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Is there more to Dirac's gamma matrices than meets the eye? It turns out that gamma zero can be factorized into a product of three operators. This revelation facilitates the expansion of Dirac's space-time algebra to Clifford algebra Cl (0 , 6). The resultant rich geometric structure can be leveraged to establish a combined framework of the standard model and gravity, wherein a gravi-weak interaction between the vierbein field and the extended weak gauge field is allowed. Inspired by the composite Higgs model, we examine the vierbein field as an effective description of the fermion–antifermion condensation. The compositeness of space-time manifests itself at an energy scale which is different from the Planck scale. We propose that all the regular classical Lagrangian terms are of quantum condensation origin, thus possibly addressing the cosmological constant problem provided that we exercise extreme caution in the renormalization procedure that entails multiplications of divergent integrals. The Clifford algebra approach also permits a weaker form of charge conjugation without particle–antiparticle interchange, which leads to a Majorana-type mass that conserves lepton number. Additionally, in the context of spontaneous breaking of two global U (1) symmetries, we explore a three-Higgs-doublet model which could explain the fermion mass hierarchies. [ABSTRACT FROM AUTHOR]

Different ways of representing the group SU(3)$$ SU(3) $$ within a Geometric Algebra approach are explored. As part of this, we consider characteristic multivectors for SU(3)$$ SU(3) $$ and how these are linked with decomposition of generators into commuting bivectors. The setting for this work is within a 6d Euclidean Clifford Algebra. We then go on to consider whether the fundamental forces of particle physics might arise from symmetry considerations in just the 4d geometric algebra of spacetime—the STA. As part of this, a representation of SU(3)$$ SU(3) $$ is found wholly within the STA, involving preservation of a bivector norm. We also show how Octonions can be fully represented within the Spacetime Algebra, which we believe will be useful in making them understandable and accessible to a new community in Physics and Engineering. The two strands of the paper are drawn together in showing how preserving the octonion norm is the same as preserving the timelike part of the Dirac current of a particle. This suggests a new model for the symmetries preserved in particle physics. Following on from work by Günaydin and Gürsey on the link between quarks, and octonions, and by Furey on chains of octonionic multiplications, we show how both of these fit well within our scheme and give some wholly STA versions of the operations involved, which in the cases considered have easily understandable equivalents in terms of 4d geometry. Links with larger groups containing SU(3)$$ SU(3) $$, such as G2$$ {G}_2 $$ and SU(8)$$ SU(8) $$, are also considered. [ABSTRACT FROM AUTHOR]

An algorithm for finding a control of a linear switched system by means of Geometric Algebra is designed. More precisely, we develop a switching path searching algorithm for a two‐dimensional linear dynamical switched system with a non‐singular matrix whose integral curves are formed by two sets of centralized ellipses. It is natural to represent them as elements of Geometric Algebra for Conics and construct the switching path by calculating switching points, i.e., intersections and contact points. For this, we use symbolic algebra operations or, more precisely, the wedge and inner products that are realizable by sums of products in the coordinate form. Therefore, no numerical solver to the system of equations is needed. Indeed, the only operation that may bring in an inaccuracy is vector normalization, i.e., square root calculation. The resulting switching path is formed by pieces of ellipses that are chosen, respectively, from the two sets of integral curves. The switching points are either intersections in the first or final step of our algorithm, or contact points. This choice guarantees the optimality of the switching path with respect to the number of switches. Two examples are provided to demonstrate the search for the intersections of the conics and, consequently, a description is presented of the construction of a switching path in both cases. [ABSTRACT FROM AUTHOR]

This paper presents some new Lie groups preserving fixed subspaces of geometric algebras (or Clifford algebras) under the twisted adjoint representation. We consider the cases of subspaces of fixed grades and subspaces determined by the grade involution and the reversion. Some of the considered Lie groups can be interpreted as generalizations of Lipschitz groups and spin groups. The Lipschitz groups and the spin groups are subgroups of these Lie groups and coincide with them in the cases of small dimensions. We study the corresponding Lie algebras. [ABSTRACT FROM AUTHOR]

We develop the method of averaging in Clifford (geometric) algebras suggested by the author in previous papers. We consider operators constructed using two different sets of anticommuting elements of real or complexified Clifford algebras. These operators generalize Reynolds operators from the representation theory of finite groups. We prove a number of new properties of these operators. Using the generalized Reynolds operators, we give a complete proof of the generalization of Pauli's theorem to the case of Clifford algebras of arbitrary dimension. The results can be used in geometry, physics, engineering, computer science, and other applications. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a generalization of Vieta theorem (Vieta's formulas) to the case of Clifford geometric algebras. We compare the generalized Vieta formulas with the ordinary Vieta formulas for characteristic polynomial containing eigenvalues. We discuss Gelfand–Retakh noncommutative Vieta theorem and use it for the case of geometric algebras of small dimensions. We introduce the notion of a simple basis‐free formula for a determinant in geometric algebra and prove that a formula of this type exists in the case of arbitrary dimension. Using this notion, we present and prove generalized Vieta theorem in geometric algebra of arbitrary dimension. The results can be used in symbolic computation and various applications of geometric algebras in computer science, computer graphics, computer vision, physics, and engineering. [ABSTRACT FROM AUTHOR]

In practical engineering, the use of Pauli algebra can provide a computational advantage, transforming conventional vector algebra to straightforward matrix manipulations. In this work, the Pauli matrices in cylindrical and spherical coordinates are reported for the first time and their use for representing a three-dimensional vector is discussed. This method leads to a unified representation for 3D multivectors with Pauli algebra. A significant advantage is that this approach provides a representation independent of the coordinate system, which does not exist in the conventional vector perspective. Additionally, the Pauli matrix representations of the nabla operator in the different coordinate systems are derived and discussed. Finally, an example on the radiation from a dipole is given to illustrate the advantages of the methodology. [ABSTRACT FROM AUTHOR]

Several of the most fundamental transformations widely used by the power engineering community are strongly based on geometrical considerations. Clifford's geometric algebra (GA) is the natural language for describing concepts in Euclidean geometry. In this work, we show how Clarke, Park, and Depenbrock's FBD transformations can be derived by imposing orthogonality on the voltage and current vectors defined in a Euclidean space by using GA. This paper presents these transformations as spatial‐like rotations and projections by means of the use of special algebraic objects named GA rotors. We prove that there is no need to use complex numbers nor matrices to perform the mentioned transformations and to provide geometrical intuition for electrical quantities and their transformations. Furthermore, power properties can be described using GA terminology by means of the proposed geometric power. The manipulation of the geometric power allows a useful current decomposition for a variety of applications such active filtering, frequency estimation, or electrical machinery. We provide in this paper an alternative approach focused on power systems under the paradigm of GA. [ABSTRACT FROM AUTHOR]

In this paper, power flows in electrical circuits are modelled in a mixed time-frequency domain by using geometric algebra and the Hilbert transform for the first time. The use of this mathematical framework overcomes some of the limitations of some of the existing methodologies, in which the so-called "active current" may not lead to the lowest Root Mean Square (RMS) current under distorted supply or unbalanced load. Moreover, this current may contain higher levels of harmonic distortion compared to the supply voltage. The proposed method can be used for sinusoidal and non-sinusoidal power supplies, non-linear loads and single- and multi-phase electrical circuits, and it provides meaningful engineering results with a compact formulation. It can also serve as an advanced tool for developing algorithms in the power electronics field. Several examples have been included to verify the validity of the proposed theory. [ABSTRACT FROM AUTHOR]

Inspired by Laughlin's theory of the fractional quantum Hall effect, we explore the topological nature of the quark–gluon plasma (QGP) and the nucleons in the context of the Clifford algebra. In our model, each quark is transformed into a composite particle via the simultaneous attachment of a spin monopole and an isospin monopole. This is induced by a novel kind of meson endowed with both spin and isospin degrees of freedom. The interactions in the strongly coupled quark–gluon system are governed by the topological winding number of the monopoles, which is an odd integer to ensure that the overall wave function is antisymmetric. The states of the QGP and the nucleons are thus uniquely determined by the combination of the monopole winding number m and the total quark number N. The radius squared of the QGP droplet is expected to be proportional to m N. We anticipate the observation of such proportionality in the heavy ion collision experiments. [ABSTRACT FROM AUTHOR]

This research paper focuses on quaternion neural networks (QNNs) - a type of neural network wherein the weights, biases, and input values are all represented as quaternion numbers. Previous studies have shown that QNNs outperform real-valued neural networks in basic tasks and have potential in high-dimensional problem spaces. However, research on QNNs has been fragmented, with contributions from different mathematical and engineering domains leading to unintentional overlap in QNN literature. This work aims to unify existing research by evaluating four distinct QNN backpropagation algorithms, including the novel GHR-calculus backpropagation algorithm, and providing concise, scalable implementations of each algorithm using a modern compiled programming language. Additionally, the authors apply a robust Design of Experiments (DoE) methodology to compare the accuracy and runtime of each algorithm. The experiments demonstrate that the Clifford Multilayer Perceptron (CMLP) learning algorithm results in statistically significant improvements in network test set accuracy while maintaining comparable runtime performance to the other three algorithms in four distinct regression tasks. By unifying existing research and comparing different QNN training algorithms, this work develops a state-of-the-art baseline and provides important insights into the potential of QNNs for solving high-dimensional problems. • Clifford Multilayer Perceptrons outperform other quaternion neural network methods. • Multilayer GHR Calc. networks approximate real nonlinear functions reasonably well. • Classic QNN backpropagation techniques are outdated and should be avoided. [ABSTRACT FROM AUTHOR]

Directional solvent extraction (DSE) has been gaining interest as a water treatment technology in recent years. DSE utilizes the process of micellization for the purposes of species separation between water and complex chemical systems. In this article, we develop a conformal geometric algebra‐based formulation that models surfactants, their solubilities, and critical micelle concentration (CMC), with relation to temperature and pressure. Molecules are represented as spatially distributed networks embedded in R4,1 space, and the mathematical characterizations of these molecules are shown to be effective in modelling CMC as a function of temperature and pressure. One of the contributions of this work is the utilization of this formulation to develop a governing expression, in the form of a three‐dimensional relationship, between CMC, pressure, and temperature for a general surfactant. In prior works, the CMC–temperature plane and CMC–pressure plane expressions have been extensively documented for sodium alkyl sulfates. In this work, we extend the formulation to model the CMC of decanoic acid, sodium octyl sulfate, sodium decyl sulfate, sodium dodecyl sulfate, and sodium tetradecyl sulfate. Using this theoretical model, a relationship between CMC and the directional solubility of water in a surfactant is determined. Directional solubility is related to temperature and pressure, and on this basis, we devise a directional solubility–pressure–temperature expression for an arbitrary surfactant to improve the state of the art for DSE. From this expression, we propose a novel isothermal DSE process for water treatment. [ABSTRACT FROM AUTHOR]

We present a formulation of the Boolean Satisfiability Problem in spinor language that allows to give a necessary and sufficient condition for unsatisfiability. With this result we outline an algorithm to test for unsatisfiability with possibly interesting theoretical properties. [ABSTRACT FROM AUTHOR]

Majorana spinors are constructed in terms of the multivectors of relativistic Cl1,3 algebra. Such spinors are found to be multiplied by primitive idempotents which drastically change spinor properties. Running electronic waves are used to solve the real Dirac-Majorana equation transformed to Cl1,3 algebra. From the analysis of the solution it is concluded that free Majorana particles do not exist, because relativistic Cl algebra requires the massive Majorana particle to move with light velocity. [ABSTRACT FROM AUTHOR]

T. Qian As it will turn out in this paper, the recent hype about most of the Clifford-Fourier transforms is not thoroughly worth the pain. Almost everyone that has a real application is separable, and these transforms can be decomposed into a sum of real valued transforms with constant multivecor factors. This fact makes their interpretation, their analysis, and their implementation almost trivial. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

To analyze the electromagnetic wave propagation in a medium the Maxwell equations should be supplemented by constitutive relations. At present the classification of linear constitutive relations is well established in tensorial-matrix and exterior p -form calculus. Here the constitutive relations are found in the context of Clifford geometric algebra. For this purpose Cl 1,3 algebra that conforms with relativistic 4D Minkowskian spacetime is used. It is shown that the classification of linear optical phenomena with the help of constitutive relations in this case comes from the structure of Cl 1,3 algebra itself. Concrete expressions for constitutive relations which follow from this algebra are presented. They can be applied in calculating the propagation properties of electromagnetic waves in any anisotropic, linear and nondissipative medium. [ABSTRACT FROM AUTHOR]

To have a closed system, the Maxwell electromagnetic equations should be supplemented by constitutive relations which describe medium properties and connect primary fields (E, B) with secondary ones (D, H). J.W. Gibbs and O. Heaviside introduced the basis vectors {i, j, k} to represent the fields and constitutive relations in the three-dimensional vectorial space. In this paper the constitutive relations are presented in a form of Cl3,0 algebra which describes the vector space by three basis vectors {σ1, σ2, σ3} that satisfy Pauli commutation relations. It is shown that the classification of electromagnetic wave propagation phenomena with the help of constitutive relations in this case comes from the structure of Cl3,0 itself. Concrete expressions for classical constitutive relations are presented including electromagnetic wave propagation in a moving dielectric. [ABSTRACT FROM AUTHOR]

The known Complex Step Derivative (CSD) method allows easy and accurate differentiation up to machine precision of real analytic functions by evaluating them with a small imaginary step next to the real number line. The current paper proposes that derivatives of holomorphic functions can be calculated in a similar fashion by taking a small step in a quaternionic direction instead. It is demonstrated that in so doing the CSD properties of high accuracy and convergence are carried over to derivatives of holomorphic functions. To demonstrate the ease of implementation, numerical experiments were performed using complex quaternions, the geometric algebra of space, and a 2 × 2 matrix representation thereof. [ABSTRACT FROM AUTHOR]

Abstract: Connection between optical Mueller matrices and geometrical (Clifford) algebra multivectors is established. It is shown that starting from 3-dimensional (3D) Cl 3,0 algebra and using isomorphism between Cl 3,0 and even Cl 3,1 + subalgebra one can generate canonical Mueller matrices and their combinations that describe an optical system. It appears that representation of polarization devices in terms of geometric algebra is very compact and, in contrast to Mueller matrix approach, there is no need for speculative physical restrictions. If needed, properties of media can be logically introduced into Maxwell equation in a form of Clifford algebra via constitutive relations. Since representation of polarization by Cl 3,1 algebra is Lorentz invariant it allows to include relativistic effects of moving bodies on light polarization as well. In this paper only simple examples of connection between Mueller matrices and geometric algebra multivectors is presented. [Copyright &y& Elsevier]

Abstract: Matrix multiplication was first introduced by Arthur Cayley in 1855 in agreement with the composition of linear transformations. We explore an underlying geometric framework in which matrix multiplication naturally arises from the product of numbers in a geometric (Clifford) algebra. Consequently, all invariants of a linear operator become geometric invariants of the multivectors that they represent. Two different kinds of bases for matrices emerge, a spectral basis of idempotents and nilpotents, and a standard basis of scalars, vectors, bivectors, and higher order k-vectors. The Kronecker product of matrices naturally arises when considering the block structure of a matrix. Conformal geometry of is expressed in terms of the concept of an h-twistor, which is a generalization of a Penrose twistor. [Copyright &y& Elsevier]

A general formula for inversion in a relativistic Clifford–Dirac algebra has been derived. Identifying the base elements of the algebra as those of space and time, the first order differential equations over all quantities proves to encompass the Maxwell equations, leads to a natural extension incorporating rest mass and spin, and allows an integration with relativistic quantum mechanics. Although the algebra is not a division algebra, it parallels reality well: where division is undefined turns out to correspond to physical limits, such as that of the light cone. The divisor corresponds to invariants of dynamical significance, such as the invariant interval, the general invariant quantities in electromagnetism, and the basis set of quantities in the Dirac equation. It is speculated that the apparent 3-dimensionality of nature arises from a beautiful symmetry between the three-vector algebra and each of four sets of three derived spaces in the full 4-dimensional algebra. It is conjectured that elements of inversion may play a role in the interaction of fields and matter. [ABSTRACT FROM AUTHOR]

A theory in which four-dimensional space–time is generalized to a larger space, namely a 16-dimensional Clifford space (C-space) is investigated. Curved Clifford space can provide a realization of Kaluza–Klein. A covariant Dirac equation in curved C-space is explored. The generalized Dirac field is assumed to be a polyvector-valued object (a Clifford number) which can be written as a superposition of four independent spinors, each spanning a different left ideal of Clifford algebra. The general transformations of a polyvector can act from the left and/or from the right, and form a large gauge group which may contain the group U(1) × SU(2) × SU(3) of the standard model. The generalized spin connection in C-space has the properties of Yang–Mills gauge fields. It contains the ordinary spin connection related to gravity (with torsion), and extra parts describing additional interactions, including those described by the antisymmetric Kalb–Ramond fields. [ABSTRACT FROM AUTHOR]

Of the various formalisms developed to treat relativistic phenomena, those based on Clifford's geometric algebra are especially well adapted for clear geometric interpretations and computational efficiency. Here we study relationships between formulations of special relativity in the spacetime algebra (STA)Cl1,3 of the underlying Minkowski vector space, and in the algebra of physical space (APS)Cl3. STA lends itself to an absolute formulation of relativity, in which paths, fields, and other physical properties have observer-independent representations. Descriptions in APS are related by a one-to-one mapping of elements from APS to the even subalgebra STA+ of STA. With this mapping, reversion in APS corresponds to hermitian conjugation in STA. The elements of STA+ are all that is needed to calculate physically measurable quantities (calledmeasurables) because only they entail the observer dependence inherent in any physical measurement. As a consequence, every relativistic physical process that can be modeled in STA also has a representation in APS, andvice versa. In the presence of two or more inertial observers, two versions of APS present themselves. In theabsoluteversion, both the mapping to STA+ and hermitian conjugation are observer dependent, and the proper basis vectors of any observer are persistent vectors that sweep out time-like planes in spacetime. To compare measurements by different inertial observers in APS, we express them in the proper algebraic basis of a single observer. This leads to therelativeversion of APS, which can be related to STA by assigning every inertial observer in STA to a single absolute frame in STA. The equivalence of inertial observers makes this permissible. The mapping and hermitian conjugation are then the same for all observers. Relative APS gives a covariant representation of relativistic physics with spacetime multivectors represented by multiparavectors in APS. We relate the two versions of APS as consistent models within the same algebra. [ABSTRACT FROM AUTHOR]

Traditional electrical power theories and one of their most important concepts—apparent power—are still a source of debate, because they present several flaws that misinterpret the power-transfer and energy-balance phenomena under distorted grid conditions. In recent years, advanced mathematical tools such as geometric algebra (GA) have been introduced to address these issues. However, the application of GA to electrical circuits requires more consensus, improvements and refinement. In this paper, electrical power theories for single-phase systems based on GA were revisited. Several drawbacks and inconsistencies of previous works were identified, and some amendments were introduced. An alternative expression is presented for the electric power in the geometric domain. Its norm is compatible with the traditional apparent power defined as the product of the RMS voltage and current. The use of this expression simplifies calculations such as those required for current decomposition. This proposal is valid even for distorted currents and voltages. Concepts are presented in a simple way so that a strong background on GA is not required. The paper included some examples and experimental results in which measurements from a utility supply were analysed. [ABSTRACT FROM AUTHOR]

This paper presents a new framework based on geometric algebra (GA) to solve and analyse three-phase balanced electrical circuits under sinusoidal and non-sinusoidal conditions. The proposed approach is an exploratory application of the geometric algebra power theory (GAPoT) to multiple-phase systems. A definition of geometric apparent power for three-phase systems, that complies with the energy conservation principle, is also introduced. Power calculations are performed in a multi-dimensional Euclidean space where cross effects between voltage and current harmonics are taken into consideration. By using the proposed framework, the current can be easily geometrically decomposed into active- and non-active components for current compensation purposes. The paper includes detailed examples in which electrical circuits are solved and the results are analysed. This work is a first step towards a more advanced polyphase proposal that can be applied to systems under real operation conditions, where unbalance and asymmetry is considered. [ABSTRACT FROM AUTHOR]

• Upgraded geometric algebra power theory is presented for single phase circuits. • Geometric Power is redefined as the product of voltage and current reverse. • Several additions and modifications are introduced to reflect interharmonics and active current based on Fryze proposal. • This new method is validated through some real examples with linear and non-linear loads. The aim of this work is to present major upgrades to existing power theories based on geometric algebra for single-phase circuits in the frequency domain. It also embodies an interesting new approach with respect to traditionally accepted power theories, revisiting power concepts in both sinusoidal and non-sinusoidal systems with linear and nonlinear loads for a proper identification of its components to achieve passive compensation of true non-active current. Moreover, it outlines traditional power theories based on the apparent power S and confirms that these should definitively be reconsidered. It is evidenced that traditional proposals based on the concepts of Budeanu, Fryze and others fail to identify the interactions between voltage and current harmonics. Based on the initial work of Castro-Núñez and others, new aspects not previously included are detailed, modified and reformulated. As a result, it is now possible to analyze non sinusoidal electrical circuits, establishing power balances that comply with the principle of energy conservation, and achieving optimal compensation scenarios with both passive and active elements in linear and non-linear loads. [ABSTRACT FROM AUTHOR]

• Geometric algebra framework is applied to solve non sinusoidal and nonlinear circuits. • A new definition of non-active power based on geometric algebra is presented. • New current decomposition is introduced to minimize line looses and supply active power. • This new method is validated through some examples with linear and non-linear loads. A new approach for the definition of non-active power in electrical systems is presented in this paper. Through the use of geometric algebra, it is possible to define a new term called geometric non-active power, which is applicable to both sinusoidal and non-sinusoidal systems, and to both linear and nonlinear loads. The classic definitions of distortion and reactive power are compared and discussed in our proposal. We verify how geometric non-active power can appear in both purely resistive and purely reactive systems. The superiority of geometric algebra is revealed through several examples of electrical circuits previously analysed in specialised literature. Furthermore, a new geometrical current decomposition is proposed, for the first time, to provide a greater physical sense to existing geometric power. The results obtained confirm that classic concepts based on apparent power S are based on a lack of physical meaning, which is why geometric algebra theory should be adopted instead. [ABSTRACT FROM AUTHOR]

In 2007, and in a series of later papers, Joy Christian claimed to refute Bell's theorem, presenting an alleged local realistic model of the singlet correlations using techniques from geometric algebra (GA). Several authors published papers refuting his claims, and Christian's ideas did not gain acceptance. However, he recently succeeded in publishing yet more ambitious and complex versions of his theory in fairly mainstream journals. How could this be? The mathematics and logic of Bell's theorem is simple and transparent and has been intensely studied and debated for over 50 years. Christian claims to have a mathematical counterexample to a purely mathematical theorem. Each new version of Christian's model used new devices to circumvent Bell's theorem or depended on a new way to misunderstand Bell's work. These devices and misinterpretations are in common use by other Bell critics, so it useful to identify and name them. I hope that this paper can serve as a useful resource to those who need to evaluate new "disproofs of Bell's theorem". Christian's fundamental idea is simple and quite original: he gives a probabilistic interpretation of the fundamental GA equation a · b = (a b + b a) / 2 . After that, ambiguous notation and technical complexity allows sign errors to be hidden from sight, and new mathematical errors can be introduced. [ABSTRACT FROM AUTHOR]

New-generation power networks, such as microgrids, are being affected by the proliferation of nonlinear electronic systems, resulting in harmonic disturbances both in voltage and current that affect the symmetry of the system. This paper presents a method based on the application of geometric algebra (GA) to the resolution of power flow in nonsinusoidal single-phase electrical systems for the correct determination of its components to achieve passive compensation of true quadrature current. It is demonstrated that traditional techniques based on the concepts of Budeanu, Fryze or IEEE1459 fail to determine the interaction between voltage and current and therefore, are not suitable for being used as a basis for the compensation of nonactive power components. An example is included that demonstrates the superiority of GA method and is compared to previous work where GA approaches and traditional methods have also been used. [ABSTRACT FROM AUTHOR]