A Quantum Legacy

The following sections are included:

• PREFACE

• Contents

The following sections are included:

• A Brief Life of Schwinger

• Quantum Electrodynamics

• Covariant Quantum Electrodynamics

• Quantum Action Principle

• Field Theory

• Measurement Algebra

• Electroweak Synthesis

• The Nobel Prize and Reaction

• Source Theory and UCLA

• Thomas-Fermi Atom, Cold Fusion, and Sonoluminescence

• Conclusion

• References

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Attempts to avoid the divergence difficulties of quantum electrodynamics by mutilation of the theory have been uniformly unsuccessful. The lack of convergence does indicate that a revision of electrodynamic concepts at ultra-relativistic energies is indeed necessary, but no appreciable alteration of the theory for moderate relativistic energies can be tolerated. The elementary phenomena in which divergences occur, in consequence of virtual transitions involving particles with unlimited energy, are the polarization of the vacuum and the self-energy of the electron, effects which essentially express the interaction of the electromagnetic and matter fields with their own vacuum fluctuations. The basic result of these fluctuation interactions is to alter the constants characterizing the properties of the individual fields, and their mutual coupling, albeit by infinite factors. The question is naturally posed whether all divergences can be isolated in such unobservable renormalization factors; more specifically, we inquire whether quantum electrodynamics can account unambiguously for the recently observed deviations from the Dirac electron theory, without the introduction of fundamentally new concepts. This paper, the first in a series devoted to the above question, is occupied with the formulation of a completely covariant electrodynamics. Manifest covariance with respect to Lorentz and gauge transformations is essential in a divergent theory since the use of a particular reference system or gauge in the course of calculation can result in a loss of covariance in view of the ambiguities that may be the concomitant of infinities. It is remarked, in the first section, that the customary canonical commutation relations, which fail to exhibit the desired covariance since they refer to field variables at equal times and different points of space, can be put in covariant form by replacing the four-dimensional surface t=const. by a space-like surface. The latter is such that light signals cannot be propagated between any two points on the surface. In this manner, a formulation of quantum electrodynamics is constructed in the Heisenberg representation, which is obviously covariant in all its aspects. It is not entirely suitable, however, as a practical means of treating electrodynamic questions, since commutators of field quantities at points separated by a time-like interval can be constructed only by solving the equations of motion. This situation is to be contrasted with that of the Schrödinger representation, in which all operators refer to the same time, thus providing a distinct separation between kinematical and dynamical aspects. A formulation that retains the evident covariance of the Heisenberg representation, and yet offers something akin to the advantage of the Schrödinger representation can be based on the distinction between the properties of non-interacting fields, and the effects of coupling between fields. In the second section, we construct a canonical transformation that changes the field equations in the Heisenberg representation into those of non-interacting fields, and therefore describes the coupling between fields in terms of a varying state vector. It is then a simple matter to evaluate commutators of field quantities at arbitrary space-time points. One thus obtains an obviously covariant and practical form of quantum electrodynamics, expressed in a mixed Heisenberg-Schrödinger representation, which is called the interaction representation. The third section is devoted to a discussion of the covariant elimination of the longitudinal field, in which the customary distinction between longitudinal and transverse fields is replaced by a suitable co-variant definition. The fourth section is concerned with the description of collision processes in terms of an invariant collision operator, which is the unitary operator that determines the over-all change in state of a system as the result of interaction. It is shown that the collision operator is simply related to the Hermitian reaction operator, for which a variational principle is constructed.

The covariant formulation of quantum electrodynamics, developed in a previous paper, is here applied to two elementary problems—the polarization of the vacuum and the self-energies of the electron and photon. In the first section the vacuum of the non-interacting electromagnetic and matter fields is covariantly defined as that state for which the eigenvalue of an arbitrary time-like component of the energy-momentum four-vector is an absolute minimum. It is remarked that this definition must be compatible with the requirement that the vacuum expectation values of a physical quantity in various coordinate systems should be, not only covariantly related, but identical, since the vacuum has a significance that is independent of the coordinate system. In order to construct a suitable characterization of the vacuum state vector, a covariant decomposition of the field operators into positive and negative frequency components is introduced, and the properties of these associated fields developed. It is shown that the state vector for the electromagnetic vacuum is annihilated by the positive frequency part of the transverse four-vector potential, while that for the matter vacuum is annihilated by the positive frequency part of the Dirac spinor and of its charge conjugate. These defining properties of the vacuum state vector are employed in the calculation of the vacuum expectation values of quadratic field quantities, specifically the energy-momentum tensors of the independent electromagnetic and matter fields, and the current four-vector. It is inferred that the electromagnetic energy-momentum tensor, and the current vector must vanish in the vacuum, while the matter field energy-momentum tensor vanishes in the vacuum only by the addition of a suitable multiple of the unit tensor. The second section treats the induction of a current in the vacuum by an external electromagnetic field. It is supposed that the latter does not produce actual electron-positron pairs; that is, we consider only the phenomenon of virtual pair creation. This restriction is introduced by requiring that the establishment and subsequent removal of the external field produce no net change in state for the matter field. It is demonstrated, in a general manner, that the induced current at a given space-time point involves the external current in the vicinity of that point, and not the electromagnetic potentials. This gauge invariant result shows that a light wave, propagating at remote distances from its source, induces no current in the vacuum and is therefore undisturbed in its passage through space. The absence of a light quantum self-energy effect is thus indicated. The current induced at a point consists, more precisely, of two parts: a logarithmically divergent multiple of the external current at that point, which produces an unobservable renormalization of charge, and a more involved finite contribution, which is the physically significant induced current. The latter agrees with the results of previous investigations. The modification of the matter field properties arising from interaction with the vacuum fluctuations of the electromagnetic field is considered in the third section. The analysis is carried out with two alternative formulations, one employing the complete electromagnetic potential together with a supplementary condition, the other using the transverse potential, with the variables of the supplementary condition eliminated. It is noted that no real processes are produced by the first order coupling between the fields. Accordingly, alternative equations of motion for the state vector are constructed, from which the first order interaction term has been eliminated and replaced by the second order coupling which it generates. The latter includes the self action of individual particles and light quanta, the interaction of different particles, and a coupling between particles and light quanta which produces such effects as Compton scattering and two quantum pair annihilation. It is concluded from a comparison of the alternative procedures that, for the treatment of virtual light quantum processes, the separate consideration of longitudinal and transverse fields is an inadvisable complication. The light quantum self-energy term is shown to vanish, while that for a particle has the anticipated form for a change in proper mass, although the latter is logarithmically divergent, in agreement with previous calculations. To confirm the identification of the self-energy effect with a change in proper mass, it is shown that the result of removing this term from the state vector equation of motion is to alter the matter field equations of motion in the expected manner. It is verified, finally, that the energy and momentum modifications produced by self-interaction effects are entirely accounted for by the addition of the electromagnetic proper mass to the mechanical proper mass—an unobservable mass renormalization. An appendix is devoted to the construction of several invariant functions associated with the electromagnetic and matter fields.

The discussion of vacuum polarization in the previous paper of this series was confined to that produced by the field of a prescribed current distribution. We now consider the induction of current in the vacuum by an electron, which is a dynamical system and an entity indistinguishable from the particles associated with vacuum fluctuations. The additional current thus attributed to an electron implies an alteration in its electromagnetic properties which will be revealed by scattering in a Coulomb field and by energy level displacements. This paper is concerned with the computation of the second-order corrections to the current operator and the application to electron scattering. Radiative corrections to energy levels will be treated in the next paper of the series. Following a canonical transformation which effectively renor-malizes the electron mass, the correction to the current operator produced by the coupling with the electromagnetic field is developed in a power series, of which first- and second-order terms are retained. One thus obtains second-order modifications in the current operator which are of the same general nature as the previously treated vacuum polarization current, save for a contribution that has the form of a dipole current. The latter implies a fractional increase of α/2π in the spin magnetic moment of the electron. The only flaw in the second-order current correction is a logarithmic divergence attributable to an infra-red catastrophe. It is remarked that, in the presence of an external field, the first-order current correction will introduce a compensating divergence. Thus, the second-order corrections to particle electromagnetic properties cannot be completely stated without regard for the manner of exhibiting them by an external field. Accordingly, we consider in the second section the interaction of three systems, the matter field, the electromagnetic field, and a given current distribution. It is shown that this situation can be described in terms of an external potential coupled to the current operator, as modified by the interaction with the vacuum electromagnetic field. Application is made to the scattering of an electron by an external field, in which the latter is regarded as a small perturbation. It is found convenient to calculate the total rate at which collisions occur and then identify the cross sections for individual events. The correction to the cross section for radia-tionless scattering is determined by the second-order correction to the current operator, while scattering that is accompanied by single quantum emission is a consequence of the first-order current correction. The final object of calculation is the differential cross section for scattering through a given angle with a prescribed maximum energy loss, which is completely free of divergences. Detailed evaluations are given in two situations, the essentially elastic scattering of an electron, in which only a small fraction of the kinetic energy is radiated, and the scattering of a slowly moving electron with unrestricted energy loss. The Appendix is devoted to an alternative treatment of the polarization of the vacuum by an external field. The conditions imposed on the induced current by the charge conservation and gauge invariance requirements are examined. It is found that the fulfillment of these formal properties requires the vanishing of an integral that is not absolutely convergent, but naturally vanishes for reasons of symmetry. This null integral is then used to simplify the expression for the induced current in such a manner that direct calculation yields a gauge invariant result. The induced current contains a logarithmically divergent multiple of the external current, which implies that a non-vanishing total charge, proportional to the external charge, is induced in the vacuum. The apparent contradiction with charge conservation is resolved by showing that a compensating charge escapes to infinity. Finally, the expression for the electromagnetic mass of the electron is treated with the methods developed in this paper.

This paper is based on the elementary remark that the extraction of gauge invariant results from a formally gauge invariant theory is ensured if one employs methods of solution that involve only gauge covariant quantities. We illustrate this statement in connection with the problem of vacuum polarization by a prescribed electromagnetic field. The vacuum current of a charged Dirac field, which can be expressed in terms of the Green's function of that field, implies an addition to the action integral of the electromagnetic field. Now these quantities can be related to the dynamical properties of a “particle” with space-time coordinates that depend upon a proper-time parameter. The proper-time equations of motion involve only electromagnetic field strengths, and provide a suitable gauge invariant basis for treating problems. Rigorous solutions of the equations of motion can be obtained for a constant field, and for a plane wave field. A renormalization of field strength and charge, applied to the modified lagrange function for constant fields, yields a finite, gauge invariant result which implies nonlinear properties for the electromagnetic field in the vacuum. The contribution of a zero spin charged field is also stated. After the same field strength renormalization, the modified physical quantities describing a plane wave in the vacuum reduce to just those of the maxwell field; there are no nonlinear phenomena for a single plane wave, of arbitrary strength and spectral composition. The results obtained for constant (that is, slowly varying fields), are then applied to treat the two-photon disintegration of a spin zero neutral meson arising from the polarization of the proton vacuum. We obtain approximate, gauge invariant expressions for the effective interaction between the meson and the electromagnetic field, in which the nuclear coupling may be scalar, pseudoscalar, or pseudovector in nature. The direct verification of equivalence between the pseudoscalar and pseudovector interactions only requires a proper statement of the limiting processes involved. For arbitrarily varying fields, perturbation methods can be applied to the equations of motion, as discussed in Appendix A, or one can employ an expansion in powers of the potential vector. The latter automatically yields gauge invariant results, provided only that the proper-time integration is reserved to the last. This indicates that the significant aspect of the proper-time method is its isolation of divergences in integrals with respect to the proper-time parameter, which is independent of the coordinate system and of the gauge. The connection between the proper-time method and the technique of “invariant regularization” is discussed. Incidentally, the probability of actual pair creation is obtained from the imaginary part of the electromagnetic field action integral. Finally, as an application of the Green's function for a constant field, we construct the mass operator of an electron in a weak, homogeneous external field, and derive the additional spin magnetic moment of α/2π magnetons by means of a perturbation calculation in which proper-mass plays the customary role of energy.

The conventional correspondence basis for quantum dynamics is here replaced by a self-contained quantum dynamical principle from which the equations of motion and the commutation relations can be deduced. The theory is developed in terms of the model supplied by localizable fields. A short review is first presented of the general quantum-mechanical scheme of operators and eigenvectors, in which emphasis is placed on the differential characterization of representatives and transformation functions by means of infinitesimal unitary transformations. The fundamental dynamical principle is stated as a variational equation for the transformation function connecting eigenvectors associated with different spacelike surfaces, which describes the temporal development of the system. The generator of the infinitesimal transformation is the variation of the action integral operator, the space-time volume integral of the invariant lagrange function operator. The invariance of the lagrange function preserves the form of the dynamical principle under coordinate transformations, with the exception of those transformations which include a reversal in the positive sense of time, where a separate discussion is necessary. It will be shown in Sec. III that the requirement of invariance under time reflection imposes a restriction upon the operator properties of fields, which is simply the connection between the spin and statistics of particles. For a given dynamical system, changes in the transformation function arise only from alterations of the eigenvectors associated with the two surfaces, as generated by operators constructed from field variables attached to those surfaces. This yields the operator principle of stationary action, from which the equations of motion are obtained. Commutation relations are derived from the generating operator associated with a given surface. In particular, canonical commutation relations are obtained for those field components that are not restricted by equations of constraint. The surface generating operator also leads to generalized Schrödinger equations for the representative of an arbitrary state. Action integral variations which correspond to changing the dynamical system are discussed briefly. A method for constructing the transformation function is described, in a form appropriate to an integral spin field, which involves solving Hamilton-Jacobi equations for ordered operators. In Sec. III, the exceptional nature of time reflection is indicated by the remark that the charge and the energy-momentum vector behave as a pseudoscalar and pseudovector, respectively, for time reflection transformations. This shows, incidentally, that positive and negative charge must occur symmetrically in a completely covariant theory. The contrast between the pseudo energy-momentum vector and the proper displacement vector then indicates that time reflection cannot be described within the unitary transformation framework. This appears most fundamentally in the basic dynamical principle. It is important to recognize here that the contributions to the lagrange function of half-integral spin fields behave like pseudoscalars with respect to time reflection. The non-unitary transformation required to represent time reflection is found to be the replacement of a state vector by its dual, or complex conjugate vector, together with the transposition of all operators. The fundamental dynamical principle is then invariant under time reflection if inverting the order of all operators in the lagrange function leaves an integral spin contribution unaltered, and reverses the sign of a half-integral spin contribution. This implies the essential commutativity, or anti-commutativity, of integral and half-integral field components, respectively, which is the connection between spin and statistics.

The arguments leading to the formulation of the action principle for a general field are presented. In association with the complete reduction of all numerical matrices into symmetrical and antisymmetrical parts, the general field is decomposed into two sets, which are identified with Bose-Einstein and Fermi-Dirac fields. The spin restriction on the two kinds of fields is inferred from the time reflection invariance requirement. The consistency of the theory is verified in terms of a criterion involving the various generators of infinitesimal transformations. Following a discussion of charged fields, the electromagnetic field is introduced to satisfy the postulate of general gauge invariance. As an aspect of the latter, it is recognized that the electromagnetic field and charged fields are not kinematically independent. After a discussion of the field strength commutation relations, the independent dynamical variables of the electromagnetic field are exhibited in terms of a special gauge.

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The problem of calculating nonadiabatic transition probabilities is considered. It is shown that the general Güttinger equations are incorrect and lead to erroneous results in any case other than that of the rotating magnetic field, which he considered. The corrected equations are applied in the calculation of the transition probabilities between the various magnetic states of a field precessing with constant angular velocity.

The commutation relations of-an arbitrary angular momentum vector can be reduced to those of the harmonic oscillator. This provides a powerful method for constructing and developing the properties of angular momentum eigenvectors. In this paper many known theorems are derived in this way, and some new results obtained. Among the topics treated are the properties of the rotation matrices; the addition of two, three, and four angular momenta; and the theory of tensor operators.

This retrospective paper is submitted in witness of my appreciation and affection for I. I. Rabi …

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A systematic treatment is presented of the application of variational principles to the quantum theory of scattering.

Starting from the time-dependent theory, a pair of variational principles is provided for the approximate calculation of the unitary (collision) operator that describes the connection between the initial and final states of the system. An equivalent formulation of the theory is obtained by expressing the collision operator in terms of an Hermitian (reaction) operator; variational principles for the reaction operator follow. The time-independent theory, including variational principles for the operators now used to describe transitions, emerges from the time-dependent theory by restricting the discusson to stationary states. Specialization to the case of scattering by a central force field establishes the connection with the conventional phase shift analysis and results in a variational principle for the phase shift.

As an illustration, the results of Fermi and Breit on the scattering of slow neutrons by bound protons are deduced by variational methods.

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This paper is concerned with the properties of the radiation from a high energy accelerated electron, as recently observed in the General Electric synchrotron. An elementary derivation of the total rate of radiation is first presented, based on Larmor's formula for a slowly moving electron, and arguments of relativistic invariance. We then construct an expression for the instantaneous power radiated by an electron moving along an arbitrary, prescribed path. By casting this result into various forms, one obtains the angular distribution, the spectral distribution, or the combined angular and spectral distributions of the radiation. The method is based on an examination of the rate at which the electron irreversibly transfers energy to the electromagnetic field, as determined by half the difference of retarded and advanced electric field intensities. Formulas are obtained for an arbitrary charge-current distribution and then specialized to a point charge. The total radiated power and its angular distribution are obtained for an arbitrary trajectory. It is found that the direction of motion is a strongly preferred direction of emission at high energies. The spectral distribution of the radiation depends upon the detailed motion over a time interval large compared to the period of the radiation. However, the narrow cone of radiation generated by an energetic electron indicates that only a small part of the trajectory is effective in producing radiation observed in a given direction, which also implies that very high frequencies are emitted. Accordingly, we evaluate the spectral and angular distributions of the high frequency radiation by an energetic electron, in their dependence upon the parameters characterizing the instantaneous orbit. The average spectral distribution, as observed in the synchrotron measurements, is obtained by averaging the electron energy over an acceleration cycle. The entire spectrum emitted by an electron moving with constant speed in a circular path is also discussed. Finally, it is observed that quantum effects will modify the classical results here obtained only at extraordinarily large energies.

A rigorous and explicit solution is obtained for the problem of sound radiation from an unflanged circular pipe, assuming axially symmetric excitation. The solution is valid throughout the wave-length range of dominant mode (plane wave) propagation in the pipe. The reflection coefficient for the velocity potential within the pipe and the power-gain function, embodying the characteristics of the radiation pattern, are evaluated numerically. The absorption cross section of the pipe for a plane wave incident from external space, and the gain function for this direction, are found to satisfy a reciprocity relation. In particular, the absorption cross section for normal incidence is just the area of the mouth. At low frequencies of vibration, the velocity potential within the pipe is the same as if the pipe were lengthened by a certain fraction of the radius and the open end behaved as a loop. The exact value of the end correction turns out to be 0.6133.

The diffraction of a scalar plane wave by an aperture in an infinite plane screen is examined theoretically. The wave function at an arbitrary point in space is expressed in terms of its values in the aperture, and constructed so as to vanish on the screen, in accordance with the assumed boundary condition. An integral equation to determine the aperture field is obtained from the continuity requirement for the normal derivative of the wave function on traversing the plane of the aperture. Utilizing the integral equation (whose solution is generally unobtainable), the amplitude of the diffracted spherical wave at large distances from the aperture is exhibited in a form which is stationary with respect to small variations (relative to the correct values) of the aperture fields arising from a pair of incident waves. This expression is independent of the scale of the aperture fields. The transmission cross section of the aperture for a plane wave is found to be simply related to the diffracted amplitude observed in the direction of incidence. The variational formulation is applied in detail for a wave incident normally on a circular aperture. By comparison with the exact results available for this problem, it appears that the use of suitable trial aperture fields in the variational formulation yields approximate, yet accurate, expressions for the diffracted amplitude and transmission cross section over a wide range of frequencies.

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Non-Abelian vector gauge theory is given a first-order Lorentz gauge formulation and then transformed into the radiation gauge. The result agrees with the independently constructed radiation gauge theory. There is a brief discussion of the axial gauge.

A gravitational action operator is constructed that is invariant under general coordinate transformations and local Lorentz (gauge) transformations. To interpret the formalism the arbitrariness in description must be restricted by introducing gauge conditions and coordinate conditions. The time gauge is defined by locking the time axes of the local coordinate systems to the general coordinate time axis. The resulting form of the action operator, including the contribution of a spinless matter field, enables canonical pairs of variables to be identified. There are four field variables that lack canonical partners, in virtue of differential constraint equations, which can be interpreted as space-time coordinate displacements. In a physically distinguished class of coordinate system the gravitational field variables are not explicit functions of the coordinate displacement parameters. There remains the freedom of Lorentz transformation. The generators of spatial translations and rotations have the correct commutation properties. The question of Lorentz invariance is left undecided since the energy density operator is only given implicitly.

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This is the first of a series of papers dealing with many-particle systems from a unified, nonperturbative point of view. It contains derivations and discussions of various field-theoretical techniques which will be applied in subsequent papers. In a short introduction the general method of approach is summarized, and its relationship to other field-theoretic problems indicated. In the second section the macroscopic properties of the spectra of many-particle systems are described. Asymptotic evaluations are performed which characterize these macroscopic features in terms of intensive parameters, and the relationship of these parameters to thermodynamics is discussed. The special characteristics of the ground state are shown to follow as a limiting case of the asymptotic evaluations. The third section is devoted to the time-dependent field correlation functions, or Green's functions, which describe the microscopic behavior of a multiparticle system. These functions are defined, and related to intensive macroscopic variables when the energy and number of particles are large. Spectral representations and other properties of various one-particle Green's functions are derived. In the fourth section the treatment of non-equilibrium processes is considered. As a particular example, the electromagnetic properties of a system are expressed in terms of the special two-particle Green's function which describes current correlation. The discussion yields specifically a fluctuation-dissipation theorem, a sum rule for conductivity, and certain dispersion relations. The fifth section deals with the differential equations which determine the Green's functions. The boundary conditions that characterize the Green's function equations are exhibited without reference to adiabatic decoupling. A method for solving the equations approximately, by treating the correlations among successively larger numbers of particles, is considered. The first approximation in this sequence is shown to yield a generalized Hartree-like equation. A related, but rigorous, identity for the single-particle Green's function is then derived. A second approximation, which takes certain two-particle correlations into account, is shown to produce various additional effects: The interaction between particles is altered in a manner characterized by the intensive macroscopic parameters, and the modification and spread of the energy-momentum relation come into play. In the final section compact formal expressions for the Green's functions and other physical quantities are derived. Alternative equations and systematic approximations for the Green's functions are obtained.

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An earlier note¹ contains the initial stages in an evolution of the mathematical structure of quantum mechanics as the symbolic expression of the laws of microscopic measurement. The development is continued here. The entire discussion remains restricted to the realm of quantum statics which, in its lack of explicit reference to time, is concerned either with idealized systems such that all properties are unchanged in time or with measurements performed at a common time.

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When Humpty-Dumpty had his great fall nobody could put him together again. A vastly more moderate challenge is to reunite the two partial beams of a Stern–Gerlach apparatus with such precision that the original spin state is recovered. Nevertheless, as we demonstrate, a substantial loss of spin coherence always occurs, unless the experimenter is able to control the magnetic field's inhomogeneity with an accuracy of at least one part in 10⁵.

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A lecture delivered, on November 5, 1977, at a Session entitled "Physics in the Future" of a Symposium on the occasion of the fiftieth anniversary of the Pupin Laboratories (Columbia University) also honoring the many contributions of I. I. Rabi.

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A quantum field theory of magnetic and electric charge is constructed. It is verified to be relativistically invariant in consequence of the charge quantization condition eg/ħc=n, an integer. This is more restrictive than Dirac's condition, which would also allow half-integral values.

The nonrelativistic problem of the scattering of two dyons (including the case of electron scattering by magnetic monopoles) is systematically studied, both classically and quantum mechanically, with a view toward the discrimination between various combinations of electric and magnetic charges. We analyze the classical cross section with particular attention to the interesting phenomena which occur for large angle scattering, the “rainbows” and “glory,” where the cross section becomes infinite. Quantum mechanically, we find that these infinities do not occur and that, when the partial wave scattering amplitude is summed, a very elaborate structure emerges for the cross section, which depends sensitively upon the electric and magnetic charges of the particles, as well as on their relative speed. We further discuss a large modification, leading to spin flip and nonflip amplitudes, due to the dipole moments of the particles. Numerical results are presented for a variety of values of these parameters. In principle, these results could be used to distinguish the δ-ray distributions produced by the various species of electrically and magnetically charged particles. Quite apart from the experimental implications of our numerical results, we have made a number of theoretical improvements and extensions. Numbered among these are the consideration of dyons and particles having dipole moments, and the explicit demonstration, based on the methods of angular momentum, that the differential cross section is independent of the choice of singularity line.

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A new kind of particle theory is being explored, one that is intermediate in concept between the extremes of S matrix and field theory. It employs the methods of neither approach. There are no operators, and there is no appeal to analyticity in momentum space. It is a phenomenological theory, and cognizant that measurements are operations in space and time. Particles are defined realistically by reference to their creation or annihilation in suitable collisions. The source is introduced as an abstraction of the role played by all the other particles involved in such acts. Through the use of sources the production and detection of particles, as well as their interaction, are incorporated into the theoretical description. There is a creative principle that replaces the devices of other formulations. It is an insistence upon the generality, of the space-time description of the coupling among sources that is inferred from a specific spatio-temporal arrangement, in which various particles propagate between sources. Standard quantum-mechanical and relativistic requirements, imposed on the source description of noninteracting particles, imply the existence of the two statistics and the connection with spin. In this situation sources are only required to emit and absorb the mass of the corresponding particle. Particle dynamics is introduced by an extension of the source concept. It is considered meaningful for a source to emit several particles with the same total quantum numbers as a single particle, if sufficient mass is available. This is most familiar as the photon radiation that accompanies the emission of charged particles. The new types of sources introduced in this way imply new couplings among sources, which supply still further varieties of sources. This proliferation of interactions spans the full dynamical content of the initial primitive interaction. The ambition of the phenomenological source theory is to represent all dynamical aspects of particles, within a certain context, by a suitable primitive interaction. This paper is devoted to the reconstruction of electrodynamics.

Gravitational theory is reconstructed from the source description of gravitons. As indirect evidence for this starting point, the theoretical basis for the four tests of Einsteinian gravitational theory is produced by elementary arguments. Following the electromagnetic example, the source description is recast as a numerical field theory characterized by an action principle. After recognizing that the physically restricted theory possesses invariance with respect to infinitesimal coordinate transformations, it is generalized to exhibit invariance under arbitrary coordinate transformations, which supplies the primitive interactions for multigraviton processes. There is a discussion of the necessary dependence of graviton sources upon the gravitational field, and a simple model is constructed. The qualitative structure of the modification that multiparticle exchange introduces in the graviton-propagation function is exhibited, with the corresponding modification in the Newtonian potential. There are some speculative remarks about Mach's principle and the accompanying interpretation of the gravitational constant. The paper concludes by pointing out empirical scaling laws that interconnect the cosmos, the laboratory, and the atom.

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The existence of models based on smooth extrapolation from contiguous regions rather than on speculative dynamical assumptions is pointed out. Successful results already achieved in deep inelastic scattering are described and predictions are then made for the asymmetry in deep inelastic scattering of polarized particles. The predictions are consistent with the experimental values announced at this Conference. The details of the theory are outlined.

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We reconsider the Casimir (van der Waals) forces between dielectrics with plane, parallel surfaces for arbitrary temperature, using the methods of source theory. The general results of Lifshitz are confirmed, and are shown to imply the correct forces on metal surfaces. The same phenomena give rise to contributions to the surface tension and the latent heat of an idealized liquid, contributions which, unfortunately, are not well defined since they depend upon a momentum cutoff. However, with a reasonable value for this cutoff, qualitative agreement with the experimentally observed surface tension and latent heat of liquid helium at absolute zero is obtained.

Another derivation is presented, one that emphasizes the physical normalization conditions, and employs a mathematical property of a theta function that originated with S. D. Poisson.

A proper-energy method is used to find the energy of attraction between two dielectric slabs. Independent derivations of the attractive force are also presented.

Light emission produced by the reversible collapse of a cavity in a dielectric medium is given an initial, simplified treatment. The agreement between planar and spherical shapes indicates the volume nature of the effect.

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The multispinor description of particle fields and sources provides a unification of all spins and statistics. As such, it makes quite transparent the existence of transformations that alter the particle spin by , with the accompanying F(ermi)–B(ose) transformation of statistics. After an initial discussion for noninteracting particles, both massive and massless, that is restricted to the exterior of sources (on-mass shell particles), the latter restriction is removed for special examples, including spinors of the second and third ranks. Some attention is given to the spinor description of the photon. The F–B spinor transformations are then presented in more familiar representations, for particular examples. The alternative constructions of integer spins using tensors and of integer spins by means of tensor-spinors suggest transformations that change the spin by unity and retain the statistics. This is illustrated with a number of examples, including the photon and graviton. There is some additional discussion of the vector–spinor representation of spin , including the massless limit and its transformations with the graviton. The starting point in this development is the existence of easily recognizable invariance or partial invariance transformations. Such transformations form a group, but the structure of that group has not been stated a priori. It is now investigated in various situations, among which are those for which the commutator transformation is a displacement, outside the sources. As is known from prior studies, the basic multiplets here are a pair of massless particles and a quartet of massive particles. For unit spin transformations, the multiplets are those of three-dimensional angular momentum for massless particles, and of four-dimensional angular momentum for massive particles. The latter is well known in the context of the degenerate hydrogen atom levels. A coda contains a brief discussion of interacting systems, using the example of spinor electrodynamics. It is found that exact invariance of the action can be achieved, with no mass restriction, by loosening the relations between field variations and source variations. These F–B transformations maintain their form under gauge transformations.

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The correct treatment of strongly bound electrons is grafted smoothly onto the Thomas-Fermi computation of the total binding energy of neutral atoms. This provides a clearcut demonstration of the leading correction of relative order Z^−1/3 which, with effects of relative order Z^−2/3, gives an accurate account of the binding energy over a wide range of Z values. There is a brief discussion of relativistic corrections, with results that are somewhat at variance with previous numerical estimates.

A simple derivation is given for the first quantum correction to the Thomas-Fermi kinetic energy. Its application to the total binding energy of neutral atoms exploits the technique for handling strongly bound electrons that was developed in a preceding paper, and justifies the numerical value of the second correction adopted there. A proposal is made for extrapolating this improved description to the outer regions of the atom.

The following sections are included:

• Appendix Julian Schwinger — List of Publications