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In paper I, [ Ann. Phys. (N.Y.) 94 (1975)] the matrix elements of effective operators in an intrinsic frame of the spherical shell model, that were determined by an iterative, renormalization technique, were shown to be analytically equivalent to those from a deformed, unrestricted Hartree-Fock model for the case of 20 Ne with a schematic quadrupole-quadrupole Hamiltonian. This paper shows the numerical comparison of the results from the renormalized spherical shell model and the projected, deformed Hartree-Fock model in the laboratory frame with states of definite angular momentum. Good agreement is found for the full iterative, renormalization results and those from the variation-after-projection Hartree-Fock model.

Within the scope of many-body Green functions we derive a formally exact expression for the ( d,p )-stripping amplitude, which resembles the DWBA for this reaction. However, the distorted wavefunctions in the entrance and exit channel are defined with respect to the most general expressions for the optical potentials, as they are given by the mass operators of corresponding Dyson equations for correlation functions. In the same way the transition operator is a complicated many-body quantity which explicitly depends on the deuteron energy and contains for example contributions coming from the core excitations of the target nucleus. In this paper we are mainly interested in the latter effect and a numerical estimation of a crude model for the many-body effects of the transition operator reveals no pronounced (sharply resonant) energy dependence. The essential difference of our paper to other publications on the same subject is that we are dealing with the “Dyson equation” approach developed recently for higher Green functions.

A new method for treating problems involving the gas-dynamics Boltzmann equation is developed. It is analogous to the Case method, developed for the solution of the neutron transport equation. The elementary solutions have separated variables and are shown to have properties similar to the Case's ones, with some notable exceptions. The model of Gross, Bhatnagar, and Krook is used. The paper is devoted to problems at a constant temperature; the extension of the method to problems with a temperature gradient will be considered in a subsequent paper. Besides the general theory of pure shear problems in unidimensional plane geometry, a solution of the slip-flow problem is given as an illustration. An analytical expression of the slip-flow coefficient, particularly suitable to a numerical calculation, is this obtained.

The aim of this paper is to study the general response properties of a Bose liquid with condensed phase, both at T = 0°K and at finite temperature. This is accomplished by exploiting the gauge invariance requirement of the current-current correlaton functions Kμν with the aid of techniques developed in previous papers. The main result obtained is an explicit microscopic construction of the superfluid density ϱs. As an example, an approximation derived from the Bogoliubov model is explicitly worked out. Gauge and Galilei symmetry breakdowns are also discussed, showing that in an infinite volume system with long range correlations Galilei relativity principle is satisfied in a very particular way. Some results on the long-range behavior of correlation functions, first obtained by Schafroth in the special case of a free charged Bose gas, are shown to be valid also for the Bose liquid.

The generalized strong-absorption model for elastic nuclear scattering permits a straightforward extension to describe the polarization of spin - 1 2 particles strongly interacting with nuclei. Explicit expressions are derived for the differential scattering cross section σ(θ), the polarization P(θ) and the total reaction cross section σr. The general expressions reduce in special cases to simple formulas which describe a variety of possible structures of the polarization angular distribution, corresponding to different physical situations. The cases for “neutral” and charged particles are treated separately. The former applies to the entrance channel phenomena of protons and neutrons in the energy region E ⪆ 100 MeV . The latter is expected to describe possible future polarization measurements in the elastic nuclear scattering of composite spin - 1 2 particles such as He3, F19, etc. In the present paper the general formalism is developed and several special cases are considered. In two subsequent papers we shall, respectively, give an analysis of a complete set of high-energy proton scattering data, and discuss the connection between the present model and the complex potential model.

We apply the S -matrix theory and the Dashen-Frautschi technique to the Lamb shift. The formalism was developed in a first paper, in which the problem of the infrared divergence was solved and the term in α( Z α) 4 Log Z α of the Lamb shift was obtained. No ultraviolet divergences appear in the calculations. In this paper, the contributions in α( Z α) 4 of the “nonrelativistic cuts”, which correspond to the “Bethe logarithm” and to the anomalous magnetic moment term, are estimated. Our approximate value of the Bethe logarithm is 3.25 for the 2 S state, and we get Δ E 2 S = 980Mc. For the anomalous magnetic moment term, analyticity assumptions lead to an exact result. Our approximate value of Δ E 2 P 1/2 is +7 Mc. By the way, we obtain the radiative corrections to the low energy Compton amplitude (for spin 0 particles) using dispersion relations.

The paper describes a method for the calculation of the approximate solution of the Schroedinger equation of finite Fermion systems. The method consist of approaching the exact solution in successive approximations. The first step is a Hartree-Fock calculation; the second step is the calculation of two-particle correlation effects; in each successive steps higher order correlation effects are taken into account. For the description of correlation effects, two-particle, three-particle, etc., correlation functions are introduced; these are determined from equations derived using the variation method. At each step the calculated energy is an upper limit to the exact solution. The present paper treats the Hartree-Fock approximation and the two-particle correlation effects; higher order effects will be treated in the second part.

We derive exact expressions for the transport properties of a degenerate Fermi liquid. These expressions are obtained by solving certain integral equations which were first derived by Abrikosov and Khalatnikov and by Hone. In the present paper, we evaluate the coefficients of shear viscosity, thermal conductivity, diffusion, and second viscosity; a later paper deals with sound propagation. The solutions obtained here for shear viscosity and diffusion agree within 25% with those originally quoted. However the thermal conductivity is reduced by a factor of about 2. The coefficient of second viscosity does not seem to have been calculated previously; here we show that it varies with temperature like T2. Applications of the present theory to liquid 3He and 3He4He mixtures are discussed, but numerical work reported by other authors has not been duplicated. Apart from having application to Fermi liquids, the exact solution described here is of general interest because it provides transport theory with a new soluble model.

The present paper concerns the exact analytic forms of correlation functions of liquids and some of their macroscopic properties which these correlations determine. Since these functions are expressible in terms of the scattering structure factors of liquids, the macroscopic properties they define will be described in terms of the properties of the structure factors. One is then led, with some limitations, to a rigorous result on the behavior of pair potential energies of isolated stationary atoms at close separations. Part of the paper is devoted to a discussion of some of the limitations of the experimentally derived structure factors as well as the liquid correlation functions to which they give rise. It will be shown, on the basis of liquid helium data, that improvements in the empirical correlation functions as well as the various macroscopic liquid properties they define, require the knowledge of the structure factors over a far wider range of the momentum changes on scattering than obtained heretofore. The final topic concerns a detailed study of the asymptotic, large atomic separation, behavior of the correlation functions. The temperature limitations of a direct method yielding these asymptotic forms will be obtained. A new method of derivation of these forms, involving the scattering structure factors, will be described. This same method also leads to the asymptotic form of the structure factors themselves, at large values of the momentum changes on scattering. These latter asymptotic structure factors retain their validity over the whole temperature interval of existence of liquids.

In a previous paper (F. Calogero and A. Degasperis, J. Math. Phys. 9 (1968), 90) an approach to the inverse problem for the nonrelativistic S -wave potential scattering has been developed with the aim of obtaining explicit and exact expressions of the potential in terms of the phase shift and of the bound states parameters. Since this method, based on the high energy asymptotic expansion, yields the expression of the potential and of all its derivative at the origin only if these quantities are bounded, physically interesting interactions such as the Yukawa potential could not be considered. In this paper this approach has been generalized to potentials with a simple pole at the origin and the expression of the Laurent coefficients of the potential in a neighborhood of the origin has been obtained as an explicit functional of the experimental data. Furthermore the general structure of the high energy expansion of the phase shift has been derived and discussed. The requirement that the potential has only a simple pole at the origin turns out to imply strong constraints on the experimental data. These conditions on the experimental data have been explicitely expressed as a set of infinitely many nonlinear sum rules involving only the phase shift and the energies of the bound states.

In a series of papers, we shall develop and apply formal operator techniques for radiative level shift calculations. This first paper is concerned with the general formulation of the method and its application to the evaluation of the lowest order Lamb shift. An effort has been made to keep it pedagogically coherent, and a detailed review of the various theoretical contributions and comparison with experiment has been included.

This paper is a report of some studies leading to a new mathematical description of the condensation point for a simple class of models of first-order phase transitions. The paper consists of three main parts. In the first part it is pointed out that, although the conventional droplet model of condensation predicts that the free energy has an essential singularity at the condensation point, this singularity is so weak as to be experimentally unobservable. Furthermore, the analytic continuation of the free energy beyond the singularity describes a metastable phase according to the assumptions of the model. The second part of the paper is devoted to the study of a soluble functional integral that exhibits an essential singularity similar to that found in the droplet model. A method is developed for computing the singular properties of such integrals in cases where it is not possible to evaluate the integrals exactly. In the third part of the paper this method is applied to a simple model of a ferromagnet at temperatures well below the Curie point. Most of the really characteristic features of the droplet model are recovered in this calculation. The detailed results have a bearing on problems of phase coexistence, surface energies, and possibly even condensation rates.

In the first part of this paper, the relativistic significance of the projective curvature tensor Whijk is discussed. It is found that the vanishing of the divergence of this tensor in an electrovac universe implies a purely electric field. By breaking Whijk into its antisymmetric and symmetric parts, Phijk and Qhijk, respectively, it is shown that the vanishing of the symmetric part is a necessary and sufficient condition for the space to be an Einstein space. In an electromagnetic field Phijk is expressible linearly in terms of the Riemann curvature tensor Rhijk and the conformal curvature tensor Chijk. Phijk has all the symmetry and antisymmetry properties of Rhijk and also it possesses the cyclic property Ph[ijk] = 0. The vanishing of the contracted tensor Pij implies an Einstein space. This enables us to extend the Pirani formalism of gravitational waves to the Einstein space with the help of Whijk. In the second part of the paper, the electromagnetic significance of the cylindrically symmetric metric of Marder is considered. The Rainich algebraic relations give rise to four possible cases of electromagnetic fields, three of which correspond to electromagnetic wrenches, in the terminology of flat spacetime, along one of the spatial coordinate lines. The complexion vector vanishes implying a purely electric field for the cylindrically symmetric electrovac universes. The conditions for the fields in question to be of Petrov type II have been found out. None of the well-known cylindrical electrovac universes satisfies these conditions. The fourteen scalar invariants of order two have been evaluated for the four cases of electromagnetic fields. Two scalar invariant relations and a tensor relation exist as necessary conditions for electromagnetic fields of cylindrical symmetry. The groups of motions admitted by the fields are discussed. An exact nonstatic solution of the Einstein-Maxwell equations in vacuo is obtained and some of its properties are studied.

In an earlier paper we extracted an asymptotic series solution in half-integral powers of (kTE) from the Boltzmann transport equation for neutrons coming into thermal equilibrium with a moderating material. In this paper we consider the series in detail for moderators having simple crystalline structure, or in which vibrational modes predominate, and give the coefficients of the series explicitly in terms of moments of the frequency spectrum of crystal vibrations. Our calculations indicate that the neutron density in the asymptotic region increases as one hardens the spectrum of crystal vibrations, whence the “hardness” of the neutron spectrum increases, too. We also discuss some aspects of the mass-expansion for these systems, and compare our calculations with those of other workers, and with typical experimental data for thermalization in water. Finally, we discuss the asymptotic expansion for moments of the scattering kernel and for ξσ_s, the slowing down power.

We introduce certain functions which may be considered as generalizations of the usual surface harmonics. They enable us to express in concise form the relation between relativistic wave functions as given in the linear momentum basis and those given in the angular momentum basis for particles of any spin and any mass. Representations of finite rotations about any axis orthogonal to the z-axis can also be expressed simply in terms of these generalized surface harmonics. Completeness and orthogonality relations will be given for these functions as well as other properties such as expansions of products. The new functions form the basis of a ray representation of the rotation group in a manner similar to that which is formed by the usual surface harmonics. In addition to their usefulness in relating the linear momentum basis to the angular momentum basis for relativistic wave functions, in later papers we shall use the generalized surface harmonics to quantize the electromagnetic vector potential in terms of an angular momentum basis which is a close analogue to the expansions of classical radiation in terms of multipoles used by Blatt and Weisskopf. Also in a later paper we shall use the generalized surface harmonics to obtain the Clebsch-Gordan coefficients for the direct product of two massless representations of the proper, orthochronous Lorentz group.

In the introduction the nonlinear system of functional equations for the matrix elements of the S-matrix is formulated, which we call the Chew-Low type equations. A short review of some papers devoted to equations of this type is given, and the advantages of the approach of the present paper are discussed. In part II, Section 1 the transition to the projective coordinates in the space of the matrix elements of the S-matrix and the linearization of the unitarity conditions are performed. An interpretation of the system of functional equations as a transformation in the (n − 1) dimensional real space is given. It is shown that the some of the solutions of the initial system of equations are contained on the invariant hypersurfaces of this space (Section 2). In Section 3 a method of the local construction of the invariant subspaces is proposed. In part III the method suggested is applied to the Chew-Low equations with the 3 × 3 crossing matrix. It is established that, if the Chew-Low equations possess a solution, then the arbitrariness of the solutions of the class (2.12), being the generalization of the familiar β-arbitrariness, is not exhaustive.

In a previous paper the irreducible representations of the rotation group were given in terms of a parametrization of the rotation through the axis and angle of rotation. In the present paper we give the orthogonality relations between the matrices of two such irreducible representations. We also show how the infinitesimal generators of the rotation group act in the parameter space and show how finite rotations in the parameter space are to be described. Thus the previous paper and the present one can be used to replace completely the usual theory of irreducible representations in terms of the Euler angles by the far more convenient ones in terms of the angle and axis of rotation.

The aim of this paper is to draw attention to an apparently forgotten paper ( 1 ) on the inverse scattering problem in quantum mechanics and to add some generalizations and explanations. In the enormous literature ( 2, 3 ) which in the course of 15 years has grown up around this problem, nobody appears to have taken advantage of the simple principles on which that paper was based. Several inconveniences and some weak points of current theories are pointed out.

This is the first of the series of papers in which it is shown that the iteration of the kernel in the Lippmann-Schwinger equation is the basic manipulation by which a meaningful result is obtained for any finite strength of the potential, no matter whether the Born series converges or not. In order that this statement may be true the potential must be free from a long range tail and a strong singularity. In this paper the conditions for a generalized potential, which can be nonlocal and/or energy-dependent, to have these properties when transformed into the configuration space are discussed directly in terms of its matrix element in momentum space. These conditions are interpreted as equivalent to the restriction that the potential decreases faster than 1 r 3 at a large distance and is less singular than 1 r 2 at the origin. It follows from these conditions that all the individual terms of the Born series as well as the traces of the kernel and its iteration to an arbitrarily higher order are bounded.

This paper continues the study of the two-level model utilized in the preceding paper. First the boson expansion method is reformulated from a viewpoint which emphasizes its relationship to the phenomenological description of vibrations. In this version a general method for reaching the limit of large particle numbers is described and carried out, resulting in a complete theory for finding the asymptotic form (semiclassical limit) of any collective parameter. An exact formulation of the theory in terms of single-particle coefficients of fractional parentage is then described. The semiclassical limit of this theory is recognized as a version of a generalized self-consistent cranking model or time dependent Hartree-Fock theory. The two semiclassical limits are shown to be identical. The equivalence of time-dependent Hartree-Fock theory to the asymptotic form of more complete theories of collective motion is argued to be general.

It has been known for some time that, when no magnetic field is present, transport coefficients have an extremal nature. In this paper, it is shown that transport coefficients in the presence of a magnetic field or an alternating electric field also exhibit an extremal property. A mini-max (saddle point) principle is obtained which is used to solve the first order Chapman-Enskog transport equation with a Fokker-Planck collision term for a fully ionized gas. The thermoelectric coefficients and the viscosity when an arbitrary magnetic field is present, and the ac conductivity for the case of no magnetic field are evaluated for a singly ionized two component gas to well within the accuracy justified by the basic equations (∼10%). It is emphasized that this variational formalism rests upon two simple general properties of the collision integral, and hence is capable of considerably wider application than the particular example offered in the present paper. The physical significance of the variational principles is briefly discussed, and they are related to the rate of entropy production and amount of entropy stored in the system.

This paper treats a linearized problem in Couette flow which has been derived from a simple version of the Krook statistical model for intermolecular collisions. The problem is reduced to the solving of Fredholm equations which are shown to converge rapidly for all values of the Knudsen number. The methods are based on singular integral equations and were first developed in neutron transport theory. Cercignani, in applying Case's method to the slip flow problem, has also studied singular equations dual to those of this paper.

An investigation is made of the possibility that the S-matrix cannot be unambiguously determined from scattering experiments. It is shown that the helicity formalism of Jacob and Wick is suited to treat this question. In the first part of the paper a derivation of this formalism from the channel spin formalism of Blatt and Biedenharn is given. For this derivation only elementary methods are used. The second (main) part of the paper contains the derivation and description of ambiguities of a certain type, so-called torsions. The possibility or impossibility of a physical interpretation of these torsions is discussed. The cases of spin 12-spin 0, spin 1-spin 0, spin 32-spin 0, and spin 12-spin12 scattering are explicitly treated. In particular nucleon-nucleon scattering with charge-independent interaction is considered.

The paper begins with a review of the extraction of the rotational energy and corresponding collective coordinate within the formalism of the random-phase approximation (RPA) for a two-dimensional deformed “nucleus.” An intrinsic difficulty is observed: The random-phase approximation to the angular-momentum operator appears to have a continuous rather than discrete spectrum. Correspondingly, the RPA Hamiltonian has a continuum of unnormalizable eigenstates. One of the main results of the paper is a diagnosis of the origin of this difficulty and a method to deal with it. It then becomes possible to calculate matrix elements of static and transition operators, and illustrative examples are given. Similar difficulties are also encountered in mechanical small-oscillation theory; simple examples are used to illustrate and motivate the procedures applied to the nuclear RPA problem. In the latter connection, the recent general formulation of rotational motion due to Villars is used as a convenient formal framework. Next, the relation between the RPA and the self-consistent cranking (constrained Hartree-Fock) model is simply exhibited, and subsequently it is shown that Kelson's “external consistency criterion” is fulfilled up to terms that are of the order of corrections to the RPA itself.

The effects of nuclear Fermi motion on the extraction of high energy neutron cross sections from deuterium data are discussed in some detail. We consider three examples: (i) resonance scattering, (ii) asymptotic hadron scattering and (iii) deep inelastic electron scattering. In the first example we show in a very simple way how a nucleon resonance is broadened when viewed in the deuteron. Simple qualitative expressions are derived for the depression of the peak and the broadening of the width due to the Doppler shifting of the incident energy. For asymptotic hadron scattering, where total cross sections are generally very slowly varying, we show that there is a subtle effect due to the threshold constraint on the total nucleon cross sections which tends to deplete the total deuteron cross section. The size of this defect depends crucially upon the size of the high momentum components in the deuteron wavefunction. The effect is capable of explaining the apparent energy and process dependence of the mean inverse square radius parameter which occurs in the Glauber screening formula. An estimate of the effect in total photoabsorption cross sections indicates that the proton-neutron difference may be considerably smaller than expected. In the third example, deep inelastic electron scattering, this threshold effect increases near the threshold region. This together with the conventional smearing effect is shown to simulate the shape of the apparent difference between the proton and neutron. The size of the overall effect again depends crucially upon the high momentum components in the deuteron. However, reasonable estimates indicate that there is a real proton-neutron difference, especially in the so-called low ω region; the difference probably vanishes in the large ω region. Several other effects in deep inelastic electron scattering are discussed: We show that a value of ω exists (which we call the cross-over point, ω c ) at which the overall effect vanishes. That is, below ω c the effect reverses sign and the neutron cross section now appears larger than it really is. We also derive an expression for the threshold behavior of the deuteron cross section in terms of the threshold behavior of the nucleon cross sections and the asymptotic form of the deuteron elastic form factor. Finally we derive a series of wavefunction independent sum rules which relate the overall integrated depletion to the integrated deuteron cross section below the quasi-elastic peak. Such sum rules could be extremely useful in “measuring” the effects of the Fermi motion. Most of the discussions in the paper are of a qualitative nature; quantitative details will follow in a future publication. The paper has been written with the experimentalist reader in mind.

Two previous papers from this laboratory have reported measurements of heat conduction and fountain pressure for liquid He II flowing through narrow slits (0.3 μ d μ ) for temperature differences as large as 1°K. For the lower, yet appreciable, temperature differences the linear two-fluid equations of london and Zilsel were quantitatively verified; integration over the temperature interval was required. The present paper extends the analysis of the measurements to still larger ΔT 's, for which the linear equations are no longer applicable. For this purpose integrated solutions of the Gorter-Mellink nonlinear thermohydrodynamic equations, based on the concept of mutual friction, are derived with special emphasis placed on the assumptions and restrictions necessitated by the model. The integrals for heat flow and fountain pressure have been solved numerically using a high-speed computer and the results are compared with the experiments. When Vinen's values of the mutual friction parameter A ( T ) are employed in the solutions, the comparison is quite good, except near the λ-point; it is also shown that other values of A ( T ) are not compatible with the observations. An explanation in terms of the vortex line model is proposed for the deviations near T λ . Despite the agreement between the vortex line theory and experiment obtained here, several as yet unresolved difficulties are associated with flow phenomena in small slits; certain aspects of these problems are discussed, most notably the criteria for the onset of the nonlinear dissipation effects.

The question of the localizability of the energy in gravitational fields is discussed once more. It is shown that the expression for the energy density, derived in an earlier paper γ, doet not have all the properties required for a satisfactory description of the energy distribution. An essential correction to one of the statements in the mentioned paper is made. A careful analysis of the desirable properties of the notions in question is carried through, the conclusion being that none of the expressions derived so far satisfies all the requirements necessary for a satisfactory description of the localization of the energy. At the end, it is indicated how the “four-leg” formulation of the gravitational theory might provide a solution of the problem.

This paper is concerned with the oscillations which appear in the smoothed density of eigenvalues when the smoothing width is relatively small. The existence of these oscillations is demonstrated by evaluating exactly the smoothed eigenvalue density for simple shapes of the volume—flat parallelepiped, sphere. A general theory of the density oscillations is then developed on the basis of the time-independent Green function formalism used in Part I of this work. An asymptotic evaluation of the various terms of the multiple reflection expansion is carried out by means of the principle of stationary phase, or by means of a contour integration method. A systematic investigation of the various resulting contributions shows that the dominant oscillations are associated with the closed classical trajectories, i.e., the closed polygons having their vertices on the boundary surface S and such that mirror reflections on S take place at each vertex. Various possible geometrical configurations of these closed paths are discussed—degenerate, accidentally degenerate, repeated paths. Several cases of cancellations between various contributions are indicated. The second part of the paper is devoted to applications and examples. In the case of the parallelepiped, it is easy to compare the expansion given by the present theory with the exact density. The case of the sphere is discussed numerically, in order to determine the number of terms of the expansion which are actually necessary to reproduce the exact distribution of eigenvalues. In view of practical applications, detailed expressions are given for axially symmetrical surfaces. The contour integration method is applied to an axially symmetrical surface having a complex stationary path. Finally, density oscillations are discussed and interpreted in terms of the semiclassical picture.

In the formulas given in the previous papers by Froman and by Froman and Myhrman on the eigenvalue problem of the double oscillator, a certain quantity σ occurs, which is negligible for energies lying far from the top of the barrier but which is important for energy levels in the immediate neighborhood of the top of the barrier. For the higher-order phase-integral approximations the quantity σ, in fact, cancels a singularity appearing in certain contour integrals when the energy approaches the top of the barrier. The numerical results reported in the papers mentioned were obtained disregarding σ, since convenient approximate expressions for σ, pertaining to the higher order phase-integral approximations used, were not known when the papers were published. As a consequence, the energy levels in the immediate neighborhood of the top of the barrier were not good. This deficiency is now remedied, and the purpose of the present paper is to give definitive formulas obtained by the quantal treatment of the energy eigenvalue problem of the double oscillator by means of the phase-integral approximations in question, up to the fifth order. The great accuracy of these formulas, also for energies close to the top of the barrier, is illustrated numerically.

Nonlinear effects in alternating-gradient synchrotrons are responsible for resonances in the betatron-oscillation mechanism over and above those which arise in linear theory. These effects are here investigated by means of a perturbation procedure which leads directly to practical formulas. The larger part of the paper is given to “static” theory which ignores the acceleration process and concomitant synchrotron oscillation. Results are represented diagrammatically in three ways: by invariant contours in the phase plane; by “configuration diagrams” which are convenient transformations of the preceding diagrams; and by curves which relate the amplitudes of closed orbits to the characteristic phase. The resonances are associated with the following critical relations between the characteristic phases Θx, Θy, which denote the phase-change of radial and vertical oscillations, respectively, over one revolution: rxΘx + ryΘy = 2kπ, where rx, ry, k are integers. In an ideal machine, composed of N identical sections, the above resonances are restricted to values of k which are integral multiples of N; the remaining resonances must therefore be attributed to departures of the machine from its ideal form due to constructional inaccuracies. For this reason, an analysis of nonlinear effects is necessary for the evaluation of tolerances. The resonance with “coordinates” rx, ry, is due principally to fluctuations in the nth radial derivative of the magnetic field strength, ∂n H y/∂xn, where n = |rx| + |ry| − 1. The following general observations may be made: (i) The relation |rx| + |ry| ≤ 3 determines values of the characteristic phases in some neighborhood of which betatron oscillations of indefinitely small amplitude will grow to finite, possible large, amplitude. If |rx| + |ry ≥ 5, there is no such neighborhood. Resonances associated with |rx| + |ry| = 4 may have the characteristics of either of the preceding categories, depending upon the “quartic phase-independent term” Q, which is due principally to ∂n H y/∂x3, but there is reason to expect that in practice its properties will be those of the second category. Static theory therefore suggests that resonances for which |rx| + |ry| ≥ 4 may be ignored. (ii) Among the coupling resonances, for which rx, ry are both nonzero, those for which rx, ry have opposite signs lead only to an exchange of betatron-oscillating energy, which in alternating-gradient synchrotrons is harmless. (iii) Since deviations of the magnetic field which violate the (mirror) symmetry of the accelerator will be slight compared with those which are compatible with this symmetry, a resonance associated with an even value of ry and odd value of rx is more serious than that for which these values are interchanged. Calculation of the effect of fluctuations in the nonlinear contribution to the magnetic field may be based upon first-order perturbation theory. However, the consideration of a specific example shows that the combined influence of linear fluctuations with the average nonlinear contribution may be of comparable importance. Calculation of such effects involves second-order perturbation theory, formulas for which are derived in an appendix. Some attempt is made to consider the “dynamic” problem which takes account of the influence upon the betatron-oscillation mechanism of synchrotron oscillation and progressive saturation of the magnets. The two extremes of slow and rapid traversal of a resonance are discussed: the former should be appropriate to the possible crossing of low-order resonances near ejection due to magnet saturation; the latter should be appropriate to the inevitable crossing of high-order resonances during the complete acceleration cycle, and to the possible crossing of low-order resonances due to the large amplitude of synchroton oscillation near injection. Slow traversal of a resonance will give drastic expansion of at least part of the beam for at least one of the two directions, but the effect may be reduced by increasing Q. The effect of repeated rapid traversal may, under certain conditions, be approximated to a diffusion process which leads to progressive expansion of the entire beam; this effect is independent of Q and receives contributions from resonances of all orders. According to the simple theory of this paper, this effect is sufficiently serious to warrant more thorough investigation.

Thermodynamics is usually subdivided into a theory dealing with equilibrium and into one concerned with irreversible processes. In the present paper this subdibvision is carried further and the Gibbsian thermodynamics of phase equilibrium is distinguished from the thermodynamics of Clausius and Kelvin. The latter was put into an axiomatic form by Caratheodory; the present paper attempts a similar task for the Gibbs theory. The formulation of this theory as an autonomous logical structure reveals characteristic aspects that were not evident until the two logical structures were differentiated. The analysis of the basic assumptions of the Gibbs theory allows the identification and removal of defects that marred the classical formulation. In the new theory thermodynamic systems are defined as conjunctions of spatially disjoint volume elements (subsystems), each of which is characterized by a set of additive conserved quantities (invariants): the internal energy, and the mole numbers of the independent chemical components. For the basic theory, it is convenient to assume the absence of elastic, electric, and magnetic effects. This restriction enables us to define thermodynamic processes as transfers of additive invariants between subsystems. Following Gibbs, we postulate that all thermostatic properties of system are contained in a fundamental equation representing the entropy as a function of the additive invariants. Geometrically, this equation is represented as a surface in a space to which we refer as Gibbs space. In order to make the information contained in the fundamental equation complete, we have to use, in many cases, additional quasi-thermodynamic variables to specify the intrinsic symmetry properties of the system. The walls, or boundaries limiting thermodynamic systems are assumed to be restrictive or nonrestrictive with respect to the transfer of the various invariants. The manipulations of the boundary conditions (imposition and relaxation of constraints) are called thermodynamic operations. In systems with nonrestrictive internal boundaries, the constraints are consistent with infinitely many distributions of the invariants over the subsystems. These virtual states serve as comparison states for the entropy maximum principle. This principle allows us to identify the state of thermodynamic equilibrium, attained asymptotically by real systems. The thermostatic extremum principle is the basis of a theory of stability. Stability may be normal or critical. In the latter case, the compliance coefficients (specific heat c p , expansion coefficient, and isothermal compressibility) tend, in general, to infinity. The phases of thermodynamic systems are each represented by a primitive surface in Gibbs space, the points of these surfaces correspond to modifications of the phase. The actual distribution of the invariants of a system over phases is determined by the entropy maximum principle. Among the fundamental theorems are the two phase rules. The first rule specifies the dimension of the set of points in the space of intensities, in which a given number of modifications can coexist. The second rule specifies the dimensionality of the set of critical points. The phase rules are somewhat more general than those of Gibbs because of our use of symmetry considerations. At a critical point two modifications become identical, and we obtain critical points of two kinds: (i) the modifications differ in densities, as for liquid and vapor; (ii) the modifications differ in symmetry, as the two directions of ordering in the Ising model. The second case relates to the well-known λ-points and λ-lines in the p - T diagram, hence these phenomena fit into the framework of the theory without ad hoc assumptions. For the case of liquid helium, this interpretation requires that the superfluid ground stage be degenerate. This conclusion is not inconsistent with the third law, but it requires substantiation by quantum-mechanical methods. Another fundamental theorem is the principle of thermostatic determinism: a reservoir of given intensities determines the densities (energy and components per unit volume) of a system which is in equilibrium with it and, conversely, the densities of the system determine the intensities. The mutual determination is unique, except at critical points and at absolute zero. A more satisfactory description of these singular situations calls for the use of statistical methods. The present approach leads to several statistical theories, the simplest of which is developed in the second paper of this series.

In this paper we continue the investigation of a model for the description of irreversible processes which we had proposed in an earlier publication. This model permits the construction of Gibbs-type ensembles for open systems not in equilibrium. The internal dynamics of the system that is engaged in a nonequilibrium process is assumed to be described fully by its Hamiltonian. Its interaction with its surroundings, i.e. the driving reservoirs, is described in terms of impulsive interactions (collisions). The reservoirs themselves possess definite temperatures, are inexhaustible, and are free of internal gradients (i.e. they are temperature baths). The ensemble obeys an integro-differential equation in Γ-space, containing both the terms of the Liouville equation and a stochastic integral term that describes the collisions with the reservoirs. It is shown in this paper that, under very general assumptions, all distributions approach each other in the course of time. If there exists a stationary solution, it will be unique and will be approached asymptotically by every time-dependent solution. In general the stationary state does not represent thermodynamic equilibrium; the ensemble remains unchanged only because its surroundings maintain temperature gradients inside the thermodynamic system. Only if these surroundings are all at one temperature, i.e. if the system is in contact with but one reservoir, then the stationary state will correspond to the canonical distribution. As a result, the stochastic integral kernel that describes the effect of collisions with the reservoir will satisfy certain symmetry conditions. A detailed investigation of our micro-model shows that these conditions are indeed satisfied if the reservoir components are themselves in a canonical distribution prior to collision. In the presence of several reservoirs at slightly different temperatures, the Onsager reciprocal relations are satisfied by the stationary distribution. In our model the Onsager relations are thus obtained without an appeal to fluctuation theory, and without the assumption that detailed balancing holds for the elementary stochastic processes, i.e. for the interactions between system and reservoir. In the latter part of the paper, finally, we consider reservoirs that maintain thermodynamic potentials in addition to the temperature, including chemical potentials. It turns out that our principal results are unaffected by this generalization.

Among the unitary irreducible representations of the Poincare group discovered by Wigner, there occurs a class of mass zero representations which are, for historical reasons, usually referred to as continuous spin representations, and are labeled by values of the Casimir invariants P2 = 0, WμWμ = −ϱ2, with ϱ > 0. A corresponding particle can exist in a denumerable set of helicity states λ, with λ taking either all integer values λ=0, ±1, ±2,… or all half-odd integer values λ = ± 1 2 , ± 3 2 ,…. We construct free quantum fields for such particles and also study their interactions to lowest order in the coupling constant. The fields are of necessity infinite component fields, and their commutator is found to vanish at most in proper subsets of the spacelike region. Particular attention is paid to the behavior of the theory for small ϱ. In the limit ϱ → 0, the fields go over into those given by Bender, Frishman and Itzykson for the description of a single ordinary massless particle with fixed helicity. The limiting behavior of S-matrix elements is also studied, and it is demonstrated how predictions of conventional finite component field theory, including the helicity selection rules, can be recovered in the limit ϱ → 0. The interactions studied in this paper are at least quadratic in the fields of continuous spin particles. Currents and interaction Hamiltonians which contain them only linearly need a separate investigation. This will be carried out in a subsequent paper.

The hierarchy of distribution-function kinetic equations for a many-body system is usually derived from the Liouville equation, incorporating the elegance of ensemble theory. In this paper emphasis is placed upon the physics of the interactions among particles and the kinetic equations are obtained by considering the time rate of change of the number of particles in phase space due to the flow of particles over phase-space boundaries. In this treatment the connection with experimental requirements is made clear and the distribution functions are identified with coarse grained space and time averages. This paper deals only with nonrelativistic, classical, structureless particles, although guidelines are given for the general treatment whenever approximations are made. The general approach developed here is used to generate kinetic equations for the one-body, the two-body equal-time, and the two-body different-time distribution functions. The remainder of the paper treats the case of an infinite homogeneous system, with a constant ionic background and only Coulomb interactions between electrons. Effective screening potentials are defined whose efficacy is greatest near thermal equilibrium. The one- and two-body kinetic equations are shown to have conservation properties and are analyzed in an energy-angular momentum phase space. The kinetic equation for the correlation function is analyzed to estimate the phase space region of validity for the truncated adiabatic form. Using the effective potentials, a formal solution for g is obtained which leads to a one-particle kinetic equation with no divergence difficulties that is a Boltzmann equation with a distribution function-dependent cross section. This kinetic equation is shown to have an H-theorem to lowest significant order in the plasma parameter. Two phase-space regions are identified, based upon the influence of the effective potential, and an approximate solution for the distribution function is obtained for each region. This analysis shows that, away from thermal equilibrium, the correlation function has an inverse-power asymptotic behavior rather than exponential. Finally, the one-body kinetic equation is shown to reduce in the appropriate region to the Lenard-Balescu form.

We describe the steady-state electron distribution of a high temperature, spatially inhomogeneous semiconductor subject to a large magnetic and an infinitesimal electric field, using classical mechanics (strictly valid if the range of the potential describing the inhomogeneities is considerably larger than the thermal electron wavelength), by a modified Boltzmann equation retaining both spatial and velocity variables. Such a classical transport equation can also describe the behavior of electrons in a plasma subject to given spatially dependent density fluctuations. In the case of the semiconductor the collision term refers to collisions with phonons which we treat in this paper in the relaxation time approximation with a constant relaxation time, this being our assumption (1). For mathematical convenience we restrict ourselves in this paper to a stratified medium in which the inhomogeneities are distributed in sheets perpendicular to say the x-axis, this being our assumption (2). We obtain the asymptotic solution of the transport equation in reciprocal powers of the strength of the magnetic field H . The velocity moments corresponding to the first few terms of the solution are identical with the solution of the velocity moment (4, 10, etc.) equations obtained by retaining only 2, 3, etc. generalized Hermite polynomials in the electron velocity, in the representation of the solution by these polynomials. These moment solutions provide us, in principle, with a method of solving approximately a general linear transport equation for which assumption (1) is dropped. The leading term of the asymptotic solution corresponds to a macroscopic theory of the high field magnetoresistance obtained by Herring. The transverse magnetoresistance does not saturate but increases indefinitely with H 2, while there in no longitudinal magnetoresistance in our stratified medium. The general first correction to the macroscopic theory results are explicity obtained. The mathematical methods can in certain instances be extended to more realistic inhomogeneity distributions, i.e., not subject to assumption (2). These extensions will be the subject of a subsequent paper. The physical significance of these mathematical results is noted.

In this first paper, the scattering amplitude given by its partial waves expansion is shown to obey an integral representation of the form of a Bessel transform. This integral representation appears to be valid for all energies and in the whole angular domain (0,π) and coincides with the expression previously proposed by Blankenbecler and Goldberger for high energy amplitudes. The expression of the spectral function in terms of the partial waves is given together with its inversion. Decomposing the amplitude in its even and odd part (with respect to cosθ), an alternate equivalent integral representation is similarly deduced which exhibits in a simpler way the even/odd character of the amplitude. The connections between the two representations is then discussed. Spin complications are ignored throughout the whole paper.

The exchange operator 1 (1 − K) , introduced in an earlier paper, is discussed. To this end, the eigen-problem for the K-matrix is examined and examples are given. The role of the exchange operator is interpreted in the context of exchange scattering and the Pauli principle with emphasis placed on the treatment of spurious levels arising from these effects. An effective Schrodinger equation for open-channel amplitudes alone is constructed from a truncated set of coupled equations and the components of the Hamiltonian in this equation are examined and illustrated. Two calculations are discussed, one applying the formal treatment of this and the earlier paper to a shell-model-type calculation of n-−16O elastic scattering, the second treating the same problem but omitting certain exchange corrections. The results of the two calculations are compared.

In this and the accompanying paper, we investigate the Kinetic Theory of collisionless systems in closed, anisotropic (Bianchi Type IX) cosmologies including the effects of rotation. In this paper we show, for collisionless radiation which was produced in a thermal process (i.e. which had a thermal distribution at one instant), that the energy density T00 can be written ϱν[1 + Vν(β, t)], where ϱν∞R−4 and Vν(β, t)≧0 is a function of the anisotropy, β and of the time. The exact form of Vν cannot be given explicitly, but we give several limiting cases, including the small anisotropy limit. We discuss the behavior of the distribution function for two special cases (subsets of the full rotating situation). We present a Lagrangian for the behavior of the anisotropy in these models. The contribution of the collisionless radiation to the motion of the anisotropy is given by the appearance of a term proportional to Vν in the Lagrangian. We discuss the meaning of the angular momentum constraints embodied in the T0i = G0i equations. We analyze the behavior of the models in the large anisotropy regime in terms of “wall collisions.” We discuss the geodesic equation, and show that it can be reduced to a one-dimensional time dependent Hamiltonian system. We show that in the large anisotropy regime, certain parameters of the geodesic motion are essentially constant, if cosmological parameters were the same in the distant past, as they are today. (This is taken to suggest that dissipation may have occurred in the interim to change those parameters). The accompanying paper will give an explicit calculation in the small anisotropy regime, and will apply these results to the observations of the microwave background radiation.

We integrate the infinitesimal generators of the proper, orthochronous, inhomogeneous Lorentz group for the nonzero mass case when the infinitesimal generators are given in the Foldy-Shirokov form. We also integrate the infinitesimal generators for the zero mass case when they are given in the Lomont-Moses form. The results are essentially the same as those given by Ritus, but we use a notation which we believe is more convenient for our purposes. The formulas of the present paper will be used to obtain properties of generalized spherical harmonics and to obtain the relativistic quantization of the electromagnetic vector potential in an angular momentum basis in papers which we shall write shortly.

The one-dimensional molecular crystal model of polaron motion, developed in parts I and II, is suitably generalized to consider the existence of a Hall effect. As in II, the treatment is confined to the case for which the electronic-overlap term of the total Hamiltonian is a small perturbation. In zeroth order—i.e., in the absence of this term—the electron is localized at a given site. The vibrational state of the system is specified by a set of quantum numbers, Nk, giving the degree of excitation of each vibrational mode. The existence of a nonvanishing electronic bandwidth then gives rise to transitions to neighboring sites. Of principal interest in the present paper is the high temperature regime (T > Tt, as defined in II) where polaron motion is predominantly by means of random jumps between neighboring sites. Although the lowest order jump rate is adequate in considering the polaron drift mobility, higher order processes, involving the occupation of (at least) three sites, must be taken into account in treating the Hall effect. In particular, it is demonstrated the relative probability of the electron, initially located on one of the three sites, hopping to one or the other of the remaining two sites, is modified by a contribution which, both in sign and magnitude, is linearly proportional to the applied magnetic field. This effect is shown to arise from the interference between the amplitude for the direct transition from the initial to the final site, and the amplitude for an indirect, second order transition, involving intermediate occupancy of the third site. The (magnetic) field induced components of the jump rates, corresponding to the above processes, are first calculated by a classical occurrence-probability approach which treats the lattice vibrational coordinates as given functions of time. The second approach presents a full quantum mechanical calculation of the jump rates. The results of this treatment agree with those of the occurrence-probability approach in the classical limit (T ⪢ ΘDe bye). The order of magnitude of the calculated Hall Coefficients are found to be greater than or comparable to the “normal” result ( R = −1 nec ) depending on whether or not the three sites involved in the transition are mutually nearest neighbors. A final note corrects an error in a previous paper by one of the authors (T. H.) concerning the sign of the Hall effect in impurity conduction.

This paper is a sequel to an earlier paper that described nonrelativistic quantum electrodynamics in terms of the time symmetric theory of direct interparticle action. The restriction to a nonrelativistic treatment is removed in the present paper. The path integral approach to quantum mechanics is extended to include relativistic particles with spin as a preliminary to achieving this end. The response of the Universe can be obtained in two ways, by an extension of the method of the previous paper or by using a general condition analogous to the general condition, Σparticles[Fret − Fadv] = 0, of Wheeler and Feynman. The second method is used in the present paper since it avoids the need to calculate the detailed properties of the absorber. The rules of quantum electrodynamics in the form stated by Feynman are derived, thereby showing that all the practical results of the usual theory of quantum electrodynamics can also be obtained from the direct particle theory. Radiation corrections are discussed. The direct particle theory throws some light on the processes of renormalization. The point of view that both mass and charge should be finite before as well as after renormalization is partially developed.

The nuclear physics of the spallation reactions leading to the formation of lithium, beryllium, and boron from proton bombardment on light nuclei is reviewed. A large amount of experimental data is gathered, including in particular several measurements made at Orsay with the 150-MeV synchrocyclotron. A simple rule is devised which classifies quite neatly the cross sections in term of their isotopic spin parameters. This rule can be used to predict the results of unmeasured reactions. For 12C and 16O target nuclei, more accurate determinations are made with the help of calculations based on statistical models. All this information is used to determine the formation ratios of the Li, Be, and B isotopes in CNONe, first by protons of fixed energies, second by cosmic ray protons with an extended energy spectrum. We do not know of course the exact shape of the spectrum giving rise to these Li Be B, but we may assume that they are of the type commonly found in cosmic ray physics: exponential in rigidity or power of kinetic energy. Fortunately, the isotopic ratios thereby obtained are largely insensitive to the shape of the spectrum. Different ratios could of course be obtained from certain special types of spectra (for instance from much steeper spectra). This possibility is only briefly discussed here; it will be more fully discussed in a companion paper. The nuclear reactions possibly responsible for the alteration of the ratios after their formation are discussed next. Finally, the observations of these elements in natural settings are summarized, compared and discussed in terms of the formation and destruction rates, and some conclusions are drawn. The astronomical information gathered so far is consistent with the following picture: each star generates its own Li, Be, B, by proton irradiation of the stellar atmosphere. The isotopes 6Li and 7Li are later partly depleted by proton capture reactions, presumably at the bottom of the convective zone. One of the main results of this paper is that, within the uncertainty of the evaluation, the meteoritic 11B to 10B ratio is equal to its formation ratio on CNONe. This result therefore leads, for the formation of the light elements in the solar system, to a model somewhat different from the FGH one. The 7Li to 6Li formation ratio has a value smaller than the meteoritic one, but the alteration can be ascribed to (p, α) reactions at temperatures between 2 and 4 million degrees at the bottom of the surface convective zone of the young sun, and this process would not alter the 11B10B ratio. A tentative picture of the history of the solar system would go as follows: from energetic considerations, we are led to think that the Li Be B have been formed even before the Hayashi (fully convective) phase of the sun. The 7Li6Li ratio would then have been altered during the Hayashi phase. This material would still later have been separated from the sun. An alternative picture is also sketched, based on the assumption of a special type of proton spectrum, of the kind mentioned above.