The equation is analysed with respect to the following consequences. I the group theoretical structure of the equation is studied. The equation is invariant under a number of continous transformations: the inhomogeneous Lorentzgroup, the transformations of PAULI, GÜRSEY and TOUSCHEK, and the scale transformation [ϰ→η x or ψ → η⅔ψ(ϰη, l η)]. The PAULI-GÜRSEY group is used for the interpretation of the isospin; the γ5-transformation of TOUSCHEK establishes a quantum number IN, and the scale transformation leads to a quantum number IN, which both are connected with the baryonic and the leptonic number. The strangeness s=lN-lQ is suggested to be connected with the discrete groups of the equation and could then be defined and conserved only modulo 4. Of the discrete groups only the well known transformations P, C and T and the reversal of l (l→ - l) are briefly discussed. In II the vacuum expectation values of products of two field operators are studied. These values are considered to be only in a first approximation invariant under the Isospingroup. The deviations from the PAULI-GÜRSEY symmetry in higher approximations are supposed to be due to the replacement of the state “vacuum” by an idealised state “world”, which possesses an infinite isospin; the strange particles are consequently interpreted as states which “borrow” an isospin 1/2 or 1 from the ground state “world”. The concept of “One particle-wavefunctions” is discussed in III. The fermions of finite mass belong to wavefunctions obeying a KLEIN-Gordon-spinor equation instead of a DIRAC equation. The connection with the conventional formalism of the DIRAC equation is treated in detail. The process of ^-conjugation springing from these discussions is used for a variation of the methods of approximation needed later on for the determination of mass values and the pion-nucleon coupling constant. In IV the TAMM-DANCOFF method is applied in two different forms for an estimate of the masses of nucleons and π-mesons. The masses and the symmetry properties of the particles agree qualitatively with the experimental results. The scattering of π-mesons from nucleons is treated in V by a method related more closely to the BETHE-SALPETER theory than to the TAMM-DANCOFF method; the theory leads to a relativistic pseudovector-coupling as the main term and to a value of the coupling constant of the right order of magnitude. In VI the interaction for β-decay is analysed with respect to its symmetry properties. The theory leads to cs =cT= cp=0 and, in the lowest approximation, to cA= -cV, while in higher approximations the ratio cA/cv will be somewhat altered. In VII some mathematical questions are discussed that have been raised by PAULI at the Geneva conference 1958. For the renormalized operators of the Lee model an integro-differential equation is given, that contains only the arbitrarily small time interval Δt. It is further shown in detail why a linear differential equation leads to δ-functions on the light cone for the propagator, while a non linear differential equation can produce there a different kind of singularity.
Dirac Equation under Scalar, Vector, and Tensor Cornell Interactions
