Formulation Of Quantum Electrodynamics Research Articles

On the γ-Decay of Neutral Meson

Recently Tomonaga and Schwinger have independently developed a covariant formulation of quantumelectrodynamics (super-many-time theory), and have successfully applied it to the explanation of the Lamb shift in the hydrogen atom and the anomalous magnetic moment of the electron. They have shown that, although the present theory of fields, in general, gives infinte answers to such field reaction problems, since it cannot be formulated in a Lorentz- and gauge-covariant way without introduction of the singular delta function of Jordan and Pauli, it is nevertheless possible to avaid the divergences by amalgamating them into the mass and charge, and that the remaining finite term can well account for the experimental facts. But there remains the question, whether or not such finite term, as is separated from infinity, can be free from any ambiguity arising from the pathological character of delta function of Jordan and Pauli. One typical example of the appearance of such an ambiguity is the photon self energy. As first pointed out by Schwinger, the photon self energy should be zero from the gauge covariant point of view, while, completely against Schwinger's prediction, the recent calculation by Wentzel leads to a finite but non-gauge covariant result for the photon self energy. It is evident that this inconsistent result comes from the mathematical difficulty of obtaining a definite expression for the singularity of the light cone of Jordan-Pauli's delta function. A very similar situation is also encountered in the γ decay of neutral meson. By using the method of evaluation which has been applied by Schwinger to the calculation of the anomalous magnetic moment of the electron, we have obtained the convergent but non-gauge covariant result for the γ decay of neutral meson. Further our result is inconsistent with the recent discussions of Dyson, Sawada and Nanbu on the equivalence between the pseudoscalar and pseudovector couplings of the pseudoscalar meson field. Thus, in the present state of the field theory, we cannot give an unambiguous life-time for neutral meson. It is very desirable to find some general prescription which enables one to get the definite answer to field reaction problams.

Masses of fundamental particles.

IN considering the self-energy problem on the basis of the multiple-time formalism of quantum electrodynamics, Wentzel1 has obtained the relation where mn(s) is the rest-mass of a field-free particle, mn the corresponding mass in an electromagnetic field and s the particle extension. If we now assume that mn(s) and mn refer to the masses of two fundamental particles M1 and M2 (expressed in terms of the electron mass m) it follows directly from equation (1) that

Quantum Electrodynamics. II. Vacuum Polarization and Self-Energy

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.

On the Self-Energy of a Bound Electron

The electromagnetic shift of the energy levels of a bound electron has been calculated on the basis of the usual formulation of relativistic quantum electrodynamics and positron theory. The theory gives a finite result of 1052 megacycles per second for the shift $2^{2}S_{\frac{1}{2}}\ensuremath{-}2^{2}P_{\frac{1}{2}}$ in hydrogen, in close agreement with the non-relativistic calculation by Bethe.

Quantum Electrodynamics. I. A Covariant Formulation

Attempts to avoid the divergence difficulties of quantum electrodynamics by multilation of the theory have been uniformly unsuccessful. The lack of convergence does indicate that a revision of electrodynamic concepts at ultrarelativistic 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 $t=\mathrm{const}$. 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 Schr\"odinger 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 Schr\"odinger 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 convariant and practical form of quantum electrodynamics, expressed in a mixed Heisenberg-Schr\"odinger 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 covariant 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.

New Aspects of the Photon Self-Energy Problem

A finite but non-vanishing value for the self-energy of the photon, corresponding to a finite rest-mass, can be deduced from the new invariant formulation of quantum electrodynamics developed by Tomonaga and Schwinger, in the ${e}^{2}$ order approximation. The implications of this result are discussed.

Applied Mathematics

Electromagnetic theory was under discussion at the Second Symposium on Applied Mathematics of the American Mathematical Society held at MIT on July 29–31. Papers were presented on the status of the new covariant formulation of quantum electrodynamics, reformulations of electromagnetic field theory along classical lines, the solutions of several types of boundary value problems related to the Maxwell field theory, methods of solving boundary value problems, orbit theory and its applications, the response of linear electrical networks, nonlinear electrical networks, the abstract formulation of problems in the communication of information, and on random noise.

The theory of neutral particles

Several authors (1, 2, 3) have discussed the formulation of quantum electrodynamics in the configuration space of the light quanta; they conclude that no meaning can be given to the position of a light quantum, but it is not made clear whether this situation is due to the vanishing of the charge or of the rest mass. However, since the previous discussion (2, 3), various neutral particles of non-vanishing rest mass have been postulated, and it is of interest to consider what meaning can be given to the position of such particles. We shall at first consider integral spin and Bose statistics; the case of spin ½ and Fermi statistics is simpler and is briefly discussed in the last section.