Neutrino reactions and restrictions on the O(4) × U(1)-gauge theory of weak and electromagnetic interactions

In this paper we have analysed and restricted the freedom of constructing gauge models of Pais type, based on the symmetry group O(4) × U(1) using the latest experimental data from neutrino reactions. The hadrons are made out of the fractionally charged Gell-Mann-Zweig quarks. The left-handed quarks and leptons are restricted to (a) 4-vector (b) 4-spinor and (c) adjoint representations of O(4). The right-handed quarks and leptons are taken as scalars of O(4). It is found that the model based on 4-spinorL-leptons and 4-spinorL-quarks agrees the best with the known phenomenology of weak interactions.     

Infinite-dimensional representations of the Einstein group

The representations of the general linear group GL(4.R) are described. This group corresponds to the space-time transformations discussed in the general theory of relativity. Besides the well-known tensor representations, the group is also characterized by infinite-dimensional representations with integral and half-integral spins. This fact opens up a natural possibility, in principle, of constructing a covariant theory of particle fields. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 5, 309–312 (10 September 1996)     

Realizations of causal manifolds by quantum fields

Quantum mechanical operators and quantum fields are interpreted as realizations of timespace manifolds. Such causal manifolds are parametrized by the classes of the positive unitary operations in all complex operations, i.e., by the homogenous spacesD(n)=GL(C _R ⁿ )/U(n) withn=1 for mechanics andn=2 for relativistic fields. The rankn gives the number of both the discrete and continuous invariants used in the harmonic analysis, i.e., two characteristic masses in the relativistic case. ‘Canonical’ field theories with the familiar divergencies are inappropriate realizations of the real 4-dimensional causal manifoldD(2). Faithful timespace realizations do not lead to divergencies. In general they are reducible, but nondecomposable—in addition to representations with eigenvectors (states, particle), they incorporate principal vectors without a particle (eigenvector) basis as exemplified by the Coulomb field. In theorthogonal andunitary groupsO(N ₊,N ₋), respectively, thepositive orthogonal and unitary ones areO(N) andU(N), respectively.     

Matter as Spectrum of Spacetime Representations

Bound and scattering state Schrödinger functions of nonrelativistic quantum mechanics as representation matrix elements of space and time are embedded into residual representations of spacetime as generalizations of Feynman propagators. The representation invariants arise as singularities of rational representation functions in the complex energy and complex momentum plane. The homogeneous space GL(²)U(2) with rank 2, the orientation manifold of the unitary hypercharge-isospin group, is taken as model of nonlinear spacetime. Its representations are characterized by two continuous invariants whose ratio will be related to gauge field coupling constants as residues of the related representation functions. Invariants of product representations define unitary Poincaré group representations with masses for free particles in tangent Minkowski spacetime.     

Running coupling in electroweak interactions of leptons from <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>)-gravity with torsion

The f(R)-gravitational theory with torsion is considered for one family of leptons; it is found that the torsion tensor gives rise to interactions having the structure of the weak forces, while the intrinsic non-linearity of the f(R) function provides an energy-dependent coupling: in this way, torsional f(R) gravity naturally generates both structure and strength of the electroweak interactions among leptons. This implies that the weak interactions among the lepton fields could be addressed as a geometric effect due to the interactions among spinors induced by the presence of torsion in the most general f(R) gravity. Phenomenological considerations are given.     

Localization of Unitary Braid Group Representations

Governed by locality, we explore a connection between unitary braid group representations associated to a unitary R-matrix and to a simple object in a unitary braided fusion category. Unitary R-matrices, namely unitary solutions to the Yang-Baxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations from a unitary braided fusion category with possibly different positive multiplicities to get representations that are uniformly equivalent to the ones from a unitary R-matrix. Such an equivalence will be called a localization of the unitary braid group representations. We show that the q = e ^πi/6 specialization of the unitary Jones representation of the braid groups can be localized by a unitary 9 × 9 R-matrix. Actually this Jones representation is the first one in a family of theories (SO(N), 2) for an odd prime N > 1, which are conjectured to be localizable. We formulate several general conjectures and discuss possible connections to physics and computer science.     

Nonstandard GLh(n) quantum groups and contraction of covariant q-bosonic algebras

GL_h(n) × GL_h(m)-covariant h-bosonic algebras are built by contracting the GL_q(n) × GL_q(m)-covariant q-bosonic algebras considered by the present author some years ago. Their defining relations are written in terms of the corresponding R _h-matrices. Whenever n = 2, and m = 1 or 2, it is proved by using U_h(sl(2)) Clebsch-Gordan coefficients that they can also be expressed in terms of coupled commutators in a way entirely similar to the classical case. Some U_h(sl(2)) rank-(1/2) irreducible tensor operators, recently constructed by Aizawa in terms of standard bosonic operators, are shown to provide a realization of the h-bosonic algebra corresponding to n = 2 and m = 1.     

Letter: Riemannian Metric of the Invariance Quantum Group of the Fermion Algebra

We calculate the bi-invariant metric of FIO(2), the inhomogeneous invariance quantum group of the fermion algebra. We find that this metric is identical to that of the bi-invariant metric of GL(2, R) ⁺ × SU (1, 1). However, the quantum group manifold is restricted to a region of the GL(2, R) manifold.     

Finite Dimensional Unitary Representations of Quantum Anti–de Sitter Groups at Roots of Unity

We study irreducible unitary representations of U _q (SO(2,1)) and U _q (SO(2,?3)) for q a root of unity, which are finite dimensional. Among others, unitary representations corresponding to all classical one-particle representations with integral weights are found for , with M being large enough. In the “massless” case with spin bigger than or equal to 1 in 4 dimensions, they are unitarizable only after factoring out a subspace of “pure gauges” as classically. A truncated associative tensor product describing unitary many-particle representations is defined for . Received: 27 November 1996 / Accepted: 28 July 1997     

Master Analytic Representations and unified representation theory of certain orthogonal and pseudo-orthogonal groups

The representation theory of the groupsSO(5),SO(4, 1),SO(6) andSO(5, 1) is studied using the method of Master Analytic Representations (MAR). It is shown that a single analytic expression for the matrix elements of the generators ofSO(n+1) andSO(n, 1) in anSO(n) basis yields all the unitary representations (forn=4,5); and that the compact and non-compact groups have essentially the same analytic representation. Once the MAR of a group is worked out, the search for the unitary irreducible representations is reduced to a purely arithmetic operation. The utmost care has been exercised to conduct the discussions at an elementary level: knowledge of simple angular momentum theory is the only prerequisite.Work supported in part by the National Science Foundation.Work supported in part by the U.S. Atomic Energy Commission.     

Analysis of Weinberg's algebraic relations

Implications of Weinberg's algebraic relations when only p-wave pions are taken into account are investigated. It is shown that decay rates and mass spectra of hadrons assigned to certain classes of unitary irreducible representations of theSU(4) group are in a strong disagreement with experimental data.     

Geometrical basis for the Standard Model

The robust character of the Standard Model is confirmed. Examination of its geometrical basis in three equivalent internal symmetry spaces-the unitary planeC ², the quaternion spaceQ, and the real spaceR ⁴—as well as the real spaceR ³ uncovers mathematical properties that predict the physical properties of leptons and quarks. The finite rotational subgroups of the gauge groupSU(2) _L ×U(1) _Y generate exactly three lepton families and four quark families and reveal how quarks and leptons are related. Among the physical properties explained are the mass ratios of the six leptons and eight quarks, the origin of the left-handed preference by the weak interaction, the geometrical source of color symmetry, and the zero neutrino masses. The (u, d) and (c, s) quark families team together to satisfy the triangle anomaly cancellation with the electron family, while the other families pair one-to-one for cancellation. The spontaneously broken symmetry is discrete and needs no Higgs mechanism. Predictions include all massless neutrinos, the top quark at 160 GeV/c ², theb quark at 80 GeV/c ², and thet quark at 2600 GeV/c ².     

Nonstandard U q(so3) and U q(so4): tensor products of representations, oscillator realizations and roots of unity

Nonstandard q-deformed algebras U _q(so₃) and U _q(so₄), which can be embedded into U _q(sl₃) and U _q(sl₄) and are coideals in them, are considered. It is shown how to multiply finite dimensional representations of U _q(so₃) when q is positive. Homomorphisms from U _q(so₃) and U _q(so₄) to the q-oscillator algebras are given. By making use of these homomorphisms, irreducible representations of U _q(so₃) and U _q(so₄) for q equal to a root of unity are obtained.     

Representations of Multiparameter Quantum Groups

We construct representations of the quantum algebras U_q,q(gl(n)) and U_q,q(sl(n)) which are in duality with the multiparameter quantum groups GL_qq(n), SL_qq(n), respectively. These objects depend on n(n − 1)/2+ 1 deformation parameters q, q_ij (1 ≤ i< j ≤ n) which is the maximal possible number in the case of GL(n). The representations are labelled by n − 1 complex numbers r_i and are acting in the space of formal power series of n(n − 1)/2 non-commuting variables. These variables generate quantum flag manifolds of GL_qq(n), SL_qq(n). The case n = 3 is treated in more detail.

     

Representations of Spacetime as Unitary Operation Classes; or Against the Monoculture of Particle Fields

Spacetime is modeled as a homogeneous manifoldgiven by the classes of unitary U(2) operations in thegeneral complex operations GL( ). The residual representations of thisnoncompact symmetric space of rank two are characterized by two continuousreal invariants, one invariant interpreted as a particlemass for a positive unitary subgroup and the second onefor an indefinite unitary subgroup related to nonparticle interpretable interaction ranges.Fields represent nonlinear spacetimeGL( )/U(2) by theirquantization and include necessarily nonparticlecontributions in the timelike part of their flat-space Feynman propagator.     

The decomposition of the product of a momentum-zero- and a mass-zero-representation of the Poincaré group

The irreducible unitary representations of a twofold covering groupE(2) of the euclidean group of the plane are set up. The product of two irreducible unitary representations ofE(2) and the irreducible unitary representations of SL(2,C) restricted to a subgroupE～(2) isomorphic toE(2) are reduced to irreducible unitary components. The results are applied to the decomposition of the product of a momentum-zero- and a mass-zero-representation of the proper orthochronous quantummechanical Poincaré group.     

Relativistic spin selection rules obtained by SL(3, R) symmetry

All irreducible (unitary or not) ray representations of SL(3, R) obeying the Δj = 2 selection rule imposed by Regge trajectories are constructed. They provide irreducible ray representations of $\text{SL(4,}\text{R}\text{) · T}{}^4$ which restricted to the Poincaré subgroup yield unitary representation of real mass and of spin spectrum which statisfies the Δj = 2 selection rule.     

Characters of the positive energy UIRs of <Emphasis Type="Italic">D</Emphasis> = 4 conformal supersymmetry

We give character formulas for the positive energy unitary irreducible representations of the N-extended D = 4 conformal superalgebras su(2, 2/N). Using these, we also derive decompositions of long superfields as they descend to the unitarity threshold. These results are also applicable to irreps of the complex Lie superalgebras sl(4/N). Our derivations use results from the representation theory of su(2, 2/N) developed as early as the 1980s. The text was submitted by the author in English.