Higher-dimensional effective-potential model of quarks and leptons

Abstract

By cloning the Dirac operator -i )V in four extra dimensions i?4 and trapping fermions in a harmonic-oscillator potential in the extra dimensions to yield an eight dimensional field equation (refer to book) Bryan managed to obtain 'internal' quantum numbers such as weak isospin, particletype (quark or lepton), color, and generation, as well as the ordinary quantum numbers. (Here quantities related to ordinary space M4, extra dimensions i?' or both are denoted : normally, with tildes, bold-face, respectively.) To include chirality into the model, using the solutions of the equation above as basis functions, we propose the eight-dimensional field equation The general solution of this equation in the limit of high momentum is which links the right-handed solutions of the ordinary Dirac equation only to the 'right-handed' extra- dimensional solutions, and the left-handed solution of the ordinary Dirac solutions only to the 'left-handed' extra- dimensional solutions. If we treat the 'generalized' electro- weak bosonic fields as vector-isovector weak-bosonic fields and a vector-isoscalar B-field, and ignore the Cab'bbo- Kobayash' Maskawa mixing, then we can show that through the electro-weak interaction, weak isospin, chirality, particle- type, color and generation number are conserved.