Approximately Relativistic Equations

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.