Authors:Viktor Abramov Pages: 9 - 16 Abstract: Abstract We propose a notion of a super n-Lie algebra and construct a super n-Lie algebra with the help of a given binary super Lie algebra which is equipped with an analog of a supertrace. We apply this approach to the super Lie algebra of a Clifford algebra with even number of generators and making use of a matrix representation of this super Lie algebra given by a supermodule of spinors we construct a series of super 3-Lie algebras labeled by positive even integers. PubDate: 2017-03-01 DOI: 10.1007/s00006-015-0604-3 Issue No:Vol. 27, No. 1 (2017)

Authors:Alexandre Trovon; Osamu Suzuki Pages: 59 - 70 Abstract: Abstract In this paper we introduce the idea of Galois extension for a class of associative algebras and discuss binary and ternary Clifford algebras by such an algebraic construction. Nonion algebra is characterized by Galois extensions and a ternary structure is proposed for \({\mathfrak{su}(3)}\) leading to a duality for certain binary and ternary differential operators. PubDate: 2017-03-01 DOI: 10.1007/s00006-015-0565-6 Issue No:Vol. 27, No. 1 (2017)

Authors:Sirkka-Liisa Eriksson; Heikki Orelma Pages: 99 - 110 Abstract: Abstract We are studying a function theory of k-hypermonogenic functions connected to k-hyperbolic harmonic functions that are harmonic with respect to the hyperbolic Riemannian metric $$ds_{k}^{2} = x_{n}^{\frac{2k}{1-n}} \left(dx_{0}^{2} + \cdots + dx_{n}^{2} \right)$$ in the upper half space \({\mathbb{R}_{+}^{n+1} = \left\{\left(x_{0}, \ldots,x_{n}\right)\, \,x_{i} \in \mathbb{R}, x_{n} > 0\right\}}\) . The function theory based on this metric is important, since in case \({k = n-1}\) , the metric is the hyperbolic metric of the Poincaré upper half space and Leutwiler noticed that the power function \({x^{m}\,(m \in \mathbb{N}_{0})}\) , calculated using Clifford algebras, is a conjugate gradient of a hyperbolic harmonic function. We find a fundamental \({k}\) -hyperbolic harmonic function. Using this function we are able to find kernels and integral formulas for k-hypermonogenic functions. Earlier these results have been verified for hypermonogenic functions ( \({k = n-1}\) ) and for k-hyperbolic harmonic functions in odd dimensional spaces. PubDate: 2017-03-01 DOI: 10.1007/s00006-015-0629-7 Issue No:Vol. 27, No. 1 (2017)

Authors:Gehová López-González; Nancy Arana-Daniel; Eduardo Bayro-Corrochano Pages: 647 - 660 Abstract: Abstract This work presents a parallelization method for the Clifford support vector machines, based in two characteristics of the Gaussian Kernel. The pure real-valued result and its commutativity allows us to separate the multivector data in its defining subspaces. These subspaces are independent from each other, so we can solve the problem using parallelism. The motivation is to present an easy approach that can be explained using the more common known concepts of complex numbers and quaternions, because in general there exists a lack of familiarity with geometric algebra. PubDate: 2017-03-01 DOI: 10.1007/s00006-016-0726-2 Issue No:Vol. 27, No. 1 (2017)

Authors:Sergio Giardino Abstract: Abstract We have built a constrained four-dimensional quaternion-parametrized conformal field theory using quaternion holomorphic functions as the generators of quaternionic conformal transformations. With the two-dimensional complex-parametrized conformal field theory as our model, we study the stress tensor, the conserved charge, the symmetry generators, the quantization conditions and several operator product expansions. Future applications are also addressed. PubDate: 2017-04-17 DOI: 10.1007/s00006-017-0781-3

Authors:Xinxue Chai; Qinchuan Li Abstract: Abstract It has been challenging to obtain an analytical or closed-form expression of the motion space of a Bennet linkage because it is over-constrained and has complex geometric conditions. This paper presents an analytical mobility analysis of the Bennet linkage using geometric algebra. Frist, the Bennet linkage is regarded as a 2-RR (R: revolute joint) parallel mechanism and the limb motion space is the join of two twists associated with the two revolute pairs. Then, the motion space of the output link of the Bennet linkage is obtained by using meet operator to calculate the intersection of the two limb motion space. Compared to the constrained screw-based method, the use of a meet operator results in fewer steps, thus simplifying the computation. This intersection presents an analytical or symbolic expression of the motion space of the Bennet linkage with straightforward geometric interpretations. PubDate: 2017-04-11 DOI: 10.1007/s00006-017-0778-y

Authors:Werner Benger; Wolfgang Dobler Abstract: Abstract Geometric algebra (GA) is a promising approach to address interoperability bottlenecks that are particularly prominent in the big data era. Similar to how GA unites and simplifies otherwise distinct mathematical branches it may also help to unite software via common interfaces between otherwise distinct applications. The promising potential of GA would be best exhibited by the ability of seamless integration into existing, complex applications. To achieve this vision various constraints have to be considered. Particularly having C++ in focus, we discuss the “wish list” that an optimal C++ implementation should provide. We find that to cover the various constraints an hybrid approach benefiting from multiple programming paradigms, ranging from generic to object-oriented programming, will be needed. C++ is a very suitable platform providing all these capabilities and promising approaches like Generative Programming and Active Libraries provide technology highly desirable for universally promoting GA to an extensive range of application domains. PubDate: 2017-04-10 DOI: 10.1007/s00006-017-0780-4

Authors:Antônio M. Moya; Waldyr Alves Rodrigues; Samuel A. Wainer Abstract: Abstract In this paper we provide using the Clifford and spin-Clifford formalism and some few results of the extensor calculus a derivation of the conservation laws that follow directly from the Dirac–Hestenes equation (DHE) describing a Dirac–Hestenes spinor field (DHSF) in interaction with an external electromagnetic field without using the Lagrangian formalism. In particular, we show that the energy-momentum and total angular momentum extensors of a DHSF is not conserved in spacetime regions permitting the existence of a null electromagnetic field F but a non null electromagnetic potential \(A \) . These results have been used together with some others recently obtained (e.g., that the classical relativistic Hamilton–Jacobi equation is equivalent to a DHE satisfied by a particular class of DHSF) to obtain the correct relativistic quantum potential when the Dirac theory is interpreted as a de Broglie–Bohm theory. Some results appearing in the literature on this issue are criticized and the origin of some misconceptions is detailed with a rigorous mathematical analysis. PubDate: 2017-04-04 DOI: 10.1007/s00006-017-0779-x

Authors:Mauricio Cele Lopez Belon; Dietmar Hildenbrand Abstract: Abstract The usage of Geometric Algebra motors instead of Euclidean vectors for describing the position and orientation of points on a surface has promising applications in Computer Science and Engineering. Common geometric transformations, such as rotations and translations of Euclidean points, are also applicable to motors. However, encoding vertex positions and orientations as motors adds the capability of computing motor interpolation on surfaces. Thanks to that, general curves and surfaces can be generated by a motor interpolation process using different basis functions and parameterizations. In applications, the generated surfaces can be visually manipulated and deformed in a predictable way by changing the motors. In this paper we look inside the theory behind those applications as well as practical details on how Geometric Algebra algorithms can be computed efficiently. We show that geometric deformations can be computed at interactive rates on surface models with millions of vertices using the GPU. PubDate: 2017-04-01 DOI: 10.1007/s00006-017-0777-z

Authors:Dharam Vir Ahluwalia Abstract: Abstract About a decade ago the present author in collaboration with Daniel Grumiller presented an ‘unexpected theoretical discovery’ of spin one-half fermions with mass dimension one (Ahluwalia-Khalilova and Grumiller in Phys Rev D 72:067701 arXiv:hep-th/0410192, 2005, JCAP 0507:012, arXiv:hep-th/0412080, 2005). In the decade that followed a significant number of groups explored intriguing mathematical and physical properties of the new construct. However, the formalism suffered from two troubling features, that of non-locality and a subtle violation of Lorentz symmetry. Here, we trace the origin of both of these issues to a hidden freedom in the definition of duals of spinors and the associated field adjoints. In the process, for the first time, we provide a quantum theory of spin one-half fermions that is free from all the mentioned issues. The interactions of the new fermions are restricted to dimension-four quartic self interaction, and also to a dimension-four coupling with the Higgs. A generalised Yukawa coupling of the new fermions with neutrinos provides an hitherto unsuspected source of lepton-number violation. The new fermions thus present a first-principle dark matter partner to Dirac fermions of the standard model of high energy physics with contrasting mass dimensions—that of three halves for the latter versus one of the former without mutating the statistics from fermionic to bosonic. PubDate: 2017-03-30 DOI: 10.1007/s00006-017-0775-1

Authors:Cornelia-Livia Bejan; Şemsi Eken Meriç; Erol Kılıç Abstract: Abstract Clifford algebras are used in theoretical physics and in particular, in the general theory of relativity, where Einstein’s equations are rewritten in Girard (Adv Appl Clifford Algebras 9(2):225–230, 1999) within a Clifford algebra. Let M be a manifold with a torsion-free connection which induces on its cotangent bundle \(T^{*}M\) , a semi-Riemannian metric \(\bar{g}\) , called the natural Riemann extension, Kowalski and Sekizawa (Publ Math Debrecen 78:709–721, 2011). The main result of the present paper gives a necessary and sufficient condition for \(\bar{g}\) restricted to certain hypersurfaces of \(T^{*}M\) to be Einstein. PubDate: 2017-03-21 DOI: 10.1007/s00006-017-0774-2

Authors:Abhijit Banerjee Abstract: Abstract We investigate the complete analytical solutions of quantum mechanical harmonic and isotonic oscillators formulated in the commutative ring of bicomplex numbers. We obtain the explicit closed form expressions for the excited eigenstates and corresponding energy eigenvalues associated with the harmonic and isotonic oscillator potentials by extending the formalism adopted in Banerjee (Ann Phys 377:493, 2017) recently to find the analytical closed form solutions for ground states. PubDate: 2017-03-16 DOI: 10.1007/s00006-017-0772-4

Authors:Stéphane Breuils; Vincent Nozick; Laurent Fuchs Abstract: Abstract This paper presents an efficient implementation of geometric algebra, based on a recursive representation of the algebra elements using binary trees. The proposed approach consists in restructuring a state of the art recursive algorithm to handle parallel optimizations. The resulting algorithm is described for the outer product and the geometric product. The proposed implementation is usable for any dimensions, including high dimension (e.g. algebra of dimension 15). The method is compared with the main state of the art geometric algebra implementations, with a time complexity study as well as a practical benchmark. The tests show that our implementation is at least as fast as the main geometric algebra implementations. PubDate: 2017-03-16 DOI: 10.1007/s00006-017-0770-6

Authors:Hakan Simsek; Mustafa Özdemir Abstract: Abstract In this paper the generalization of the rotations on any lightcone in Minkowski 3-space \({\mathbb {R}}_{g}^{1,2}\) is given. The rotation motion on the lightcone is examined by means of a bilinear form and Lorentzian notions. We use the corresponding Rodrigues and Cayley formulas and benefit from the hyperbolic split quaternion product to obtain the corresponding rotation matrix. PubDate: 2017-03-16 DOI: 10.1007/s00006-017-0771-5

Authors:Ekaterina Pervova Abstract: Abstract We consider the diffeological pseudo-bundles of exterior algebras, and the Clifford action of the corresponding Clifford algebras, associated to a given finite-dimensional and locally trivial diffeological vector pseudo-bundle, as well as the behavior of the former three constructions (exterior algebra, Clifford action, Clifford algebra) under the diffeological gluing of pseudo-bundles. Despite these being our main object of interest, we dedicate significant attention to the issues of compatibility of pseudo-metrics, and the gluing-dual commutativity condition, that is, the condition ensuring that the dual of the result of gluing together two pseudo-bundles can equivalently be obtained by gluing together their duals, which is not automatic in the diffeological context. We show that, assuming that the dual of the gluing map, which itself does not have to be a diffeomorphism, on the total space is one, the commutativity condition is satisfied, via a natural map, which in addition turns out to be an isometry for the natural pseudo-metrics on the pseudo-bundles involved. PubDate: 2017-03-14 DOI: 10.1007/s00006-017-0769-z

Authors:Sergio Giardino Abstract: Abstract A quaternionic analog of the Aharonov–Bohm effect is developed without the usual anti-hermitian operators in quaternionic quantum mechanics. A quaternionic phase links the solutions obtained to ordinary complex wave functions, and new theoretical studies and experimental tests are possible for them. PubDate: 2017-03-07 DOI: 10.1007/s00006-017-0766-2

Authors:Waldyr Alves Rodrigues; Samuel A. Wainer Abstract: Abstract Using Clifford and Spin–Clifford formalisms we prove that the classical relativistic Hamilton Jacobi equation for a charged massive (and spinning) particle interacting with an external electromagnetic field is equivalent to Dirac–Hestenes equation satisfied by a class of spinor fields that we call classical spinor fields. These spinor fields are characterized by having the Takabayashi angle function constant (equal to 0 or \(\pi \) ). We also investigate a nonlinear Dirac–Hestenes like equation that comes from a class of generalized classical spinor fields. Finally, we show that a general Dirac–Hestenes equation (which is a representative in the Clifford bundle of the usual Dirac equation) gives a generalized Hamilton–Jacobi equation where the quantum potential satisfies a severe constraint and the “mass of the particle” becomes a variable. Our results can then eventually explain experimental discrepancies found between prediction for the de Broglie–Bohm theory and recent experiments. We briefly discuss de Broglie’s double solution theory in view of our results showing that it can be realized, at least in the case of spinning free particles.The paper contains several appendices where notation and proofs of some results of the text are presented. PubDate: 2017-03-07 DOI: 10.1007/s00006-017-0768-0

Authors:Mircea Martin Abstract: Abstract The goal of this article is to introduce a concept of Clifford structures on vector bundles as natural extensions of the standard complex and quaternionic structures, and to determine the derivations and linear connections on smooth Clifford vector bundles compatible with their Clifford structures. The basic object used to get such descriptions is an involution on the space of derivations of a Clifford vector bundle explicitly defined in terms of the specific Clifford structure. That involution is actually derived from an operation called the Clifford conjugation relative to a Clifford structure, which is defined in a purely algebraic setting as an involution on the space of derivations of a Euclidean Clifford algebra. Its definition essentially relies on the use and a complete description of the geometric concept of tangent Clifford structures of a Euclidean Clifford algebra. PubDate: 2017-03-03 DOI: 10.1007/s00006-017-0767-1