Weinberg-Salam Theory Based on a Z_2-Graded Algebra : Particles and Fields

概要

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Generalized differential calculus on discrete space M_4 X Z_2 which is an underlying space-time in the non-commutative geometry for the standard model is reformulated in terms of a Z_2-graded algebra, even and odd elements of which being pairs of complex matrices defined over Minkowski space-time with different properties of product and involution.It is shown that the Z_2-grading is equivalent to that of Coquereaux et al. if the pair is represented by 2 × 2 matrices, although our formalism has closer contact with the differential calculus on the discrete space. A graded differential algebra is then defined,in which the exterior derivative with respect to the pair is assumed to determine the pattern of symmetry breaking of the theory. On the basis of it the Weinberg-Salam theory in both bosonic and fermionic sectors is constructed. It is pointed out that, in contrast to usual assertion in non-commutative geometry,the Weinberg angle and the Higgs mass in the tree level are not fixed separately but related through m_H=2√<2>εm_wsinθ_w. Connes' prescription of constructing gauge-invariant lagrangian, which is based on the assumption that there arise only logarithmic divergences from one-loop diagrams, corresponds to the case ε=1. In principle, however, the parameter ε is arbitrary due to possible presence of Sitarz' linear term so that non-commutative geometry alone says nothing about the Higgs mass.
理論物理学刊行会の論文
1994-08-25