Approximately Relativistic Equations for Nuclear Particles

TLDR: In this paper, a relativistic theory of nuclear forces has been proposed, which is based on the electron-neutrino field and is relativistically invariant for transformations involving low velocities.

Abstract: Cosmic-ray showers indicate that at high energies interaction between nuclear particles is concerned with the creation and destruction of matter. It may, therefore, be expected that a complete relativistic theory of nuclear forces will involve explicit reference to the phenomena described at present as the electron-neutrino field. Theories of this kind are still too incomplete and self-contradictory to be reliable in practical work. It is, nevertheless, possible to set up equations which are relativistically invariant for transformations involving low velocities. Such equations form the subject of the present report. They are restricted to energies of relative motion that are small in comparison with the rest mass. By means of them it should be possible to discuss relativistic effects for ordinary nuclear energy levels. Possible forms of classical equations contain an interaction energy between two particles in the form given by Eq. (13.2). Here $a$, $b$ are arbitrary real constants. The vector from particle 1 to particle 2 is r, the velocities of the particles are ${\mathrm{v}}_{1}$, ${\mathrm{v}}_{2}$, the velocity of light is $c$. If $a=b=1$, one obtains a generalization of Darwin's equation, which describes the motion of electrically charged particles. For $a=\ensuremath{-}1$, $b=1$ particle 1 acts on particle 2 approximately as though it produced a scalar potential field responsible for the acceleration of particle 2. The requirement of invariance for wave equations of particles with spin (Pauli types) makes it necessary to have spin-orbit coupling which should give rise to the fine structure of nuclear levels. For ordinary interactions the spin-orbit energy may have the form given by Eq. (15.4), where $b$ is an arbitrary real constant, p is the momentum and $\mathbf{\ensuremath{\sigma}}$ is Pauli's spin matrix. For ${b}_{\mathrm{ij}}=\ensuremath{-}1$ one obtains the type of coupling taking place between extranuclear electrons. If ${b}_{\mathrm{ij}}=1$ each particle interacts only with its own orbit as though it were moving in a scalar field. It is the latter hypothesis that is simplest and corresponds to $a=\ensuremath{-}1$, $b=1$ of the classical equation. Extensions of the above classifications have been made to the Majorana [Eqs. (15.7), (15.8)] and the Heisenberg [Eq. (15.9)] ex change interactions. The simplest type in the Majorana case appears to be in satisfactory agreement with experiment for ${\mathrm{Li}}^{7}$ and agrees in order of magnitude with other cases. Extensions to Dirac's types of equations have been made. They lead one to expect coupling between spins of nuclear particles in apparent qualitative but not quantitative agreement with experiment for the deuteron. This agreement is not sufficiently good to establish a form of interaction energy but indicates a possibility of doing so in the future.