Katsusada Morita

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-Invariant Non-Commutative QED

the Ctransformation. Since Cis reduced to the conventional charge conjugation C in the commutative limit, in the same limit, the two gauge fields become identical to the photon field which couples to only four spinors, with charges ±2, ±1. Following Carlson, Carone and Zobin, our NCQED respects Lorentz invariance, employing the Doplicher-Fredenhagen- Roberts algebra instead of the usual algebra with constant θ µν . In the new version, θ µν becomes an integration variable. We show, using a simple NC scalar model, that the θ integration yields an invariant damping factor instead of the oscillating one in the nonplanar self-energy diagram in the one-loop approximation. The Seiberg-Witten map shows that the θ expansion of NCQED generates exotic but well-motivated derivative interactions beyond QED, with allowed charges being only 0, ±1, ±2.

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.

Algebraic Gauge Theory of Quarks and Leptons

The quaternionic Weinberg· Salam theory developed in the previous paper is combined with an alternative gauge theory based on local quaternions, which is naturally extended to SU(3) octonionic gauge theory, an algebraic version of QCD. The result is a reformulation of the standard theory of quarks and leptons with SU(3)X SU(2)LX SU(2)RX U(l), the family structure being incorporated with the hypercomplex algebra of octonions times quaternions. According to the superselection rule operative at present energies in the octonionic Hilbert space of GUnaydin, leptons and hadrons are classified into the associative coherent subspace, while quarks belong to the non·associative transverse component. This picture was originally proposed by GUnaydin and GUrsey, the present method offers QCD as a dynamics, leading to, say, Suuras equation for extended mesons.

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 � � � � � � .

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

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.

Is the Wess-Zumino model consistent?

Abstract It is shown that the inconsistency of the Wess-Zumino model, claimed by Krasnikov and Nicolai, cannot be justified to exist in the present framework of quantum field theory.

Quaternionic Variational Formalism for General Relativity in Riemann and Riemann-Cartan Space-Times

It is shown that there exists a 2-dimensional matrix representation of complex quaternions over real quaternions, which allows to define Pauli matrix in 4 dimensions over the quaternionic field and leads to the quaternionic spinor group previously proposed. It is also attempted to apply complex quaternions to general relativity at the level of the variational formalism. Linear gravitational Lagrangian in Riemann-Cartan space-time U4 is derived using quaternion caluculus; namely scalar curvature in U4 is put into a quaternionic form. Consequently, Einstein-Hilbert Lagrangian in Riemann space R4 is also defined over quaternions, as first shown by Sachs. The matter fields coupled to gravity are assumed to be the scalar and the Dirac fields. The quaternionic variational formalism corresponds to the firstorder formalism but with a limited pattern of allowed fields such that the quaternionic fields carry only coordinate tensor indices but no local Lorentz indices which are contracted with that possessed by the basis of complex quaternions. In particular, both the quaternionic vierbein field and Lorentz gauge field (corresponding to the spin connection) are regarded as coordinate vectors which are independently varied, obtaining Einstein and Cartan equations, respectively. It is incidentally shown that the consistent condition of Einstein equation in U4 is proved via the variational formalism and the anti-symmetric part of Einstein equation together with Cartan equation in U4 leads to an identity which expresses the anti-symmetric part of the enegy-momentum tensor by means of the covariant divergence of the spin angular momentum tensor, both of Dirac field. We also present pedagogical proofs of Bianchi and Bach-Lanczos identities in U4 using the quaternionic formalism.

Note on a New Quaternionic Approach to Relativity and the Dirac Theory

The purpose of this note is to provide a convincing argument that the quaternionic spinor analysis1) leads to the existence of an extra imaginary unit commuting with Hamilton’s imaginary units of quaternions so that the conventional quaternionic approach2) to relativity, which makes use of complex quaternions, is derived from our new quaternionic approach1) defined over the field of quaternions, H. The proof is based on the fact that the Dirac representation is the only fundamental representation of the quaternionic spinor group Spin(2,H) ⊂ SL(2,H) but is defined as a direct sum of two fundamental representations (two inequivalent Weyl representations) of SL(2,C). It is explicitly shown that 2-component quaternionic Dirac spinor when transformed into 2-component complex quaternionic Dirac spinor via complexification is decomposed into 4-component Dirac spinor and its Dirac adjoint in the Weyl representation of Dirac matrices linearly. Let us first summarize the quaternionic spinor analysis1) in which we associate an arbitrary space-time point xμ with an hermitian quaternionic matrix

Temperature-Dependent Pauli Matrix

The thermal average of magnetization of charged spin 1/2 particles is calculated in terms of a pure state. This leads to the introduction of a temperature-dependent Pauli matrix. A new Bogoliubov transformation for a spin system in thermofield dynamics is then proposed. This transformation clarifies the physical significance of the temperature-dependent Pauli matrix. Generalization to the case of arbitrary spin is straightforward.