Hiromi Kase

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.

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

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 � � � � � � .

STOCHASTIC QUANTIZATION OF NAMBU-JONA-LASINIO MODEL

Stochastic quantizations of the Nambu-Jona-Lasinio model with and without the bare mass term are carried out. It is shown that, although the linear fermion Langevin equation is not chiral-invariant, one obtains the chiral-invariant effective potential in the equilibrium limit when the bare mass vanishes. Numerical solutions of the growing order parameter and two kinds of effective potentials, which take qualitatively different forms if the bare mass is nonvanishing, are presented.

Regularization in the Minkowski stochastic quantization scheme. Scalar model and QED

Abstract Using the functional method of Fukuda and Higurashi we show that, for a markovian stochastic regularization, there exists an equilibrium limit of the regularized generating functional in Minkowski stochastic quantization of scalar theory. The method is also applied to define Minkowski stochastic quantization of QED in the covariant α -gauge. It is pointed out that the present perturbation scheme needs Pauli-Villars regularization performed at the Langevin level.