Approximately Relativistic Equations

Abstract

The stationary states of the system of extranuclear electrons in an atom can be treated without making much explicit use of electrodynamics and, as is well known, the present lack of understanding of the electromagnetic field is relatively unimportant in practical calculations of spectroscopic energy terms. It suffices, in such applications, to deal with wave equations which are correct only to the order $\frac{{v}^{2}}{{c}^{2}}$ times the term value, where $v$ stands for the velocity of the electron. The difficulties involved in forming a completely satisfactory quantum electrodynamics or any quantum field theory appear at the moment to be so formidable that one may expect their solution along lines rather different from those attempted so far. It has been a lucky circumstance for the development of atomic theory that the $\frac{{v}^{0}}{{c}^{0}}$ approximations sufficed for the gross treatment of energy levels while the $\frac{{v}^{2}}{{c}^{2}}$ approximation apparently gives a satisfactory account of their fine structure. The development of the theory of nuclear physics has paralleled that of atomic physics inasmuch as the gross structure of nuclear levels has been of primary interest. This has been done with an apparent sacrifice of even approximate agreement with relativity through the introduction of potentials varying in an arbitrary way with the distance. The present paper is a continuation of a previous attempt to improve this state of affairs. While in atomic theories, Maxwell's equations can be used as a guide in the setting up of a wave theory, no field concept of comparable certainty is as yet available for nuclear interactions. Fortunately, however, it turns out that the requirement of relativistic invariance to the order $\frac{{v}^{2}}{{c}^{2}}$ together with the known symmetries of the electromagnetic field are sufficient to determine the $\frac{{v}^{2}}{{c}^{2}}$ approximations to the wave equations in the electronic case. Even though the retardation of electromagnetic potentials is involved in the problem, its complete wave mechanical understanding can thus be partly replaced by requirements of invariance to order $\frac{{v}^{2}}{{c}^{2}}$. In the present as well as in the previous paper the possibilities of making analogous extensions are investigated for arbitrary interactions.In the previous work the possibilities in classical relativistic dynamics have been the starting point. The classical approximately relativistic equations have then turned out to be invariant in the sense that for each particle the motion, as obtained by means of the equations in one system, is the transform of the motion as obtained by the same equations in another system. Using the picture of wave packets subjected to small accelerations, some necessary conditions for wave equations have been also derived and the corresponding forms of spin orbit interactions have been obtained.In the present paper approximately relativistic equations are investigated using two different methods of approach. One of them consists in using the first approximation of Born's method for the description of the collision process. It is then possible to devise matrix elements that give an invariant description of the collision process not only in the $\frac{{v}^{2}}{{c}^{2}}$ approximation but in all orders of $\frac{v}{c}$. These matrix elements are not applicable, however, to ordinary physical systems in higher approximations of Born's method. The second method of approach consists in working only with the $\frac{{v}^{2}}{{c}^{2}}$ approximation. It is then found possible to have relatively simple equations that give an invariant description of the collision process exactly, i.e., independently of Born's first approximation. The equations correct to order $\frac{{v}^{2}}{{c}^{2}}$ are set up for Wigner, Majorana, Heisenberg and Wheeler forces using wave functions with two components per particle. It is also found possible to have equations for exchange forces with four components per particle. An equation of this type has been used by Share and the writer in a calculation of the relativistic effects in the deuteron and shows that spin-spin interactions of relativistic origin can be appreciable and should be expected to have a range of force different from the nonspin dependent part. The interpretation of wave equations is discussed. It is shown how, even in approximately relativistic discussions, these equations should be considered only as approximations to equations with four components per particle. The limitations of the classical spin model arising from this cause are pointed out and the magnetic interaction energy of the deuteron is discussed from this point of view. It is concluded that most of the existing estimates are not sound, since the assignment of a magnetic moment to an elementary particle has to be defined in a physical way, and since the spin current, as obtained from Dirac's equation, is the particle current rather than the electric current.