Discrete Symmetries In Lorentz-Invariant Non-Commutative QED

It is pointed out that the usual $\theta$-algebra assumed for non-commuting coordinates is not $P$- and $T$-invariant, unless one {\it formally} transforms the non-commutativity parameter $\theta^{\mu\nu}$ in an appropriate way. On the other hand, the Lorentz-covariant DFR algebra, which `relativitizes' the $\theta$-algebra by replacing $\theta^{\mu\nu}$ with a second-rank antisymmetric tensor operator \htheta^{\mu\nu}, is $C$-, $P$- and $T$-invariant. It is then proved that $C, P$ and $T$ are separately conserved in Lorentz-invariant Non-Commutative QED.Comment: 19page

Non-Commutative Differential Geometry and Standard Model

We incorporate Sogami's idea in the standard model into our previous formulation of non-commutative differential geometry by extending the action of the extra exterior derivative operator on spinors defined over the discrete space-time; four dimensinal Minkovski space multiplyed by two point discrete space. The extension consists in making it possible to require that the operator become nilpotent when acting on the spinors. It is shown that the generalized field strength leads to the most general, gauge-invariant Yang-Mills-Higgs Lagrangian even if the extra exterior derivative operator is not nilpotent, while the fermionic part remains intact. The proof is given for a single Higgs model. The method is applied to reformulate the standard model by putting left-handed fermion doublets on the upper sheet and right-handed fermion singlets on the lower sheet with generation mixing among quarks being taken into account. We also present a matrix calculus of the method without referring to the discrete space-time.Comment: 27 page

Lagrangian Formulation of Connes' Gauge Theory

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.Comment: 9 pages, PTPTEX.st

Gauge Theories Coupled to Fermions in Generation

Gauge theories coupled to fermions in generation are reformulated in a modified version of extended differential geometry with the symbol $\chi$. After discussing several toy models, we will reformulate in our framework the standard model based on Connes' real structure. It is shown that for the most general bosonic lagrangin which is required to also reconstruct N=2 super Yang-Mills theory Higgs mechanism operates only for more than one generation as first pointed out by Connes and Lott.Comment: 18 pages, ptptex.st