Yoshitaka Okumura

Lorentz-Invariant Non-Commutative Space-Time Based on DFR Algebra

It is argued that the familiar algebra of 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 operator ˆ µν . The simplest among them is the 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, ˆ θ µν . It is shown that we then recover the Carlson-Carone-Zobin (CCZ) formulation of the Lorentz-invariant non-commutative gauge theory with no need for the 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 the division of the θ-space into two disjoint spaces not connected by any Lorentz transformation, so that the CCZ covariant moment formula holds 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 M4 × Z2 obtained in the commutative limit.

Lorentz Invariance and the Unitarity Problem in Non-Commutative Field Theory

It is shown that the one-loop two-point amplitude in Lorentz-invariant non-commutative (NC) φ 3 theory is finite after subtraction in the commutative limit and satisfies the usual cutting rule. This eliminates the unitarity problem in Lorentz-non-invariant NC field theory in the approximation considered.

Quark/lepton mass and mixing in

Weak bases of flavors (u, c), (d,s), (e, \mu), (\nu_e,\nu_\mu) are assumed as the S_3 doublet and t, b, \tau, {\nu_\tau} are the S_3 singlet and further there are assumed S_3 doublet Higgs (H_1, H_2) and S_3 singlet Higgs H_S. We suggest an S_3 invariant model in which the Yukawa interactions constructed from these S_3 doublets and singlets are S_3 invariant. In this model, we can explain the quark sector mass hierarchy, quark mixing V_{CKM} and measure of CP violation naturally. In the leptonic sector, neutrino masses are assumed to be constructed through the see-saw mechanism from the Majorana mass. The tri-bimaximal-like character of neutrino mixing V_{MNS} can be explained dynamically without any other symmetry restrictions. It is predicted that a quasi-degenerate mass spectroscopy of neutrino is favorable, and values of |V_{MNS}|_{13}, CP violation invariant measure J and the effective Majorana mass | | in neutrino-less double beta decay are not so tiny.

S_3

The quantization of spontaneously broken gauge theories in a noncommutative geometry (NCG) has been sought for some time, because quantization is crucial for making the NCG approach a reliable and physically acceptable theory. Lee, Hwang, and Ne{close_quote}eman recently succeeded in realizing the BRST quantization of gauge theories in a NCG in the matrix derivative approach proposed by Coquereaux and co-workers. The present author has proposed a characteristic formulation to reconstruct a gauge theory in a NCG on the discrete space {ital M}{sub 4}{times}{ital Z}{sub {sub {ital N}}}. Since this formulation is a generalization of the differential geometry on the ordinary manifold to that on the discrete manifold, it is more familiar than other approaches. In this paper, we show that within our formulation we can obtain the BRST-invariant Lagrangian in the same way as Lee, Hwang, and Ne{close_quote}eman and apply it to the SU(2){times}U(1) gauge theory. {copyright} {ital 1996 The American Physical Society.}

invariant model and CP-violation of neutrino

Abstract The effective potential for q q condensation is examined also taking account of contributions from the non-perturbative region. An electronic computer is used to solve the nonlinear differential equation. We reach the conclusions that contributions from the non-perturbative region play an important role in explaining spontaneous chiral symmetry breaking quantitatively, and that effective coupling should be strong to some extent in the non-perturbative region.

BRST-invariant Lagrangian of spontaneously broken gauge theories in a noncommutative geometry.

Connes’ gauge theory is defined on noncommutative space-time. It is applied to formulate a noncommutative Weinberg-Salam (WS)model in the leptonic sector with νR. It is shown that the model has two Higgs doublets and a gauge boson sector after the Higgs mechanism contains the massive charged gauge fields, two massless and two massive neutral gauge fields. It is also shown that, at the tree level, the neutrino couples to one of two ‘photons’, the electron interacts with both ‘photons’, and there exists a nontrivial νR-interaction on noncommutative space-time. To investigate the commutative limit of the model at the Lagrangian level, we generalize the charge conjugation transformation in QED to that in noncommutative QED. We show that there are two different generalizations, the C and � � � � � � .

Spontaneous chiral symmetry breaking in particle physics

A gravity-incorporated standard model is constructed in a generalized differential geometry (GDG) on R4×X2. Here, R4 and X2 are the four-dimensional Riemann space and two-point discrete space, respectively. A GDG on R4×X2 is constructed by adding the basis χn (n = 1,2) of the differential form on X2 to the ordinary basis dxµ on R4, and so it is a direct generalization of the differential geometry on the continuous manifold. A GDG is a version of non-commutative geometry (NCG). We incorporate gravity by simply replacing the derivative ∂µ by the covariant derivative ∂µ + ωµ for a general coordinate transformation in the definition of the generalized gauge field on R4×X2, keeping other parts unchanged. The Yang-Mills-Higgs Lagrangian for the standard model is obtained by taking the inner product of two generalized field strengths, whereas the Einstein-Hilbert gravitational Lagrangian is created by the inner product of a generalized field strength and a tensor {Ea}b on local Lorentz space.

Connes' Gauge Theory and Weinberg-Salam Model of Leptons on Noncommutative Space-Time

SummaryThe scheme previously proposed by the present authors is modified to incorporate the strong interaction by affording the direct product internal symmetry. We do not need to prepare the extra discrete space for the colour gauge group responsible for the strong interaction to reconstruct the standard model and the left-right symmetric gauge model (LRSM). The approach based on non-commutative geometry leads us to present many attractive points such as the unified picture of the gauge and Higgs field as the generalized connection on the discrete spaceM4×ZN. This approach leads us to unified picture of gauge and Higgs fields as the generalized connection. The standard model needsN=2 discrete space for reconstruction in this formalism. LRSM is still alive as a model with the intermediate symmetry of the spontaneously brokenSO(10) grand unified theory (GUT).N=3 discrete space is needed for the reconstruction of LRSM to include two Higgs bosonsφ andξ which are as usual transformed as (2, 2*, 0) and (1, 3, −2) underSU(2)L×SU(2)R×U(1)Y, respectively.ξ is responsible to makeνR Majorana fermion and so well explains the seesaw mechanism. Up and down quarks have different masses through the vacuum expectation value ofφ.

Gravity-incorporated standard model in a generalized differential geometry

BRST symmetry is very important to quantize gauge theory and to construct a covariant canonical formulation. It may also be important for gauge theory on noncommutative (NC) spacetime. The BRST symmetry of U(N) gauge theory on NC spacetime has recently been studied by Soroush, who manipulated the component fields of the gauge and fermion fields. We define the BRST transformations of the representation fields of gauge and fermion fields and show its nilpotency in a transparent way. The scale symmetry of ghost fields is also investigated. BRST charge and FP ghost charge are derived and shown to be the generators of the respective symmetries. As a byproduct, it is shown that Noether’s theorem is recovered on NC spacetime by redefining the current.

Reconstruction of the spontaneously broken gauge theory in non-commutative geometry

The left?right symmetric gauge model with the symmetry of SU(3)c ? SU(2)L ? SU(2)R ? U(1) is reconstructed in the generalized differential geometry (GDG) on the product space of Minkowski space M4 and four-point discrete space XN with N = 4. Ordinary gauge fields are regarded as the connection on M4; on the other hand, Higgs boson fields are regarded on the discrete space XN. This is a common view in the non-commutative geometry initiated by Connes. Gauge and Higgs fields are described in this paper as connections with the 1-form base on M4 ? XN, a situation which is more similar to the ordinary differential geometry. We explain this formulation based on the matrix representation by use of the N ? N matrix form. In addition to the generalized field strength obtained from the generalized gauge field, the anomalous field strength is introduced to yield sufficient Higgs potential and interacting terms by combining with corresponding terms in the generalized field strength. The resultant Higgs potential terms deduce the particular mass pattern such that among the Higgs doublet bosons, one CP-even scalar boson survives in the weak energy scale and the other four scalar bosons acquire the mass of the SU(2)R ? U(1) breaking scale.