Total extensions of effect algebras

It is shown that if the natural order on a total extension of an effect algebra coincides with the order on, then is unique. The structure of is called a QI-algebra. It is shown that a QI-algebra is less general than a QMV-algebra, but that a QI-algebra is equivalent to a quasi-linear QMV-algebra. Some examples are given and the properties of these structures are studied.     

Idempotents of Clifford Algebras

A classification of idempotents of Clifford algebras C ^p,q is presented. It is shown that using isomorphisms between Clifford algebras C ^p,q and appropriate matrix rings, it is possible to classify idempotents in any Clifford algebra into continuous families. These families include primitive idempotents used to generate minimal one-sided ideals in Clifford algebras. Some low-dimensional examples are discussed.     

Comments on a paper by Marchildon,Antippa, and Everett about tachyons

Marchildonet al. [Phys. Rev. D 27, 1740 (1983)] expressed criticisms about old versions of some approaches to the theory of tachyons. We refute those criticisms somewhatbriefly, since suitable answers werealready present in the literature at the time when the paper here commented upon was being prepared.     

Algebraic and geometric properties of Bethe Ansatz eigenfunctions on a pentagonal magnetic ring

The exact solution of the eigenproblem of the Heisenberg Hamiltonian for the XXX model in the case of a magnetic ring with N=5 nodes is presented in a closed algebraic form. It is demonstrated that the eigenproblem itself is expressible within the extension of the prime field Q of rationals by the primitive fifth root of unity, whereas the associated Bethe parameters, i.e. pseudomomenta, phases of scattering, and spectral parameters, require some finite field extensions, such that the nonlinearity remains algebraic rather than transcendental. Classification of exact Bethe Ansatz eigenstates in terms of rigged string configurations is presented.     

The Charge–Mass–Spin Relation of Clifford Polyparticles, Kerr–Newman Black Holes and the Fine Structure Constant

A Clifford-algebraic interpretation is proposed of the charge, mass, spin relationship found recently by Cooperstock and Faraoini, which was based on the Kerr–Newman metric solutions of the Einstein–Maxwell equations. The components of the polymomentum associated with a Clifford polyparticle in four dimensions provide for such a charge, mass, spin relationship without the problems encountered in Kaluza–Klein compactifications which furnish an unphysically large value for the electron charge. A physical reasoning behind such charge, mass, spin relationship is provided, followed by a discussion on the geometrical derivation of the fine structure constant by Wyler, Smith, Gonzalez-Martin and Smilga. To finalize, the renormalization of electric charge is discussed and some remarks are made pertaining the modifications of the charge–scale relationship, when the spin of the polyparticle changes with scale, that may cast some light into the alleged Astrophysical variations of the fine structure constant.     

Self-adjoint extensions of Coulomb systems in 1, 2 and 3 dimensions

We study the nonrelativistic quantum Coulomb hamiltonian (i.e., inverse of distance potential) in Rⁿ, n=1,2,3. We characterize their self-adjoint extensions and, in the unidimensional case, present a discussion of controversies in the literature, particularly the question of the permeability of the origin. Potentials given by fundamental solutions of Laplace equation are also briefly considered.     

Nonholonomic LL systems on central extensions and the hydrodynamic Chaplygin sleigh with circulation

We consider nonholonomic systems whose configuration space is the central extension of a Lie group and have left invariant kinetic energy and constraints. We study the structure of the associated Euler–Poincaré–Suslov equations and show that there is a one-to-one correspondence between invariant measures on the original group and on the extended group. Our results are applied to the hydrodynamic Chaplygin sleigh, that is, a planar rigid body that moves in a potential flow subject to a nonholonomic constraint modeling a fin or keel attached to the body, in the case where there is circulation around the body.     

Space inversion of spinors revisited: A possible explanation of chiral behavior in weak interactions

We investigate a model in which spinors are considered as being embedded within the Clifford algebra that operates on them. In Minkowski space M_1,3$M_{1,3}$, we have four independent 4-component spinors, each living in a different minimal left ideal of Cl(1,3)$\mathit{Cl}(1,3)$. We show that under space inversion, a spinor of one left ideal transforms into a spinor of another left ideal. This brings novel insight to the role of chirality in weak interactions. We demonstrate the latter role by considering an action for a generalized spinor field ψαi${\psi }^{\alpha i}$ that has not only a spinor index α but also an extra index i   running over four ideals. The covariant derivative of ψαi${\psi }^{\alpha i}$ contains the generalized spin connection, the extra components of which are interpreted as the SU(2) gauge fields of weak interactions and their generalization. We thus arrive at a system that is left–right symmetric due to the presence of a “parallel sector”, postulated a long time ago, that contains mirror particles coupled to mirror SU(2) gauge fields.     

Irreducible Representations of a Class of Current Algebras of Etingof and Frenkel

A class of representations is described for the central extensions, found by Etingof and Frenkel of current algebras over Riemann surfaces. Their irreducibility is proved. The possibility/impossibility to obtain integrable representations within that class is discussed briefly.     

Cohomological obstructions to the equivariant extension of closed invariant forms

We study under what condition a closed invariant form on a manifold with a group action admits an equivariant extension. We derive a sequence of obstructions in the cohomology groups of the Lie algebra with coefficients in appropriate modules. We illustrate the result with two specific examples. We then discuss when such cohomological obstructions vanish. Finally, we compare our analysis with the spectral sequence point of view.     

Further investigation of mass dimension one fermionic duals

In this paper we proceed into the next step of formalization of a consistent dual theory for mass dimension one spinors. This task is developed approaching the two different and complementary aspects of such duals, clarifying its algebraic structure and the so called τ-deformation. The former regards the mathematical equivalence of the recent proposed Lorentz preserving dual with the duals of algebraic spinors, from Clifford algebras, showing the consistency and generality of the new dual. Moreover, by revealing its automorphism structure, the hole of the τ-deformation and contrasting the action group orbits with other Lorentz breaking scenarios, we argue that the new mass dimension one dual theory is placed over solid and consistent basis.     

Quantum Field Theories on an Algebraic Curve

We formulate quantum field theories on an algebraic curve and outline a 'paradigm' interpreting Ward identities as reciprocity laws.     

Geometric and Algebraic Approach to Classical Dynamics of a Particle with Spin

We investigate the Frenkel equation and alternative approaches aimed to describe classically the dynamics of a particle with a spin degree of freedom. For all those theories unphysical solutions are shown to exist but to be removable via formal use of geometric perturbation theory. With an Clifford algebraic representation of the employed geometric concepts we put forward the modified Frenkel equation as more intuitive as alternative approaches.     

Quantum-Gravity Thermodynamics, Incorporating the?Theory of Exactly Soluble Active Stochastic Processes, with Applications

A re-visitation of QFT is first cited, deriving the Feynman integral from the theory of active stochastic processes (Glueck and Hueffler, Phys. Lett. B. 659(1–2):447–451, 2008; Hueffel and Kelnhofer, Phys. Lett. B 588(1–2):145–150, 2004). We factor the lie group “generator” of the inverse wavefunction over an entropy-maximizing basis. Performing term-by-term Ito-integration leads us to an analytical, evaluable trajectory for a charged particle in an arbitrary field given a Maximum-Entropy distribution. We generalize this formula to many-body electrodynamics. In theory, it is capable of predicting plasma’s thermodynamic properties from ionic spectral data and thermodynamic and optical distributions. Blessed with the absence of certain limitations (e.g., renormalization) strongly present in competing formalisms and the incorporation of research related to many different phenomena, we outline a candidate quantum gravity theory based on these developments.     

A comment on generalized electromagnetism and dirac algebra

We show that, contrary to a recent claim by G. Fryberger, it is not possible within the Clifford bundle formalism to write a local Lagrangian which gives simultaneously the field equations and the equations of motion of electric charges and (non-topological), magnetic monopoles, and the generalized field equations.     

A remark on extensions of quasi-free derivations on the CAR-algebra

Let δ be a quasi-free derivation of the CAR algebra, and let be a closed *-derivation which is an extension of δ. We use Price's techniques from [6] to show that if the polynomials in the linear field operators a(f)→a ^* (f) in D( ) is a core for , then is quasi-free.