Quark theory with internal coordinates

Abstract

The free-particle Dirac wave function $\ensuremath{\psi}(x)$ is generalized to $\ensuremath{\psi}(x){\ensuremath{\xi}}^{a}(z)$. Here, $z$ denotes a set of three complex coordinates, called internal coordinates, in an abstract complex three-dimensional space, called internal space, $a$ runs from 1 to 3, and ${\ensuremath{\xi}}^{a}(z)$ is assumed to contain a representation of the state of a quark triplet. The mass in the free-particle Dirac equation is replaced by a second-order operator $\ensuremath{-}{\ensuremath{\partial}}_{a}^{b}$ operating on ${\ensuremath{\xi}}^{a}(z)$. The so-modified Dirac equation is assumed to include a description of a free-quark triplet. Subsequently, symmetry-preserving interactions in space-time as well as in the internal space between two quark triplets are introduced. Two S${\mathrm{U}}_{3}$-symmetry-breaking interactions, one transforming like the eighth component of an S${\mathrm{U}}_{3}$ octet vector and the other like the S${\mathrm{U}}_{3}$ charge operator, are also introduced. A similarly generalized Bethe-Salpeter equation in the ladder approximation is obtained. This equation is treated in greater detail in the following paper in which the Gell-Mann-Okubo formula for pseudoscalar mesons is derived with the coefficients determined by given relations. Then, spherical coordinates and corresponding spherical harmonics in the internal space are introduced. Finally, the equation for a one-quark system is briefly treated.