The Irreducible Volume Character of Events. I. A Theory of the Elementary Particles and of Fundamental Length

In this series of papers a theory of elementary particles is presented based on the postulates that the elementary particles are a material process and that with every such process there is associated an inherent irreducible volume of space-time. The equations of motion are obtained from the wave equations of Dirac, Maxwell, etc., through a two-step correspondence principle consisting of replacing these equations by self-adjoint partial difference equations and then integrating these over the group manifold of the closed real orthogonal group $G$ in four-dimensional space. This introduces a new fundamental constant of the dimensions of a length, the difference step $\omega$. The resulting equations are relativistically invariant integro-difference equations in a complex four-dimensional space. They give a quantized mass spectrum when $\omega$ is purely imaginary. This paper treats only the Dirac equation. Section 1 sketches the basic physical ideas from which we work. Section 2 carries out the mathematical realization of the ideas of Section 1, obtains the fundamental integro-difference equation (2.2) for spin ½ processes, points out that space is a fourdimensional complex continuum, and defines the dynasphere. Section 3 transforms the integro-difference equation into an equivalent partial differential equation of infinite order (3.10). The operator functions in (3.10) are the Bessel functions $J_1\mathrm{}\left(z\right)\mathrm{}$ and $J_2\mathrm{}\left(z\right)\mathrm{}$ when the radius ($i\omega$) of the dynasphere is purely imaginary. Section 4 gives the general solution of (3.10) and deduces the mass quantization condition. In keeping with our basic postulates all elementary charged particles of spin ½ must be embraced by this mass quantization condition which is the case for the electron, proton, and $\mu$-meson (predicted mass 218.76 e.m.u. [electron mass units]) with the fundamental length $\omega =4.538\mathrm{}\times {10}^{-13\mathrm{}}$ cm (one-half percent larger than the classical electron diameter). This possibility is due to the rather remarkable properties of the mass quantization condition (4.4). There is an infinite number of roots not uniformly spaced and indeed not uniformly non-uniform in this respect: first there is one very small positive root which gives the electron; the next positive root reaches way out to give on a mass scale the $\mu$-meson at 218 e.m.u.; the distance on the mass scale to the next root suffers a sudden drop to 139 e.m.u., the spacing thereafter dropping uniformly to the asymptotic value of 133.69 e.m.u. The proton is the thirteenth in the series. Table I of this section lists the first sixteen masses. All of these particles have the same magnitude of electronic charge since they are just different states of the same matter equation. Section 5 gives two alternatives for the neutral particles. In one we simply set the charge to zero and obtain one zero mass particle and an infinite number close in mass to those of the charged particles (the electron excluded). In the other we further change to a real radius for the dynasphere, in which case we have only one real zero mass. On this alternative we have only one elementary neutral particle of spin ½—the neutrino—and (combined with Section 4) the only elementary particles with non-vanishing rest mass are charged (and conversely), the universality of charge being simply identified with the existence of the particle. Since there is no neutral particle between the neutrino and the $\mu$-meson on either alternative we can predict that the ${\mu }^0$-particle postulated by Tiomno and Wheeler for the decay of the $\mu$-meson must be a neutrino, that is, $\mu \to e+2\mathrm{}\nu$.