Monopoles and family-replicated unification

The present theory is based on the assumption that, at very small (Planck scale) distances our spacetime is discrete, and this discreteness influences the Planck scale physics. Considering our (3+1)-dimensional spacetime as a regular hypercubic lattice with a parameter a=λ_Pl, where λ_Pl is the Planck length, we have investigated a role of lattice artifact monopoles, which is essential near the Planck scale if the family-replicated gauge group model (FRGGM) is an extension of the Standard Model (SM) at high energies. It was shown that monopoles have N times smaller magnetic charge in the FRGGM than in the SM (N is the number of families in the FRGGM). These monopoles can give an additional contribution to β functions of the renormalization-group equations for the running fine structure constants α_i(μ) (i=1, 2, 3 correspond to the U(1), SU(2), and SU(3) gauge groups of the SM). We have used the Dirac relation for renormalized electric and magnetic charges. Also, we have estimated the enlargement of a number of fermions in the FRGGM leading to the suppression of the asymptotic freedom in the non-Abelian theory. The different role of monopoles in the vicinity of the Planck scale gives rise either to anti-GUT or to the new possibility of unification of gauge interactions (including gravity) at the scale μ_GUT≈10^18.4 GeV. We discussed the possibility of the [SU(5)]³ SUSY or [SO(10)]³ SUSY unifications.     

Finite-dimensional representations of the euclidean group in the plane

In this article two theorems are given which permit, together with the concept of a representation vector diagram, to classify all (linear) finite-dimensional representations of the algebra and group E ₂ which are induced by a master representation on the place of the universal enveloping algebra of the algebra E ₂. Apart from a classification of the finite-dimensional representations, the two theorems make it possible to obtain the matrix elements of these representations for both, algebra and group, in explicit form. The material contained in this letter forms part of an analysis of indecomposable (finite- and infinite-dimensional) representations of the algebra and group E ₂ which is contained in Reference [1]. No proofs will be given in this letter. We refer instead to [1].     

Geometrical aspects of magnetic monopoles

The possible topological structures of elementary particles have been investigated to explore the possibility of the existence of magnetic monopoles. It is pointed out that when an elementary charged particle is depicted as an extended body such that the orientation of the internal space (internal helicity) defines the fermion number, the global conservation of this does not allow the existence of a magnetic monopole. Again it is argued that when anisotropy is introduced in the microlocal space-time depicting the internal space of hadrons, this gives rise to the internal symmetry algebra and no non-Abelian gauge fields and Higgs scalars are necessary to have a grand unified scheme of interactions. This avoids theSU ₂ and GUT monopoles. Besides, in this formalism, baryon number corresponds to the orientation or internal helicity of the composite system and the global conservation of this quantum number is found to be a consequence of Lorentz invariance. This forbids the existence of any sort of cosmological monopole in this Lorentz invariant Universe.     

Derivation of the B (3) Field and Concomitant Vacuum Energy Density from the Sachs Theory of Electrodynamics

The archetypical and phaseless vacuum magnetic flux density of O(3) electrodynamics, the B ⁽³⁾ field, is derived from the irreducible representation of the Einstein group and is shown to be accompanied by a vacuum energy density which depends directly on the square of the scalar curvature R of curved spacetime. The B ⁽³⁾ field and the vacuum energy density are obtained respectively from the non-Abelian part of the field tensor F and the non-Abelian part of the metrical field equation. Both of these terms are given by Sachs [5].     

Monopoles without strings: a conflict between the one-photon condition and duality invariance

We prove a new no-go theorem in the Dirac-algebra formulation of generalized electromagnetic theory, which includes magnetic monopoles and uses two potentialsA andM : It is impossible to construct a Lagrangian which is duality invariant and satisfies the one-photon assumption, from which Maxwell's equations and the equations of motion can be derived. Such a Lagrangian can be found only if either duality invariance or the one-photon assumption is sacrificed. These constraints as well as others discussed here are based on recently published results on monopoles without strings in the Dirac algebra, but they do not arise from any artificial restrictions in the Dirac-algebra formulation.     

Nonstandard 2+1 conformal Lie bialgebras and their quantum deformations

The nonstandard and so(2, 2) Lie bialgebras are generalized to the so(3, 2) case in two natural ways by considering this algebra as the conformal algebra of the 2+1 Minkowskian spacetime. Lie bialgebra contractions are analyzed providing conformal bialgebras of the 2+1 Galilean and Carroll spacetimes. The corresponding quantum Hopf so(3, 2) algebras are presented and contractions are performed at the quantum level.     

Imaginary numbers are not real—The geometric algebra of spacetime

This paper contains a tutorial introduction to the ideas of geometric algebra, concentrating on its physical applications. We show how the definition of a geometric product of vectors in 2-and 3-dimensional space provides precise geometrical interpretations of the imaginary numbers often used in conventional methods. Reflections and rotations are analyzed in terms of bilinear spinor transformations, and are then related to the theory of analytic functions and their natural extension in more than two dimensions (monogenics), Physics is greatly facilitated by the use of Hestenes' spacetime algebra, which automatically incorporates the geometric structure of spacetime. This is demonstrated by examples from electromagnetism. In the course of this purely classical exposition many surprising results are obtained—results which are usually thought to belong to the preserve of quantum theory. We conclude that geometric algebra is the most powerful and general language available for the development of mathematical physics.The title of this paper is inspired by David Hestenes, who is known to have a fondness for deliberate ambiguity.⁽¹⁾ Supported by a SERC studentship.     

Monopole charges for arbitrary compact gauge groups and Higgs fields in any representation

The topological invariants of monopoles are described for an arbitrary compact gauge groupG and Higgs field in any representation. The results generalize those obtained recently for compact and simply connectedG and in the adjoint representation. The cases when the residual symmetry group isH=U(1) orH=U(3) are worked out explicitly. This latter is needed to accommodate fractional electric charge with monopoles having one Dirac unit magnetic charge.The general theory is illustrated on the SU(5) monopole.     

Conceptual Unification of Gravity and Quanta

We present a model unifying general relativity and quantum mechanics. The model is based on the (noncommutative) algebra on the groupoid Γ=E×G where E is the total space of the frame bundle over spacetime, and G the Lorentz group. The differential geometry, based on derivations of , is constructed. The eigenvalue equation for the Einstein operator plays the role of the generalized Einstein’s equation. The algebra , when suitably represented in a bundle of Hilbert spaces, is a von Neumann algebra ℳ of random operators representing the quantum sector of the model. The Tomita–Takesaki theorem allows us to define the dynamics of random operators which depends on the state φ. The same state defines the noncommutative probability measure (in the sense of Voiculescu’s free probability theory). Moreover, the state φ satisfies the Kubo–Martin–Schwinger (KMS) condition, and can be interpreted as describing a generalized equilibrium state. By suitably averaging elements of the algebra , one recovers the standard geometry of spacetime. We show that any act of measurement, performed at a given spacetime point, makes the model to collapse to the standard quantum mechanics (on the group G). As an example we compute the noncommutative version of the closed Friedman world model. Generalized eigenvalues of the Einstein operator produce the correct components of the energy-momentum tensor. Dynamics of random operators does not “feel” singularities.     

Development of the Evans Wave Equation in the Weak-Field Limit: The Electrogravitic Equation

The Evans wave equation [1-3] is developed in the weak-field limit to give the Poisson equation and an electrogravitic equation expressing the electric field strength E in terms of the acceleration g due to gravity and a fundamental scalar potential ⁽⁰⁾ with the units of volts (joules per coulomb). The electrogravitic equation shows that an electric field strength can be obtained from the acceleration due to gravity, which in general relativity is non-Euclidean spacetime. Therefore an electric field strength can be obtained, in theory, from scalar curvature R. This inference is supported by recent experimental data from the patented motionless electromagnetic generator [5].     

The Spectral Shift Function and Spectral Flow

At the 1974 International Congress, I. M. Singer proposed that eta invariants and hence spectral flow should be thought of as the integral of a one form. In the intervening years this idea has lead to many interesting developments in the study of both eta invariants and spectral flow. Using ideas of [24] Singer’s proposal was brought to an advanced level in [16] where a very general formula for spectral flow as the integral of a one form was produced in the framework of noncommutative geometry. This formula can be used for computing spectral flow in a general semifinite von Neumann algebra as described and reviewed in [5]. In the present paper we take the analytic approach to spectral flow much further by giving a large family of formulae for spectral flow between a pair of unbounded self-adjoint operators D and D +  V with D having compact resolvent belonging to a general semifinite von Neumann algebra and the perturbation . In noncommutative geometry terms we remove summability hypotheses. This level of generality is made possible by introducing a new idea from [3]. There it was observed that M. G. Krein’s spectral shift function (in certain restricted cases with V trace class) computes spectral flow. The present paper extends Krein’s theory to the setting of semifinite spectral triples where D has compact resolvent belonging to and V is any bounded self-adjoint operator in . We give a definition of the spectral shift function under these hypotheses and show that it computes spectral flow. This is made possible by the understanding discovered in the present paper of the interplay between spectral shift function theory and the analytic theory of spectral flow. It is this interplay that enables us to take Singer’s idea much further to create a large class of one forms whose integrals calculate spectral flow. These advances depend critically on a new approach to the calculus of functions of non-commuting operators discovered in [3] which generalizes the double operator integral formalism of [8–10]. One surprising conclusion that follows from our results is that the Krein spectral shift function is computed, in certain circumstances, by the Atiyah-Patodi-Singer index theorem [2].     

On the relationship between Twistors and Clifford algebras

Basis p-forms of a complexified Minkowski spacetime can be used to realize a Clifford algebra isomorphic to the Dirac algebra of matrices. Twistor space is then constructed as a spin space of this abstract algebra through a Witt decomposition of the Minkowski space. We derive explicit formulas relating the basis p-forms to index one twistors. Using an isomorphism between the Clifford algebra and a space of index two twistors, we expand a suitably defined antisymmetric index two twistor basis on p-forms of ranks zero, one, and four. Together with the inverse formulas they provide a complete passage between twistors and p-forms.     

Dirac monopoles on quantized vortices in superfluid3He

Quantized vortices in superfluid³He display a variety of novel structures that have previously not been observed in any other quantum fluids; their basic experimental manifestations and theoretical features have been reviewed by Hakonen, Lounasmaa and Simola [Physica B160 (1989) 1] and by Salomaa and Volovik [Rev. Mod. Phys.59 (1987) 533]. In order not to repeat these reviews in this paper, here we choose to discuss the theme of the possible pointlike orderparameter singularities, monopoles, that can occur on the quantized³He vortices. Such monopoles, mathematical analogues of the magnetic Dirac monopole, may exist in superfluid³He due to the complicated order-parameter structure, which makes it possible to have several different types of quantized vortex lines and phase boundaries between the superfluid states. Analogs of Dirac monopoles, and also monopoles with 1/2 and 1/4 of the magnetic charge of the Dirac monopole, may exist at the points of intersections of quantized vortex lines and phase boundaries — or along vortices if they change their quantum state; several examples are discussed.Invited talk at the International Conference on Macroscopic Quantum Phenomena, Smolenice Castle, Czechoslovakia, September 18–22, 1989.I want to thank G. E. Volovik for a useful discussion and O. V. Lounasmaa for encouragement. This research has been supported through the Award for the Advancement of European Science by the Körber-Stiftung (Hamburg, FRG) and by the Academy of Finland.     

Indecomposable representations of the algebra su(2,1)

Representations of the Lie algebra sl(3) with highest weight are analyzed. Invariant subspaces of indecomposable representations are determined. We study the decomposition of these representations with respect to the subalgebras su(2) and su(1,1) (in their obvious imbedding in su(2,1)).For special cases this decomposition gives indecomposable non multiplicity free representations (indecomposable pairs) with highest weight. These were discussed in [1] and appear also in the decomposition so(3,2) su(1,1) of the Rac representation, [7].     

Affine connection structure of the charged symplectic 2-form

It is shown that the charged symplectic form in Hamiltonian dynamics of classical charged particles in electromagnetic fields defines a generalized affine connection on an affine frame bundle associated with spacetime. Conversely, a generalized affine connection can be used to construct a symplectic 2-form if the associated linear connection is torsion-free and the antisymmetric part of theR ^4* translational connection is locally derivable from a potential. Hamiltonian dynamics for classical charged particles in combined gravitational and electromagnetic fields can therefore be reformulated as aP(4)=O(1, 3)R ^4* geometric theory with phase space the affine cotangent bundleAT ^* M of spacetime. The sourcefree Maxwell equations are reformulated as a pair of geometrical conditions on the ^4* curvature that are exactly analogous to the source-free Einstein equations.     

Drinfeld–Sokolov Reduction for Difference Operators and Deformations of -Algebras¶I. The Case of Virasoro Algebra

We propose a q-difference version of the Drinfeld-Sokolov reduction scheme, which gives us q-deformations of the classical -algebras by reduction from Poisson-Lie loop groups. We consider in detail the case of SL ₂ . The nontrivial consistency conditions fix the choice of the classical r-matrix defining the Poisson-Lie structure on the loop group LSL ₂ , and this leads to a new elliptic classical r-matrix. The reduced Poisson algebra coincides with the deformation of the classical Virasoro algebra previously defined in [19]. We also consider a discrete analogue of this Poisson algebra. In the second part [31] the construction is generalized to the case of an arbitrary semisimple Lie algebra. Received: 20 April 1997 / Accepted: 22 July 1997     

From Dynamical to Non-Dynamical Twists

We provide a construction which gives a twisting element for a universal enveloping algebra starting from a certain dynamical twist. This construction is a quantization of the analogous quasi-classical process given in [Karolinsky and Stolin, Lett. Math. Phys. 60 (2002), 257–274]. In particular, we reduce the computation of the twisting element for the classical r-matrix constructed from the Frobenius algebra , the maximal parabolic subalgebra of related to the simple root _n-1, to the computation of the universal dynamical twist for .*Supported in part by the Royal Swedish Academy of Sciences**Supported in part by RFFI grant 02-01-00085a and CRDF grant RM1-2334-MO-02Mathematical Subject Classifications (2000). 17B35, 17B10, 17B20.     

On quantum deformations of d=4 conformal algebra

Three classes of classical r-matrices for sl(4, C) algebra are constructed in quasi-Frobenius algebra approach. They satisfy CYBE and are spanned respectively on 8,10,12 generators. The o(4, 2) reality condition can be imposed only on the eight dimensional r-matrices with dimension-full deformation parameters. Contrary to the Poincaré algebra case, it appears that all deformations with a mass-like deformation parameter (-deformations) are described by classical r-matrices satisfying CYBE.