation that do not resort to perturbation theory. The effective solution of approximate integro-differential equations of the theory encounters specific difficulties which are absent when considering a system of three-dimensional equations, cut off with respect to the number of virtual particles, obtained from covariant equations in the interaction representation (Chung–Dyson). Application of the first approximation of the theory (the so-called “old” Tamm–Dancoff method) to meson scattering by nucleons (Fubini and others) has given very encouraging results (a resonance in a state with isotopic and ordinary spins equal to \(3/2\)).

The authors considered scattering in the next approximation, taking into account the so-called minus-particles. The principal unresolved problem of the indicated method consists in finding a consistent method of renormalizing the equations. If in the approximation considered one performs mass renormalization by analogy with perturbation theory, then difficulties arise in the renormalization of the meson mass (in the integral equations “false poles” appear). These difficulties are absent, however, contrary to assertions in the literature, in the renormalization of the nucleon mass. Possible methods of renormalizing vertex parts were also discussed.

The report by V. Ya. Fainberg was devoted to a more detailed consideration of the question of renormalizations in the Tamm–Dancoff method. As is known, renormalization is especially simple to carry out for the case of the four-dimensional formulation of the equations. A connection was established between the covariant generalizing equations and the three-dimensional equations obtained in the Tamm–Dancoff method. Using the example of an equation of the Bethe–Salpeter type, it was demonstrated how the correspondence is established between equations of this type and the Tamm–Dancoff equations, and a number of questions connected with the renormalization of the equations under consideration were clarified.

In the report by E. L. Feinberg and D. S. Chernavskii, questions connected with the problem of the stability of the deiton in meson

theory. Within the framework of a nonrelativistic approximation for nucleons, a method was proposed and applied for separating out self-energy terms, not presupposing the smallness of the coupling constant. It is based on the following requirement: for two mutually distant nucleons at rest the energy must be equal to their phenomenological rest energy. From an equation for the amplitudes of states with different numbers of mesons one obtains variational equations for the values of the Hamiltonian of the system; in carrying out the variation an additional condition is imposed expressing the requirement indicated above. Owing to this, the resulting equations differ from the corresponding Tamm—Dancoff equations. In particular, the term having the meaning of the interaction potential of the nucleons decreases with distance (for vector interaction between nucleons and mesons) somewhat more slowly than \(1/r^3\). This ensures the stability of the deuteron. Quantitative estimates were obtained by applying Ritz’s direct variational method. The results appear encouraging and prompt the construction of a more complete theory.

M. A. Markov gave a survey report devoted to the theory of nonlocal fields. The report noted that at the present time, within the framework of the Hamiltonian method, there is not a single internally consistent attempt to introduce into the theory a form factor of elementary particles. The introduction of a form factor leads to a violation of the conditions for the existence of the mathematical apparatus of the theory. Attempts to confine violations of relativity to a small region also have not led to positive results. The author attempts to introduce into the theory a “length” by means of new commutation relations for field operators, which proves equivalent to the idea of introducing internal degrees of freedom of elementary particles. Equations of elementary particles with internal degrees of freedom were earlier considered by Tamm and Ginzburg for the purpose of describing excited states of nucleons. More recently, searches have revived for similar equations, though here there is a difficulty—namely, to write equations with an increasing mass spectrum in the absence of an infinite expression for the lowest state. It is possible, however, to indicate equations with an increasing spectrum of eigenmasses which contain form factors connected with the internal wave function of particles. These form factors could in principle be deformable under the influence of external interactions, in accordance with relativity. The report also discussed various questions connected with the introduction of dynamically deformable form factors.

The report of B. V. Medvedev was devoted to certain questions of quantum field theory with nonlocal interaction. In connection with the difficulties which exist regarding the unitarity of the \(S\)-matrix in the usual theory, Schwinger—Bogoliubov’s method of direct construction of the \(S\)-matrix is developed; it makes it possible to construct an \(S\)-matrix whose unitarity is obvious. Such an \(S\)-matrix is uniquely determined by the form of the interaction Lagrangian. This method of constructing the \(S\)-matrix makes it possible to pass directly to the matrix \(S(\sigma)\), which solves the problem of constructing canonical variables in quantum field theory with nonlocal interaction.

In the report of M. M. Mirianashvili it was indicated that there exists a connection between a nonlocalized theory and the relativistic theory of two particles. The equation for two particles can be reduced to the form of a nonlocalized equation for one particle, the interaction operators determining the character of the corresponding form factor. By means of an alternative notation for the equations of two particles, in which instead of the product of two Dirac operators their sum enters, it is shown that this equation is equivalent to an equation for a nonlocalized particle; here the role of additional conditions determining the mass spectrum is played by the presence of an operator of interaction of the two particles.

V. L. Ginzburg gave a report on relativistic wave equations with a mass spectrum. The investigation of possible relativistic wave equations

MEETINGS AND CONFERENCES

equations for particles that can be in states with different values of mass, spin, and charge, is of interest in connection with: 1) attempts to regard at least some of the known particles as states of a certain “generalized particle,” whose mass and spin can be quantized; 2) the relativistic treatment of excited (isobaric) states of nucleons; and 3) the desire to introduce into the theory new variables necessary for constructing a nonlocal field theory. The report discussed relativistic equations for a particle that can have spins \(1/2\) and \(3/2\). The use of such equations for a semiphenomenological description of meson scattering on nucleons and of other effects is limited, in the author’s opinion, by the large width of the isobaric level. In the case of infinite-dimensional relativistic equations there is a difficulty associated with the appearance of a falling mass spectrum. This difficulty is absent, however, if certain mass values are imaginary. In this case the solutions corresponding to imaginary masses may be discarded as a result of a certain finiteness condition on the solutions. The question of the suitability of the equations obviously cannot be settled without investigating the behavior of the particle in external fields. The report also discussed other questions related to this subject.

The report by Yu. M. Shirokov was devoted to certain new possibilities in the relativistic theory of elementary particles. The main point is that, from the general conditions of relativistic invariance taking account of spatial translations, equations are derived for elementary particles of any specified spin.

The survey report by I. E. Tamm, Yu. A. Gol’fand, G. F. Zharkov, L. V. Parijskaya, and V. Ya. Fainberg was devoted to a semiphenomenological isobaric theory of meson interactions with nucleons. Since the construction of a consistent theory of mesons will evidently require considerable time, it is expedient to try to construct a semiphenomenological theory that generalizes the experimental data and makes it possible to predict the results of new experiments. On the basis of recently discovered features of the interaction of mesons with nucleons (the presence of a quasirezonance in the scattering and photoproduction of mesons), it is possible to develop a theory of this kind, based on introducing into consideration an excited isobaric state of nucleons with spin and isotopic spin \(I=J=3/2\). The authors developed a quantitative form of the isobaric theory. Relativistic equations of motion of nucleons and mesons are considered, containing four quantities to be determined by comparison with experiment (the excitation energy of the isobar and three coupling constants). Each process is considered only in the first nonvanishing approximation, with self-energy effects neglected; the influence of these effects and of higher approximations is assumed to be taken into account by including the isobar’s own state and by an appropriate choice of constants. This theory quantitatively describes, within the errors of experiment, all available data on the cross sections for scattering of \(\pi\)-mesons by nucleons and on their angular distribution, up to energies of the order of \(400\) Mev. With the aid of constants determined from the scattering data, the nuclear forces were calculated in the adiabatic approximation, and the deuteron and the cross section for collisions of nucleons at low energies were calculated. It is necessary here to introduce one more parameter, \(r_0\)—the cutoff radius of the nuclear forces at small distances, common to the singlet and triplet states. Satisfactory agreement with experiment was obtained. This theory will also be applied to a number of other problems.

In the communication by V. I. Ritus, the results were presented of applying the semiphenomenological isobaric theory to the effects of meson photoproduction and scattering of gamma rays by nucleons. To describe the interaction of a photon with a nucleon, in which the nucleon passes into an isobaric state, it is necessary to introduce a corresponding new coupling constant. The equations obtained are solved with damping taken into account by expanding the functions in

angular polynomial matrices. Satisfactory agreement has been obtained between the theory of photoproduction of mesons and experimental data. The theoretical consideration of gamma-ray scattering by nucleons makes it possible to predict a number of effects that can be tested experimentally.

S. Z. Belen’kii and A. I. Nikishev considered the theory of multiple meson production at energies of 1–2.2 Bev. The experimental results contradict the predictions of the statistical Fermi theory concerning the formation of one and two mesons in collisions of nucleons. The authors include in Fermi’s statistical theory selected states, whose existence is indicated by experiments on the scattering of \(\pi\)-mesons by nucleons. It is assumed that, in a collision of nucleons, particles are formed with a mass equal to 1.32 nucleon masses, with isotopic spin \(3/2\) and ordinary spin \(3/2\), which then rapidly decay into a nucleon and a meson. Otherwise the calculation is carried out on the basis of statistical methods, with conservation of energy, momentum, charge, and isotopic spin taken into account. As a result, for the ratio of the number of cases of formation of two and of one meson, quite satisfactory agreement with experiment is obtained.

Ning Hu (Chinese People’s Republic) reported on the multiple production of mesons in collisions of high-energy nucleons. When two high-energy nucleons collide with one another, the strong interaction will exist only in the case when these two nucleons are at a very small distance from one another. As a result of this weak connection with the nucleon, the outer part of the field of each nucleon may be torn off in the process of collision and turn into a meson shower. In this way one can explain the two-cone structure of showers observed in experiments. The distribution of mesons in the stripped-off field is calculated by an approximate method.

D. I. Blokhintsev gave a report on nonlinear field theory and the theory of relativity. The report considered the features of nonlinear theories in which the field has a certain scale \(\varphi_0\). It was shown that formally invariant nonlinear equations derived from a definite variational principle can be divided into two classes: A) those whose characteristics do not change in comparison with the characteristics of the linear theory (\(|c|=1\)); B) those whose characteristics differ from the characteristics of the linear equations. The equations of class A) do not enter into any contradiction with the theory of relativity or quantum mechanics, but contain difficulties that appear upon quantization of the theory. The equations of class B) can be divided into two groups: equations for which the speed of light \(|c|<1\), and equations for which \(|c|\) may be greater than 1. The latter equations are incompatible with the principle of causality in the theory of relativity, although formally they are invariant. In modern field theory, nonlinearity enters only through interaction. This interaction leads to a change in the laws of propagation of light, although a very small one (because of the polarization of the vacuum). This consequence of the modern theory is not in disagreement with the classical content of the theory of relativity.

D. D. Ivanenko devoted his report to certain questions of nonlinear field theory. The report stresses the desirability of considering nonlinear generalizations of all equations of elementary particles in connection with inducing the nonlinearities by vacuum effects. Nonlinear generalizations of the gravitational equations arise owing to vacuum effects independently of the arguments of the general theory of relativity. Nonlinear generalizations of mesodynamics are essential for the analysis of nuclear forces. The report also discussed a number of questions of nonlinear electrodynamics.

In the report by V. M. Brodskii and D. D. Ivanenko, an analysis was made of the question of the interaction of gravitation with the vacuum of particles. Considered

... necessarily leads to nonlinear vacuum additions to the gravitational equations. The report emphasizes the possibility of the transition of the gravitational field into ordinary matter (analogously to the way in which an electromagnetic field can pass into ordinary matter, forming electron–positron pairs).

The report by M. Ya. Shirokov was devoted to one variant of a nonlinear generalization of the equations of the electromagnetic field with higher derivatives.

Yu. B. Rumer, in his report, formulated the basic propositions of the five-dimensional field theory he is developing, which proceeds from a far-reaching symmetry of the equations of relativistic mechanics of a point in space, time, and action. This symmetry makes it possible to interpret the action attributed to a material point as the fifth coordinate of its configuration space and to present the problem of mechanics as a problem of five-dimensional optics.

The hypothesis of the five-dimensionality of physical space makes it possible to unite electrodynamics and vector mesodynamics into a single five-dimensional field theory.

V. Votruba (Czechoslovakia) gave a report on the symmetric description of particles and antiparticles within the framework of the theory of isotopic spin.

Ya. B. Zel’dovich, in his report, discussed isotopic invariance and meson corrections in the theory of β-decay. It was established that the β-decay of the proton and neutron is not isotopically invariant, which may manifest itself in certain effects depending on violations of the electron spins formed in β-decay. Certain questions connected with meson corrections to the β-interaction constant were discussed.

The report by L. I. Lapidus was devoted to the question of the conservation of isotopic spin and of the connection between the cross sections of various processes of meson interaction. Under the assumption that the law of conservation of isotopic spin holds exactly, without the use of approximate methods, a number of relations were obtained between cross sections for the processes of production of single mesons in the interaction of nucleons with nuclei and of the transformation of π-mesons into two or a larger number of mesons in interaction with nucleons. As a result of extending the hypothesis of isotopic invariance to systems including, in addition to π-mesons and nucleons, also hyperons and heavy mesons, consequences of conservation of isotopic spin were obtained for processes of formation of Λ- and θ-particles.

A. M. Baldin gave a report on one general relation that holds in the interaction of the electromagnetic field with nucleons and mesons. This relation consists in the fact that the matrix elements of that part of the Hamiltonian of the interaction of the electromagnetic field with nucleon and meson fields which commutes with the operator of rotation by 180° in isotopic space are considerably smaller than the matrix elements of the other part of the Hamiltonian, which anticommutes with the indicated rotation operator. Application of this relation to the reaction of elastic photoproduction of mesons on nuclei gives direct confirmation of the validity of isotopic invariance as applied to π⁰-mesons. A number of relations were also obtained on the basis of which there arises the possibility of experimentally ascertaining the existence of interaction between mesons.

In the report by Ya. P. Terletskii it is indicated that the hypothesis of isotopic spin for systems of nucleons and π-mesons may be regarded as one of the consequences of the hypothesis of neutron charge, introduced by the author earlier in connection with an attempt to reveal the simplest structural elements—polarons—from which elementary particles may consist. The hypothesis of neutron charge consists in the assumption of the existence of a new absolute conservation law, analogous to the law of conservation of electric charge, whereby the neutron possesses a unit positive neutron charge, while the pro-

the neutron has no neutron charge. The report discussed possibilities for extending the law of conservation of neutron charge to light particles, as well as questions of the systematics of elementary particles arising from the hypothesis of neutron charge.

In the report by G. A. Sokolik, a new formulation of fusion theory is analyzed, based on the theory of relativistic equations developed in the works of Gelfand and Yaglom. The theory of fusion in this formulation makes it possible to obtain equations of elementary particles taking into account internal degrees of freedom.

In the report by N. Kalitsyn (Bulgaria), the nonrelativistic equation of motion of a point electron is considered with allowance for the force of radiation friction and vacuum fluctuations. Calculation of the mean square displacement and mean square velocity shows that, when the radiation-friction energy is taken into account, the energy of interaction of the electron with the zero oscillations of the field has a finite value. In the calculation a certain cutoff factor was introduced.

The report by B. T. Geilikman was devoted to the development of the theory of strong coupling for meson fields. The problem is solved by diagonalizing the Hamiltonian in the zeroth approximation only with respect to spin variables. In the first approximation, the problem is solved for oscillations of the field oscillators. A close analogy with the theory of molecules is drawn. Next, perturbation theory is developed and a number of results are obtained for the problem of interacting nucleons. Questions connected with allowance for vacuum effects, with the possibility of renormalizations, and with the construction of an invariant theory in the case of strong coupling are discussed.

S. I. Pekar, in his presentation, set forth results obtained by him, also pertaining to the theory of strong coupling. The investigation uses methods developed in the theory of polarons. The concept of the adiabatic motion of a nucleon in slow oscillations of the meson field is introduced. The method differs from that used by B. T. Geilikman and leads to somewhat different results.

The report by Yu. V. Novozhilov, F. M. Kuni, and L. A. Khalfin was devoted to the method of intermediate coupling in meson theory. In the nonrelativistic approximation, a pseudoscalar symmetric variant of meson theory with pseudoscalar coupling is considered. For solving the problem the Fock functional method is used.

In the report by Yu. M. Lomsadze, general questions of defining sums of divergent series were discussed. If a functional series converges on some interval to an analytic function, then by the sum of the series on intervals on which it diverges it is proposed to understand the analytic continuation to these intervals. As an example, the pair theory of interaction of spinor fields is considered; moreover, in the author’s opinion, the application of such a definition makes the pair theory renormalizable.

In the report by V. V. Chavchanidze, several questions connected with one method of introducing nonlinear interaction into the equations of Bose–Fermi fields were discussed.

In the report by P. E. Kunin and I. M. Taksar, several relativistic features of the behavior of particles with spin \(1/2\) were discussed. Thus, when particles with spin \(1/2\) pass through a potential barrier, the Klein paradox is absent in the case of a scalar field. It is further shown that for a particle with spin \(1/2\) located in a pseudoscalar field with a potential of the type \(1/r^n\), where \(n\) is any positive number, there is no fall of the particle to the center. A classical interpretation of this effect has been found.

L. Infeld (Poland) gave a report on the repulsion of nucleons as a relativistic effect. The treatment is carried out on the basis of the equations of classical mechanics. The results are analogous to those of Kunin and Taksar, although obtained by a different method.

In the report by I. S. Shapiro, a new method was considered for the relativistically invariant classification and covariant notation of wave functions, not requiring the introduction of the concept of a center of inertia and based on the use of results from the theory of unitary representations of the Lorentz group. The results obtained in the work may be used, in particular, for the relativistically covariant formulation of the Tamm—Dancoff method.

The report by V. N. Tsytovich was devoted to certain questions connected with the adiabatic approximation in the problem of the interaction of several bodies. Relativistic equations for two and three bound bodies were obtained, and their discussion was given.

In the report by A. A. Borgardt, various methods of matrix notation for wave equations in the theory of mesons were considered.

After many of the reports there was a lively discussion.

In summing up, the chairman of the Organizing Committee of the conference, I. E. Tamm, emphasized that the conference that had taken place would contribute to the further development of research in theoretical physics in the USSR, and also to the establishment of closer ties between Soviet and foreign physicists. The conference showed that work on the theory of elementary particles is being conducted in our country on a broad front and at a high scientific level. I. E. Tamm especially noted, as a positive fact, the active participation in the work of the conference by a large number of young theoretical physicists.

G. F. Zharkov