Lorentz violation constrained by triplicity of lepton families and neutrino oscillations

In this paper we postulate an algebraic model to relate the triplet characteristic of lepton families to Lorentz violation. Inspired by the two-to-one mapping between the group SL(2, C) and the Lorentz group via the Pauli grading (the elements of SL(2, C) expressed by direct sum of unit matrix and generators of SU (2) group), we grade the SL(3, C) group with the generators of SU(3), i. e. the Gell-Mann matrices, then express the SU(3) group in terms of three SU(2) subgroups, each of which stands for a lepton species and is mapped into the proper Lorentz group as in the case of the group SL(2,C). If the mapping from group SL(3,C) to the Lorentz group is constructed by choosing one SU(2) subgroup as basis, then the other two subgroups display their impact only by one more additional generator to that of the original Lorentz group. Applying the mapping result to the Dirac equation, it is found that only when the kinetic vertex γμξμ is extended to encompass γμξμ can the Dirac-equation-form be conserved. The generalized vertex is useful in producing neutrino oscillations and mass differences.     

Duality-Induced Reflections and CPT

The linear particle-antiparticle conjugation C and position space reflection P aswell as the antilinear time reflection T are shown to be inducible by theself-duality of representations for the operation groups SU(2), SL(C²), and R forspin, Lorentz transformations, and time translations, respectively. The definitionof a color-compatible linear CP-reflection for quarks as self-duality induced isimpossible since triplet and antitriplet SU(3)-representations are not linearlyequivalent.     

光偏振态的群论描述:转动矩阵的生成元

应用偏振光描述中的变换矩阵与群论的对应关系[1]和相应的计算理论,讨论了与偏振光学系统中的Jones矩阵、Mueller矩阵相对应的SU(2)群、SO(3)群和Lorentz群的生成元问题,给出了用单位矩阵、Pauli自旋矩阵和稀疏矩阵分别作为无耗偏振光学系统中SU(2)群元(Jones矩阵)和SO(3)群元(Mueller矩阵)生成元以及部分损耗偏振光学系统中的幺模群(Jones矩阵)和Lorentz群(Mueller矩阵)生成元的具体形式;矩阵计算理论说明这些群元的生成元表示可以简化偏振光学系统的计算。     

Numerical methods for EPRL spin foam transition amplitudes and Lorentzian recoupling theory

The intricated combinatorial structure and the non-compactness of the Lorentz group have always made the computation of \(SL(2,\mathbb {C})\) EPRL spin foam transition amplitudes a very hard and resource demanding task. With sl2cfoam we provide a C-coded library for the evaluation of the Lorentzian EPRL vertex amplitude. We provide a tool to compute the Lorentzian EPRL 4-simplex vertex amplitude in the intertwiner basis and some utilities to evaluate SU(2) invariants, booster functions and \(SL(2,\mathbb {C})\) Clebsch–Gordan coefficients. We discuss the data storage, parallelizations, time, and memory performances and possible future developments.     

Area-preserving diffeomorphisms and supermembrane Lorentz invariance

The Lie algebra of area-preserving diffeomorphisms on closed membranes of arbitrary topology is investigated. On the basis of a harmonic decomposition we define the structure constants as well as two other tensors which appear in the supermembrane Lorentz generators. We derive certain identities between these tensors and analyze their validity when the areapreserving diffeomorphisms are approximated bySU(N). One of the additional tensors can then be identified with the invariant symmetric three-index tensor ofSU(N), while the second has no obvious analog. We prove that the Lorentz generators are classically conserved in the light-cone gauge for arbitrary membrane topology, as a consequence of these tensor identities. This formulation allows a systematic study of the violations of Lorentz invariance in theSU(N) approximation.     

Polarization symmetry of vector fields

A group of additional invariance (polarization symmetry) of the Prock equations is considered, whose generators satisfy the algebra SU (3) (massive field) of SU (2) (massless field). The investigative method developed in the paper is directly related to the physical content of the transformations of the symmetry being discussed: the change in the polarization field. The small Lorentz group is a sub-group of the transformations being discussed. Possible physical applications of polarization symmetry are discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, Ho. 7, pp. 77–80, July, 1982.     

Gauge theories of weak and strong gravity

A review of some recent papers on gauge theories of weak and strong gravity is presented. For weak gravity, SL(2, C) gauge theory along with tetrad formulation is described which yields massless spin-2 gauge fields (quanta gravitons). Next a unified SL(2n,C) model is discussed along with Higgs fields. Its internal symmetry is SU(n). The free field solutions after symmetry breaking yield massless spin-1 (photons) and spin-2 (gravitons) gauge fields and also massive spin-1 and spin-2 bosons. The massive spin-2 gauge fields are responsible for short range superstrong gravity. Higgs-fermion interaction can lead to baryon and lepton number non-conservation. The relationship of strong gravity with other forces is also briefly considered.     

Compensation and nonlinear theory

The existence is demonstrated of a new class of compensating fields in addition to the well-known fields of Yang-Mills type that lead to nonlinear field theory and induce a non-Euclidean geometry of spacetime. The cases of the Lorentz group, U(1), SU(2), SU(3), SU(2, 2) and coordinate transformations are considered.     

Topology of the Symmetry Group of the Standard Model

We study the topological structure of thesymmetry group of the standard model, G_SM =U(1) × SU(2) × SU(3). Locally,G_SM S¹ ×(S3)² × S⁵. For SU(3), whichis an S³-bundle over S⁵ (and therefore a local product of thesespheres) we give a canonical gauge i.e., a canonical setof local trivializations. These formulas give explicitlythe matrices of SU(3) without using the Lie algebra (Gell-Mann matrices). Globally, we prove thatthe characteristic function of SU(3) is the suspensionof the Hopf map . We also study the case of SU(n) forarbitrary n, in particular the cases of SU(4), a flavor group, and of SU(5),a candidate group for grand unification. We show thatthe 2-sphere is also related to the fundamentalsymmetries of nature due to its relation to SO⁰(3, 1), the identity component of the Lorentz group, asubgroup of the symmetry group of several gauge theoriesof gravity.     

Left-right model of Quark and Lepton masses without a scalar bidoublet

We propose a left-right model of quarks and leptons based on the gauge group SU(3)(C)xSU(2)(L)xSU(2)(R)xU(1)(B-L), where the scalar sector consists of only two doublets: (1,2,1,1) and (1,1,2,1). As a result, any fermion mass, whether it be Majorana or Dirac, must come from dimension-five operators. This allows us to have a common view of quark and lepton masses, including the smallness of Majorana neutrino masses as the consequence of a double seesaw mechanism.     

Negative dimensions and E7 symmetry

The invariant length and volume which characterize the Lorentz group are extended to a quadratic and a quartic supersymmetric invariant. The symmetry group of the Grassmann sector can be SO(2), SU(2), SU(2) × SU(2) × SU(2), Sp(6), SU(6), SO(12) or E₇, which are also possible global symmetries of extended supergravities. Diophantine conditions which yield this classification follow from the corresponding conditions in d bosonic dimensions by the replacement d → ?d.     

Is spontaneous breaking of R-parity feasible in minimal low-energy supergravity?

Spontaneous violation of lepton number without breaking Lorentz invariance can, in principle, be incorporated in models with softly broken supersymmetry. We study the situation for minimal low-energy supergravity models coming from a GUT (hence not having hierarchy destabilizing light singlets) and where the SU(2) × U(1) breaking is radiative. It is found that for this type of model, R-parity breaking requires either too heavy a top quark for a realistic superpartner spectrum or too light a superpartner spectrum for a realistic top quark, making the spontaneous violation of lepton number in the third generation incompatible with present experimental data. We do not discard the possibility of having it in a fourth, heavier, generation.     

From the Group SL(2,?C) to Gyrogroups and Gyrovector Spaces and Hyperbolic Geometry

We show that the algebra of the group SL(2, C) naturally leads to the notion of gyrogroups and gyrovector spaces for dealing with the Lorentz group and its underlying hyperbolic geometry. The superiority of the use of the gyrogroup formalism over the use of the SL(2, C) formalism for dealing with the Lorentz group in some cases is indicated by (i) the validity of gyrogroups and gyrovector spaces in higher dimensions, by (ii) the analogies that they share with groups and vector spaces, and by (iii) the demonstration that gyrovector spaces form the setting for hyperbolic geometry in the same way that vector spaces form the setting for Euclidean geometry. As such, gyrogroups and gyrovector spaces provide powerful tools for the study of relativity physics.     

A modest proposal for generating Yukawa couplings from gauge interactions

We propose that the quark and lepton Yukawa superpotential couplings to Higgs supermultiplets arise from non-perturbative gauge interactions. This is possible in models with an SU(N) × G gauge group. We present a three-generation model based on SU(8) × G, and indicate how such a scenario could lead to a realistic hierarchy of quark and lepton masses.     

SU(3)L × SU(3)R as gauge groups of weak and associated charges

Consequences of an SU(3)_L × SU(3)_R gauge group underlying weak-electromagnetic and associated interactions are studied. A specific pattern of symmetry breaking relates the lepton mass spectrum and mixing pattern, the effective (anti) neutrino-induced neutral current and (approximate) parity conservation in the neutral-current sector.     

A chiral SU(4) × SU(4) gauge theory of weak and electromagnetic interactions

A lepton hadron analogy is considered based on the gauge group SU(4)_L × SU(4)_R × U(1), which is broken entirely spontaneously. The model satisfies the physical requirements of the V-A theory, muon-electron universality, no neutral strangeness changing currents, the Cabibbo structure for the SU(3) currents, and triangle anomalies can be avoided. The contribution of the existing neutral currents to various neutrino processes are calculated.     

A horizontal model for neutrino mixing

We consider a horizontal SU(3)_H × SU(2)_L × U(1) model in which the large Majorana neutrino masses are associated with a large horizontal scale. We find that the charged lepton sector is responsible for the neutrino mixing which we calculate in the present model. We also find that the neutrino oscillation length is related to the horizontal scale.     

Relativistic spin on the Poincaré group

Classical spinning particles are interpreted in terms of an underlying geometric theory. They are described by trajectories on the Poincaré group. Upon quantization an eleven-dimensional Kaluza-Klein type theory is obtained which incorporates spin and isospin in a local SL(2, C)×U(1)×SU(2) gauge theory, unifying gravity and the pre-Higgs standard model. The relation to parametrized relativistic quantum theory is discussed.     

Global Solutions of Einstein—Dirac Equation on the Conformal Space

The difference between the Riemann and Lorentz spinor manifolds of four dimensions is that the Dirac operator of the former is elliptic and that of the latter is hyperbolic.Moreover the spinor group of the former is a compact group and that of the latter is a noncompact group,which is isomorphic to SL(2,C).Hence the results and their interpretation coming from the two theories would be different.In this short note we study only the Lorentz spinor manifold and,especially,the solutions of Einstein-Dirac equations on the conformal space,which is closely related to the AdS/CFT correspondence.