Divergence Difficulties Research Articles

Many-Body Theory for Quantum Kinetic Equations

A quantum kinetic equation is obtained which includes scattering events beyond the binary collision process of the Boltzmann equation. The time rate of change of the Wigner function due to collisions is given as the difference of two scattering rates analogous to the collisional side of the Boltzmann equation. These total rates are given in terms of partial rates which are identified with probability amplitudes for scattering events in the medium. Thus, the method is inherently free of the divergence difficulties of theories which calculate the total rates in terms of scattering events in vacuum. Special attention is devoted to the question in what sense does some form of Fermi's golden rule apply to the calculation of the partial rates for a given event. It is shown that certain modifications to this picture are necessary, but the changes may be understood from a simple physical point of view.

Study of Transport Properties of Plasma using Semi‐classical Quantum Potential

AbstractThe current evaluation of transport properties suffers from divergence difficulties resulting from the fact that the usual potentials do not adequately represent the collision mechanism in an assembly of interacting particles, the effect of surrounding particles at large and small distances being neglected. The usual way of avoiding this is by cutting off the ranges (both at upper and lower end) and completely disregarding such influences. A more appropriate procedure would be to use modified potentials (based on physical arguments) which automatically avoid such divergences. Using the Chapman‐Enskog‐Hilbert theory, we have calculated the transport properties for semi‐classical quantum potential and compared them with those of Debye shielded potential calculations. We have limited our discussion to hydrogen plasma. We have found that there is an increase in the transport properties of the plasma in the direction parallel to the original direction of a test particle and a decrease in the perpendicular direction, due to virtual induced quantum polarization.

Divergence difficulty in the viscosities of critical mixtures

Abstract A divergence difficulty encountered in studying the viscosities of binary critical mixture is described.

Free-Bound Model of the Plasma in Equilibrium

The plasma in thermodynamic equilibrium has been extensively discussed under the assumption that all particles can be classified as “bound” or “classically free”. Under this assumption simple models led to divergencies of the partition functions of bound and free particles as well as to discrepancies of the predicted level shift. On the basis of a quantum-mechanical treatment we develop an “adapted free-bound approximation” for the eigenstates and a classical approximation for the partition function of the free particles. It is a decisive feature of the analysis that it takes freebound interaction into account. The results produce values for the free-bound limit and the limit of the series continuum. They also remove the divergence difficulties.

Anomalies due to anisotropic s-d exchange interaction

Significance of studying the anisotropic s-d exchange interaction is pointed out. The most divergent logarithmic terms are summed up. Additional divergence difficulty is suggested for the case of ferromagnetic coupling.

Fredholm Theory of Scattering

The Fredholm theory of integral equations is applied to the non-relativistic theory of scattering. Divergence difficulties are overcome by using the Poincaré-Hilbert formula. It is shown that the Fredholm determinant has a zero of order 2 l + 1 corresponding to a bound state or a resonance with orbital angular momentum l . In order to sse the speed of convergence of the Fredholm determinant as a power series in the coupling constant, scattering by the Yukawa potential and the exponential potential is studied. Application of our method to the Bethe-Salpeter equation is also discussed.

A Non-Trivial Example of a Relativistic Quantum Theory of Particles without Divergence Difficulties

A Non-Trivial Example of a Relativistic Quantum Theory of Particles without Divergence Difficulties

Treatment of Overlapping Divergences in the Photon Self-Energy Function

An analysis is made of the generalization of Ward's identity to the problem of renorrnalizing the photon self-energy function in quantum electrodynamics. It is shown that it is nontrivial to select a path of differentiation in a consistent way through the different self-energy graphs, but that a suitable set of simple rules can be given which completely resolves the overlapping divergence difficulty in the photon self-energy function.

Nonperturbative solutions in a field theory withSp 6 symmetry

AnSp6-invariant nonlinear spinor equation is introduced and discussed. The covariant Fock equations are considered by using nonperturbative methods based on appropriate decompositions of many-point amplitudes. In the first part of the paper an exactSp6 symmetry is assumed taking into account the possible spontaneous breakdown of theγ5-invariance. In order to avoid the divergence difficulties, some self-consistent boundary conditions are imposed to be satisfied by the two- and three-point amplitudes for vanishing relative co-ordinates. These self-consistent boundary conditions cannot be satisfied in a trivial way due to the well-known singularities of the relativistic interactions. A kind of eigenvalue problem is obtained for the couplings and masses. The results indicate that an acceptable solution of the problem is not excluded. This possibility is connected with the simultaneous occurence of the interactions generated by trion-antitrion pairs characterized by spin zero and one. The coupling generated by pseudoscalar pairs is a derivative one. For pseudoscalar trion-antitrion amplitudes the fulfilment of the self-consistent boundary conditions is discussed in detail in the framework of the nonperturbative methods introduced in this paper. The self-consistent boundary conditions can be satisfied if the bare coupling constantg0(11) characterizing the coupling of trions toSp6 singlet trion-antitrion pairs with spin one is fixed by (g0(11))2=12π2. The coupling of trions toSp6 singlet pseudoscalar pairs has to satisfy a simple self-consistent subsidiary condition. The other couplings of trions vanish. No conflict, with the general framework of relativistic quantum field theory appears; the usual finite cut-off is not introduced. Only the action principle is rejected. In the second part of the paper the possibility of the spontaneous medium-strong breakdown of theSp6-invariance is discussed in detail.

Padé Approximant in Potential Scattering

It is suggested that the Pad\'e approximant be used as an approximation outside the radius of convergence of the Born series for scattering from a short-range potential free from a strong singularity. Following Weinberg's analysis of the Born series, one approximates the potential by a finite number of separable potentials in order to deal with a limited range of energy, and thus the associated $K$ matrix can be evaluated without further approximation. When a number $N$ of separable potentials are constructed by using the lowest $2N$ terms of the Born series, the result is identical to the ($N, N$) Pad\'e approximant. If expanded formally into a power series in the potential strength, this approximation reproduces the original Born series exactly up to the $2N\mathrm{th}$ order; but without an expansion it takes a closed form for any finite potential strength, and therefore it is well defined outside the radius of convergence of the Born series. When the Weinberg eigenvalues and eigenfunctions on the energy shell are closely approximated with a suitably chosen $N$, the divergence difficulty of the conventional Born series can be overcome by the use of the Pad\'e approximant, because the latter is correctly continued analytically outside the convergence radius of its Born series.

The three particle collision operator in quantum mechanics

We analyse the quantum mechanical collision operator for three incident free particles from the point of view of transport theory. Starting from the Liouville-von Neumann equation for the density matrix, in the form introduced previously by Prigogine and coworkers, we obtain the generalization to quantum systems of the well known Choh-Uhlenbeck result for classical genuine triple collisions. This expression, written in terms of the Heisenberg operators of motion exp[— iHt] for two and three particles, is free of the divergence difficulties occurring in standard collision theory when the limits of a large system and of long times are taken in order to define a three particle transition probability. We also discuss the connection of this result with the Lippman-Schwinger theory and show that the scattering operator for three incident particles is not expressible as the square of a scattering matrix, like in the two particle problem.

Method of Adjusting Force Constants and Its Application to H2O, H2CO, CH2Cl, and Their Deuterated Molecules

A new method for the adjustment of vibrational force constants is presented. It avoids some of the divergence difficulties which may be encountered in perturbation and least-squares procedures. The new method employs an efficient computer program to adjust an initial set of force constants in accordance with the requirement that each successive adjustment leads to a smaller sum of squared deviations between calculated and observed eigenvalues. Several variants of the procedure are given, and their application to a number of actual cases is discussed in detail.

The pair distribution function of a dilute hard sphere bose gas

Extending the theories of De Boer, Uhlenbeck, Yang and Lee and others, a quantum statistical mechanical formalism of evaluating the pair distribution function in terms of the binary kernel is presented. This formalism is more convenient for a hard sphere system than that recently developed by Fujita, Isihara and Montroll. An explicit evaluation of the pair distribution function of a hard sphere Bose gas at the lowest temperature is reported. The results improve upon those reported by Lee, Huang and Yang and others by using the pseudopotential. The normalization and divergence difficulties encountered by Fujita and Hirota are removed and the correct interpretation of the results of Yang and Lee and others is given. the cause of the difficulties is discussed. A satisfactory result is obtained by taking what we call tick-tack-toe diagrams into consideration in addition to the chain diagrams.

Many-Particle Self-Consistent Model

The number and general nature of the self-consistency equations that may arise in a universal bootstrap theory of all strongly interacting particles is discussed. There are more equations than there are variables to be determined, so that it may be possible to bypass some of the divergence difficulties of dispersion theory by making use of a sufficient number of the equations. A general method of attacking the difficulties associated with the many-particle aspect of the problem is discussed. A simple first approximation to the method is applied to a model of four multiplets (pseudoscalar meson, vector meson, baryon ground state, and $j={\frac{3}{2}}^{+}$ baryon excited state), under the assumption that unitary symmetry is approximately valid. It is argued that comparison with experiment of the calculated differences in masses of particles within the same multiplet will provide experimental tests that are meaningful in a low-order approximation to the model. It is shown that if the mass splitting of the baryon octet is assumed to be partly self-generating (i.e., not resulting completely from the mass splitting of the meson multiplets), a nondegenerate solution to the model is most likely if there is a large violation of $R$ invariance.

ASYMPTOTIC BEHAVIOUR OF π-π SCATTERING AMPLITUDE AND SINGULARITIES IN COMPLEX L-PLANE IN STRIP APPROXIMATION

The contribution of inelastic intermediate states to the singularities in complex l-planc and to the asymptotic behaviour of crossed channel are investigated in the strip approximation. The procedure is to solve the simultaneous equations suggested by Chew, Frautschi and Mandelstam by means of iteration method. The general iteration expressions for the s channel high energy amplitude f(s,t) and the t channel partial wave amplitude ft (t) are derived. It is shown that the divergence difficulty mentioned by Chew, Frautschi and Mandelstam is not present in the iteration results of the above f(s,t). The main results of first iteration are as follows: i) The total cross section falls slowly for not very high energies, but at the end tends to a constant. ii) The diffraction angular distribution will have a long tail. As the energy increases, this long tail extends towards the small angle region. iii) In the t channel a cut will appear along the real axis in complex l-plane; the right ending point of the cut is at l=1 when t=0, and moves to the left as t decreases, with a speed nearly half that of the vacuum pole. Higher order iterations are also investigated; the results are qualitatively analogous to those of first order.

Towards Unified and Convergent Theory of Elementary Particles

In order to get rid of the well-known divergence difficulty in the quantum field theory, fictitious particles with an indefinite metric are introduced. The difficulty concerning the probabilistic interpretation caused by the indefinite metric is satisfactorily avoided by making use of the of fictitious particles, characterized by the complex poles. Based on such a general possibility, we investigate the Nagoya model assuming B-matter to be a unified assembly of fictitious particles. The divergence difficulty in the quantum field theory has been a long-pending object for the theoretical physicists from its beginning and many efforts to overcome this difficulty have been made without any satisfactory solution. Among these attempts, however, the introduction of the indefinite metric in the Hilbert space seems to have a special theoretical ground/) when taking into consideration the recent investigation of the axiomatic formulation of the quantum field theory.2) As noticed by many authors, this attempt seems still to have a serious difficulty that it can not reconcile with the probabilistic interpretation in the quantum theory or with the requirement of the macroscopic causality.3),4) However, it should be noticed that the latter conclusion is based on a purely perturbational consideration. Let us consider the well-known Pauli-Villars type regularization. Fictitious particles introduced to regularize the divergence have in general a large mass and interact with physical particles. Thus, at high energies over the threshold of the production of these fictitious particles, they must come out with an indefinite metric and thus violate the probabilistic interpretation. 4 ),O) Is this consideration, however, really correct? There necessarily arises the question of the stability of these particles. Such a fictitious particle which has a large mass and interacts with physical particles in general must not be stable. However, it should be noticed that the instability of fictitious particles with indefinite metric is only superficial and has an essentially different aspect from the instability of the usual unstable particles with a definite metric. That 1S,

Regge Poles and Resonances in Strong Interaction

The scattering of two particles via the exchange of a resonance system is considered. The exchanged resonance is approximated by a pole in the Regge representation. The composite nature of the resonance is taken into account in the Regge-pole description so that one does not encounter divergence difficulties even though the exchanged resonance may be in a $J=1$ state.

KINETIC EQUATION DESCRIBING IRREVERSIBLE PROCESSES IN IONIZED GASES

The present work deals with the formulation of the theory of irreversible processes in ionized gases on the basis of the theory of Bogoliubov for neutral gases. In an extension of the work of Guernsey and Uhlenbeck, the kinetic equation is obtained for a spatially inhomogeneous system in an approximation to the "first order" in the interaction strength (e2) and to all orders in the correlation effect (e2/v, v being the specific volume). The equation, unlike the usual Vlasov equation, is not invariant with respect to time-reversal and shows a genuine "damping" in the sense of an irreversible approach to equilibrium.In the present paper, brief discussions have also been given of the nature of a theory of irreversible processes, the so-called "divergence difficulty", due to the long-range Coulomb forces, the Landau damping, the recent general theory of irreversible processes of Prigogine and Balescu, and the theory of ionized gases of Balescu.

On the Quantization of Physical Space-time Operators

§ 1. IntrDduction In the conventional field theory, space-time coordinates are treated as c-number parameters, and field operators (consequently energy-momentum density, current density and such quantities are expressed as the products of field operators also) are represented as their functions. On the other hand, many authorsl) investigated the possibility to regard space-time coordinates as operators, or to quantize them, in order to get rid of the so-called divergence difficulties. However, no promising guiding principle to quantize space-time coordinates has been found. Among many attempts, a series of investigations 2 )-4) which was begun by the famous Dirac's work 5 ) has an exceptional attraction for the present author, although these works were developed regardless of the divergence difficulties. For their quantization of the physical coordinate operators seems to express the uncertainty principle in the measurements, as will be seen in the following discussions. The purpose of this article is to make the physical meaning of the quantization clear and to analyse the nature of the extended Hilbert space. The meaning of the uncertainty principle in quantum mechanics is as follows. As long as the particle, for which the uncertainty relation between its position and momentum holds, is used in the observation of the motion of some particle, the position and the momentum of the object also cannot be measured simultaneously and exactly. If any particle which is free from the uncertainty principle does exist, we would be able to use this particle to make the simultaneous measurement of the position and the momentum of the object, whether they are elementary or not. It may in fact be possible to take the standpoint that the uncertainty relation is derived from such a theory that introduces some hidden variable free from the uncertainty principle, as Bohm

Giant Cluster Expansion Theory and Its Application to High Temperature Plasma

The conventional virial expansion of thermodynamic functions is converted into a new expansion scheme, similar to the former but more powerful. The new method is particularly suitable to dealing with the interaction of long-range character, such as Coulomb potential, since it suffers from no divergence difficulties contrary to the conventional one. As an application of the method, the equilibrium properties of high temperature plasma is studied and the term of next higher orders than the Debye-Hückel limiting law is obtained exactly. The order estimation indicates that the Debye-Hückel law is accurate within the error of a few per cent in this case. A possible extension of the present method to the theory of non-equilibrium properties of plasma or to quantum statistic is suggested.