It is argued that the familiar algebra of the non-commutative space-time with c-number θ µν is inconsistent from a theoretical point of view. Consistent algebras are obtained by promoting θ µν to an anti-symmetric tensor operatorθ µν . The simplest among them is Doplicher-Fredenhagen-Roberts (DFR) algebra in which the triple commutator among the coordinate operators is assumed to vanish. This allows us to define the Lorentz-covariant operator fields on the DFR algebra as operators diagonal in the 6-dimensional θ-space of the hermitian operators,θ µν . It is shown that we then recover Carlson-Carone-Zobin (CCZ) formulation of the Lorentz-invariant non-commutative gauge theory with no need of compactification of the extra 6 dimensions. It is also pointed out that a general argument concerning the normalizability of the weight function in the Lorentz metric leads to a division of the θ-space into two disjoint spaces not connected by any Lorentz transformation so that the CCZ covariant moment formula holds true in each space, separately. A non-commutative generalization of Connes' two-sheeted Minkowski space-time is also proposed. Two simple models of quantum field theory are reformulated on M 4 × Z 2 obtained in the commutative limit. * ) The following commutator should be added, [M µν ,p ρ ] = i(g νρpµ − g µρpν ). * * ) There exists no constant anti-symmetric tensor. If f µν (p) is a constant symmetric tensor g µν , the coordinate operators commute with each other. We consider a contrary case in what follows. * ) Hence, there is no * -product in what follows. * * ) The matrix γ 5 = iγ 0 γ 1 γ 2 γ 3 is inserted for later convenience. * ) The differential geometry on M 4 ×Z 2 was developed in Ref. 8) and we can use it for the case under consideration. The following presentation is only a translation of the result in Ref. 8).

It is shown that Connes' generalized gauge field in non-commutative geometry is derived by simply requiring that Dirac lagrangian be invariant under local transformations of the unitary elements of the algebra, which define the gauge group. The spontaneous breakdown of the gauge symmetry is guaranteed provided the chiral fermions exist in more than one generations as first observed by Connes-Lott. It is also pointed out that the most general gauge invariant lagrangian in the bosonic sector has two more parameters than in the original Connes-Lott scheme. * ) One may add two arbitrary hermitian matrices to the diagonal blocks of the mass matrix. For simplicity we shall consider only the case indicated in the text. typeset using PTPT E X.sty * ) Strictly speaking, we should write ρ(a)ψ for aψ and ρ(ai) for ai in Eq.(2.1), where the notation ρ indicates the representation of the algebra A on the Hilbert space of the spinors. For simplicity we omit the notation ρ in what follows unless necessary.

No abstract

No abstract

Connes' gauge theory on M 4 × Z 2 is reformulated in the Lagrangian level. It is pointed out that the field strength in Connes' gauge theory is not unique. We explicitly construct a field strength different from Connes' one and prove that our definition leads to the generation-number independent Higgs potential. It is also shown that the nonuniqueness is related to the assumption that two different extensions of the differential geometry are possible when the extra one-form basis χ is introduced to define the differential geometry on M 4 × Z 2 . Our reformulation is applied to the standard model based on Connes' color-flavor algebra. A connection between the unimodularity condition and the electric charge quantization is then discussed in the presence or absence of ν R .