Non-Commutative Differential Geometry on Discrete Space M_4×Z_N and Gauge Theory : Particles and fields

概要

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The algebra of non-coimnutative differential geometry (NCG) on the discrete space M_4×Z_N previously proposed by the present author is improved to give a consistent explanation of the generalized gauge field as a generalized connection on M_4×Z_N. The nilpotency of the generalized exterior derivative d is easily proved. The matrix formulation where the generalized gauge field is denoted in matrix form is shown to have the same content with the ordinary formulation using d, which helps us understand the implications of the algebraic rules of NCG on M_4×Z_N. The Lagrangian of the spontaneously broken gauge theory which has the extra restriction on the coupling constant of the Higgs potential is obtained by taking the inner product of the generalized field strength. The covariant derivative operating on the fermion field determines the parallel transformation on M_4×Z_N, which confirms that the Higgs field is the connection on the discrete space. This implies that the Higgs particle is a gauge particle on the same footing as the weak bosons. The Higgs kinetic and potential terms are regarded as the curvatures on M_4×Z_N.
理論物理学刊行会の論文
1996-11-25