Advances in Applied Clifford Algebras最新文献

Motion polynomials are a specific type of polynomial over a Clifford algebra that can conveniently describe rational motions. There exists an algorithm for the factorization of motion polynomials that works in generic cases. It hinges on the invertibility of a certain coefficient occurring in the algorithm. If this coefficient is not invertible, factorizations may or may not exist. In the case of existence we call this an irregular factorization. We characterize quadratic motion polynomials with irregular factorizations in terms of algebraic equations and present examples whose number of unique factorizations range from one to infinitely many. For two special sub-cases we show the unique existence of such polynomials. In case of commuting factors we obtain the conformal Villarceau motion, in case of rigid body motions the circular translation.

This paper extends the concepts of weighted Drazin-star (WDS) and weighted star-Drazin (WSD) matrices to domain of quaternion matrices. We develop determinantal representations for these matrices, leveraging the theory of noncommutative row-column determinants, considering both general and Hermitian cases. As specific instances, we derive the determinantal representations of the complex WDS and WSD matrices by employing minors of appropriately constructed complex matrices. Furthermore, we investigate two-sided quaternion equations, along with one-sided particular types, where the unique solutions are expressed using WDS and WSD matrices. Explicit solutions for these quaternion matrix equations are obtained using Cramer-type methods. Finally, a numerical example is provided to confirm applicability and efficacy of our findings.

The Algebra of Physical Space (APS) is used to explore the Constructive Standard Model (CSM) of particle physics. Namely, this paper connects the spinor formalism of the APS to massive amplitudes in the CSM. A novel equivalency between traditional CSM and APS-CSM formalisms is introduced, called the Scattering Algebra (SA), with example calculations confirming the consistency of results between both frameworks. Through this all, two significant insights are revealed: The identification of traditional CSM spin spinors with Lorentz rotors in the APS, and the connection of the CSM to various formalisms through ray spinor structure. The CSM’s results are replicated in massive cases, showcasing the power of the index-free, matrix-free, coordinate-free, geometric approach and paving the way for future research into massless cases, amplitude-construction, and Wigner little group methods within the APS.

This paper explores the physics of magnetic and electric flux tubes supported by current vortices in condensed matter having a superconducting state in which bosonic charge carriers flow without resistance. The starting point is that the boson wave function satisfies the Klein–Gordon equation of relativistic quantum mechanics. Next, the electromagnetic fields within the superconducting medium are assumed to obey the quasistatic Maxwell equations expressed with geometric algebra and calculus and incorporating either electric or hypothetical magnetic currents. Finally, the Fundamental Theorem of Calculus is utilized in two forms to examine flux tubes, first in electric superconductors and then in hypothetical magnetic superconductors. Geometric algebra and calculus enable a consistent treatment of both analyses and an extension from three to four spatial dimensions.

This paper introduces Lie groups in degenerate geometric (Clifford) algebras that preserve four fundamental subspaces determined by the grade involution and reversion under the adjoint and twisted adjoint representations. We prove that these Lie groups can be equivalently defined using norm functions of multivectors applied in the theory of spin groups. We also study the corresponding Lie algebras. Some of these Lie groups and algebras are closely related to Heisenberg Lie groups and algebras. The introduced groups are interesting for various applications in physics and computer science, in particular, for constructing equivariant neural networks.

The Rarita–Schwinger fields are solutions to the relativistic field equation of spin-3/2 fermions in four dimensional flat spacetime, which are important in supergravity and superstring theories. Bureš et al. generalized it to an arbitrary spin k/2 in 2002 in the context of Clifford algebras. In this article, we introduce a mean value property, a Cauchy’s estimates, and a Liouville’s theorem for null solutions to the Rarita–Schwinger operator in the Euclidean spaces. Further, we investigate boundednesses to the Teodorescu transform and its derivatives. This gives rise to a Hodge decomposition of an (L^2) space in terms of the kernel of the Rarita–Schwinger operator and it also generalizes Bergman spaces to the higher spin cases.