Proton Decay in Grand Unified Theories

Interactions which violate the conservation of baryon and lepton number represent an intrinsic part of all grand unified theories (GUTs) of strong and electroweak interactions. These new interactions — predicted within the framework of GUTs — generate B and L violating four-fermion interactions via the exchange of superheavy particles which cannot be ascribed a well-defined baryon or lepton number. The effective coupling constant of these four-fermion interactions might be large enough to make the proton decay detectable by the present generation of experiments. In this review the basic concepts of conventional as well as supersymmetric GUTs relevant for proton decay are sketched. The baryon number violating sector of grand unified theories is discussed in more detail. Special emphasis is laid on the various selection rules arising as consequences of low-energy gauge invariance and supersymmetry for proton decay. These selection rules already determine the coarse pattern of the resulting decay modes and branching ratios without any reference to or detailed knowledge of the underlying grand unified theory. Finally the numerous theoretical predictions are summarized and confronted with experiment.     

Light thresholds in Grand Unified Theories

In a generic Grand Unified Theory with a relatively small dispersion of the spectrum around the Z-boson and the unification masses, a connection is established, exact at one loop level, between M_Z, G_F, α(M_Z and the strong coupling constant α₃(M_Z. At this level of precision, this avoids the logical and phenomenological inconsistency of predicting α₃(M_Z) by means of the electroweak couplings as extracted from the data in the Standard Model rather than in the complete theory. Attention is paid to the independence of the physical results from regularization and/or renormalization schemes.As a particularly relevant example, the analysis is specialized to the case of the Minimal Supersymmetric Standard Model, with emphasis on light charginos and neutralinos.     

Quaternion Octonion Reformulation of Grand Unified Theories

In this paper, Grand Unified theories are discussed in terms of quaternions and octonions by using the relation between quaternion basis elements with Pauli matrices and Octonions with Gell Mann λ matrices. Connection between the unitary groups of GUTs and the normed division algebra has been established to re-describe the SU(5) gauge group. We have thus described the SU(5) gauge group and its subgroup SU(3) _C ×SU(2) _L ×U(1) by using quaternion and octonion basis elements. As such the connection between U(1) gauge group and complex number, SU(2) gauge group and quaternions and SU(3) and octonions is established. It is concluded that the division algebra approach to the theory of unification of fundamental interactions as the case of GUTs leads to the consequences towards the new understanding of these theories which incorporate the existence of magnetic monopole and dyon.     

Modular Group Representations and Fusion in Logarithmic Conformal Field Theories and in the Quantum Group Center

The SL(2, ℤ)-representation π on the center of the restricted quantum group at the primitive 2p^th root of unity is shown to be equivalent to the SL(2, ℤ)-representation on the extended characters of the logarithmic (1, p) conformal field theory model. The multiplicative Jordan decomposition of the ribbon element determines the decomposition of π into a ``pointwise' product of two commuting SL(2, ℤ)-representations, one of which restricts to the Grothendieck ring; this restriction is equivalent to the SL(2, ℤ)-representation on the (1, p)-characters, related to the fusion algebra via a nonsemisimple Verlinde formula. The Grothendieck ring of at the primitive 2p^th root of unity is shown to coincide with the fusion algebra of the (1, p) logarithmic conformal field theory model. As a by-product, we derive q-binomial identities implied by the fusion algebra realized in the center of .     

Renormalization Group,Effective Action and Grand Unification Theories in Curved Space-Time

Recent developments in the renormalization group approach to grand unification theories (GUT's) in curved space-time are reviewed. The new kind of asymptotical conformal invariance in “finite” GUT's in curved space-time (with torsion) is derived. A discussion of asymptotically-finite GUT's in flat and curved space-time is presented. The modifications to the renormalization group in curved space with boundary are given. Some applications of the renormalization group are discussed as well as some prospects.     

Multidimensional Unified Theories

An extended spacetime, M^4+N, is a Riemannian (4 + N)-dimensional manifold which admits an N-parameter group G of (spacelike) isometries and is such that ordinary spacetime M⁴ is the space M^4+N/G of the equivalence classes under G-transformations of M^4+N. A multidimensional unified theory (MUT) is a dynamical theory of the metric tensor on M^4+N, the metric being determined from the Einstein-Hilbert action principle: in absence of matter, the Lagrangian is (essentially) the total curvature scalar of M^4+N. A MUT is an extension of the Cho-Freund generalization of Jordan's five-dimensional theory. A MUT can be faithfully translated in four-dimensional language: as a theory on M⁴, a MUT is a gauge field theory with gauge group G. A unifying aspect of MUT's is that all fields occur as elements of the metric tensor on M^4+N. When the isometry generators are subjected to strongest constraints, a MUT becomes the De Witt-Trautman generalization of Kaluza's five-dimensional theory; in four-dimensional language, this is the theory of Yang-Mills gauge fields coupled to gravity. With weaker constraints, a MUT appears to be more natural than a Yang-Mills theory as a physical realization of the gauge principle for an exact symmetry of gauged confined color. Such weakly-constrained MUT leads to bag-type models without the need for ad hoc surgery on the basic. Lagrangian. The present paper provides a detailed introduction to the formalism of multidimensional unified gauge field theory.     

Geometrodynamics in Multidimensional Unified Theories

The unified theory of gravitation and a Yang-Mills field is formulated as a dynamical theory of (r+3) geometries presumed to be principal bundles endowed with a Riemannian metric. Beyond the usual constraint equations the second fundamental form should satisfy a third constraint equation. It is shown that they have a wormhole-type solution describing a pair of Yang-Mills charges.     

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.     

Fusion Rings of Loop Group Representations

We compute the fusion rings of positive energy representations of the loop groups of the simple, simply connected Lie groups.     

A Generally Covariant Wave Equation for Grand Unified Field Theory

A generally covariant wave equation is derived geometrically for grand unified field theory. The equation states most generally that the covariant d'Alembertian acting on the vielbein vanishes for the four fields which are thought to exist in nature: gravitation, electromagnetism, weak field and strong field. The various known field equations are derived from the wave equation when the vielbein is the eigenfunction. When the wave equation is applied to gravitation the wave equation is the eigenequation of wave mechanics corresponding to Einstein's field equation in classical mechanics, the vielbein eigenfunction playing the role of the quantized gravitational field. The three Newton laws, Newton's law of universal gravitation, and the Poisson equation are recovered in the classical and nonrelativistic, weak-field limits of the quantized gravitational field. The single particle wave-equation and Klein-Gordon equations are recovered in the relativistic, weak-field limit of the wave equation when scalar components are considered of the vielbein eigenfunction of the quantized gravitational field. The Schrödinger equation is recovered in the non-relativistec, weak-field limit of the Klein-Gordon equation). The Dirac equation is recovered in this weak-field limit of the quantized gravitational field (the nonrelativistic limit of the relativistic, quantezed gravitational field when the vielbein plays the role of the spinor. The wave and field equations of O(3) electrodynamics are recovered when the vielbein becomes the relativistic dreibein (triad) eigenfunction whose three orthonormal space indices become identified with the three complex circular indices (1), (2), (3), and whose four spacetime indices are the indices of non-Euclidean spacetime (the base manifold). This dreibein is the potential dreibein of the O(3) electromagnetic field (an electromagnetic potential four-vector for each index (1), (2), (3)). The wave equation of the parity violating weak field is recovered when the orthonormal space indices of the relativistic dreibein eigenfunction are identified with the indices of the three massive weak field bosons. The wave equation of the strong field is recovered when the orthonormal space indices of the relativistic vielbein eigenfunction become the eight indices defined by the group generators of the SU (3) group.     

超对称大统一理论及低能唯象物理

简要地说明为什么有必要扩展粒子物理标准模型.作为最有兴趣的一类扩展,将主要讨论超对称大统一理论及其低能唯象物理.     

Pseudoeffect Algebras. II. Group Representations

This paper is the continuation of the previous paper by Dvureenskij and Vetterlein (2001), Int. J. Theor. Phys. 40(3). We show that any pseudoeffect algebra fulfilling a certain property of Riesz type is representable by a unit interval of some (not necessarily Abelian) partially ordered group. The relation of pseudoeffect to pseudo-MV algebras is made clear, and the &ell-group representation theorem for the latter structure is re-proved.     

No domain wall problem inSU(N) Grand Unified Theory

It is shown that a wide class ofSU(n)×U(1)_P.Q. Grand Unified Theories can avoid the domain wall problem.     

Induced Representations of the One-Dimensional Quantum Galilei Group

We study the representations of the quantum Galilei group by a suitable generalization of the Kirillov method on spaces of noncommutative functions. On these spaces, we determine a quasi-invariant measure with respect to the action of the quantum group by which we discuss unitary and irreducible representations. The latter are equivalent to representations on ², i.e. on the space of square summable functions on a one-dimensional lattice.     

Finite- and Infinite-Dimensional Representations of the Lorentz Group

Finite- and infinite-dimensional representations of the Lorentz group are discussed and various topics in which this group is currently in use are mentioned. The infinitesimal approach of finding representations is reviewed and all finite-dimensional spinor representations of the Lorentz group are obtained. Infinite-dimensional representations are then discussed, including the principal, complementary, and complete series of representations. A generalized Fourier transformation is introduced which enables one to use the global approach to representation theory so as to express infinite-dimensional representations in terms of matrices. This method is shown to lead to a generalization of the spinor form of finite-dimensional representation to the infinite-dimensional case. However, whereas the usual spinor representations are nonunitary, the obtained new form describes both unitary and non-unitary representations, depending on the choice of certain parameters appearing in the representation formula.     

Hypercomplex Representations of the Heisenberg Group and Mechanics

In the spirit of geometric quantisation we consider representations of the Heisenberg(–Weyl) group induced by hypercomplex characters of its centre. This allows to gather under the same framework, called p-mechanics, the three principal cases: quantum mechanics (elliptic character), hyperbolic mechanics and classical mechanics (parabolic character). In each case we recover the corresponding dynamic equation as well as rules for addition of probabilities. Notably, we are able to obtain whole classical mechanics without any kind of semiclassical limit ħ→0.     

Threshold Corrections to the Minimal SUSY SU(5) Grand Unified Theory

Threshold corrections have been carried out atlow- and high-energy scales to the minimalsupersymmetric SU(5) grand unified theory. More refinedvalues for the Weinberg angle and strong couplingconstant are predicted which are fully consistent withthe latest experimental values.