Search Results (65)

We develop a comparative synthesis of quantum-information geometry beyond complex numbers, with emphasis on what different algebraic frameworks contribute to information-processing structure rather than on their formal novelty alone. The organizing idea is a layer-by-layer test of the standard complex Hilbert-space formalism: each non-complex or deformed framework modifies the scalar field, phase group, projective state space, Born-probability semantics, composition rule, measurement geometry, symmetry algebra or representation category. The central thesis is that such frameworks are physically meaningful when they identify which assumptions make complex quantum mechanics operationally stable: positive probabilities, associative multipartite composition, reversible dynamics, experimentally testable phases, locality constraints, informationally complete measurements, error bases and clear operational semantics. Real quantum theory probes the necessity of complex phases and local tomography; quaternionic quantum mechanics probes non-Abelian phase while retaining associativity and admitting complex embeddings; octonionic proposals probe the boundary where exceptional geometry survives but generic circuit composition is obstructed by non-associativity; Jordan algebras test ordered probabilistic state spaces; Clifford algebras and Bott periodicity provide the spinorial and topological grammar connecting gates, Hopf maps and periodic dimensions; and quantum-group or q-deformed constructions probe coproducts, braiding and representation categories rather than scalar amplitudes. We distinguish three roles that are often conflated: genuine hypercomplex kinematics, Hopf-fibration coordinates for ordinary complex multipartite entanglement, and deformed algebraic or categorical structures. The resulting map separates established equivalence and experimental-constraint results from useful representation tools and speculative programs, while identifying concrete open problems for non-complex quantum information. Full article

In this paper, we present the spinor structure associated with the Hyperbolic Clifford algebra of a real n-dimensional vector space V, which is denoted by $\mathcal{C}l\left(H_V\right)$. Unlike the standard Clifford algebra, the Hyperbolic Clifford algebra $\mathcal{C}l\left(H_V\right)$ simultaneously accommodates both multiforms and multivectors in a single algebraic structure, making it the natural framework—known as the “mother algebra”—for the study of superfields in theoretical physics and for generalizing the Clifford bundle formalism to hyperbolic structures arising in gravitational theories. The orthogonal groups and orthogonal transformations associated to the hyperbolic space $H_V$ are presented. The Clifford–Lipschitz group and the Pin and Spin groups associated with $\mathcal{C}l\left(H_V\right)$ are defined. Then, the frame bundle and spinor structure associated to Hyperbolic Clifford algebra is derived. Full article

The transition of the mining industry towards Industry 5.0 demands predictive models capable of strictly adhering to physical laws and modeling complex, non-Euclidean geometries—capabilities often lacking in standard graph neural networks. This systematic review, conducted under the PRISMA 2020 protocol, analyzes the emergence of “Era 5” architectures by synthesizing 96 high-impact studies from 2019 to 2026, focusing on Clifford (geometric algebra) GNNs, simplicial and cell complex neural networks, symplectic/Hamiltonian GNNs, and generative flow networks (GFlowNets). The analysis demonstrates that Clifford architectures provide superior rotational equivariance for robotic control; Simplicial networks capture high-order topological interactions critical for geomechanics; Symplectic GNNs ensure energy conservation for stable long-term simulation of structural dynamics; and GFlowNets offer a novel paradigm for generative mine planning. We conclude that shifting from data-driven approximations to these mathematically rigorous, structure-preserving architectures is fundamental for developing reliable, physics-informed digital twins that optimize structural integrity and operational efficiency in complex industrial environments. Full article

An algorithm is presented to extract the square root from a multivector (MV) in real Clifford algebras ${\mathit{Cl}}_{p,q}$, where $n=p+q\le 3$, in radicals. It is shown that in ${\mathit{Cl}}_{3,0}$, ${\mathit{Cl}}_{1,2}$, and ${\mathit{Cl}}_{0,3}$ algebras, there are up to four isolated square roots in a case of the most general (generic) MV. The algebra ${\mathit{Cl}}_{2,1}$ is an exception and, there, the MV can have up to 16 isolated roots. In addition, a continuum of roots has been found in all Clifford algebras except $p+q=1$. Examples which clarify computations are provided to illustrate the properties of roots in all $n=3$ algebras. The results may be useful in solving nonlinear equations, like for example, the Clifford–Riccati equation. Full article

The Feynman path integral plays a central role in quantum mechanics, linking classical action to propagators and relating quantum electrodynamics (QED) to Feynman diagrams. However, the path-integral formulations used in non-relativistic quantum mechanics and in QED are neither unified nor directly connected. This suggests the existence of a missing path integral that bridges relativistic action and the Dirac equation at the single-particle level. In this work, we analyze the consistency and completeness of existing path-integral theories and identify a spinor path integral that fills this gap. Starting from a relativistic action written in spinor form, we construct a spacetime path integral whose kernel reproduces the Dirac Hamiltonian. The resulting formulation provides a direct link between the relativistic classical action and the Dirac equation, and it naturally extends the scalar relativistic path integral developed in our earlier work. Beyond establishing this structural connection, the spinor path integral offers a new way to interpret the origin of classical mechanics for the Dirac equation and suggests a spacetime mechanism for spin and quantum nonlocal correlations. These features indicate that the spinor path integral can serve as a unifying framework for existing path-integral approaches and as a starting point for further investigations into the spacetime structure of quantum mechanics. Full article

Motivation. Analytic function theory on commutative complex extensions calls for an operator–theoretic calculus that simultaneously sees the algebra-induced coupling among components and supports boundary-to-interior mechanisms. Gap. While Dirac-type frameworks are classical in several complex variables and Clifford analysis, a coherent calculus aligning structural CR systems, a canonical first derivative, and a Cauchy-type boundary identity on the commutative model $\mathbb{C}\left(\mathbb{C}\right)\cong {\mathbb{C}}^4$ has not been systematically developed. Purpose and Aims. This paper develops such a calculus for $\mathcal{O}$-regular mappings on $\mathbb{C}\left(\mathbb{C}\right)$ and establishes three pillars of the theory. Main Results. (i) A fully coupled Cauchy–Riemann system characterizing $\mathcal{O}$-regularity; (ii) identification of a canonical first derivative $g^{\prime }\left(z\right)={\partial }_{x_0}g\left(z\right)$; and (iii) a Stokes-driven boundary annihilation law ${\int }_{\partial \mathrm{\Omega }}\tau g=0$ for a canonical 7-form $\tau$. On (pseudo)convex domains, $\overline{\partial }$-methods yield solvability under natural compatibility and regularity assumptions. Stability (under algebra-preserving maps), Liouville-type, and removability results are also obtained, and function spaces suited to this algebra are outlined. Significance. The results show that a large portion of the classical holomorphic toolkit survives, in algebra-aware form, on $\mathbb{C}\left(\mathbb{C}\right)$. Full article

We develop a rigorous algebraic–analytic framework for multidual complex numbers ${\mathbb{DC}}_n$ within the setting of Clifford analysis and establish a comprehensive theory of hyperholomorphic multidual complex-valued functions. Our main contributions are (i) a fully coupled multidual Cauchy–Riemann system derived from the Dirac operator, yielding precise differentiability criteria; (ii) generalized conjugation laws and the associated norms that clarify metric and geometric structure; and (iii) explicit operator and kernel constructions—including generalized Cauchy kernels and Borel–Pompeiu-type formulas—that produce new representation theorems and regularity results. We further provide matrix–exponential and functional calculus representations tailored to ${\mathbb{DC}}_n$, which unify algebraic and analytic viewpoints and facilitate computation. The theory is illustrated through a portfolio of examples (polynomials, rational maps on invertible sets, exponentials, and compositions) and a solvable multidual boundary value problem. Connections to applications are made explicit via higher-order automatic differentiation (using nilpotent infinitesimals) and links to kinematics and screw theory, highlighting how multidual analysis expands classical holomorphic paradigms to richer, nilpotent-augmented coordinate systems. Our results refine and extend prior work on dual/multidual numbers and situate multidual hyperholomorphicity within modern Clifford analysis. We close with a concise summary of notation and a set of concrete open problems to guide further development. Full article

The Clifford–Weyl algebra ${𝒜}_q^{\pm }$, as a class of solvable polynomial algebras, combines the anti-commutation relations of Clifford algebras ${𝒜}_q^{+}$ with the differential operator structure of Weyl algebras ${𝒜}_q^{-}$. It exhibits rich algebraic and geometric properties. This paper employs Gröbner–Shirshov basis principles in concert with Poincaré–Birkhoff–Witt (PBW) basis methodology to delineate the iterated skew polynomial structures within ${𝒜}_q^{+}\mathrm{and}{𝒜}_q^{-}$. By constructing explicit PBW generators, we analyze the structural properties of both algebras and their modules using constructive methods. Furthermore, we prove that ${𝒜}_q^{+}\mathrm{and}{𝒜}_q^{-}$ are Auslander regular, Cohen–Macaulay, and Artin–Schelter regular. These results provide new tools for the representation theory in noncommutative geometry. Full article

This work addresses a significant gap in the existing literature by developing integral representation formulas for extended quaternion-valued functions within the framework of Clifford analysis. While classical Cauchy-type and Borel–Pompeiu formulas are well established for complex and standard quaternionic settings, there is a lack of analogous tools for functions taking values in extended quaternion algebras such as split quaternions and biquaternions. The motivation is to extend the analytical power of Clifford analysis to these broader algebraic structures, enabling the study of more complex hypercomplex systems. The objectives are as follows: (i) to construct new Cauchy-type integral formulas adapted to extended quaternionic function spaces; (ii) to identify explicit kernel functions compatible with Clifford-algebra-valued integrands; and (iii) to demonstrate the application of these formulas to boundary value problems and potential theory. The proposed framework unifies quaternionic function theory and Clifford analysis, offering a robust analytic foundation for tackling higher-dimensional and anisotropic partial differential equations. The results not only enhance theoretical understanding but also open avenues for practical applications in mathematical physics and engineering. Full article

Parkinson’s disease (PD) is a chronic neurodegenerative disorder that progressively impairs motor functions. Clinical assessments have traditionally relied on rating scales such as the Movement Disorder Society Unified Parkinson Disease Rating Scale (MDS-UPDRS); however, these evaluations are susceptible to rater-dependent variability and may miss subtle motor changes. This study explored objective and quantitative methods for assessing motor function in PD patients using the Quantum Metaglove, a sensory glove produced by MANUS^®, which was used to record finger movements during three tasks: finger tapping, hand gripping, and pronation–supination. Classic and geometric motor features (the latter based on Clifford algebra, an advanced approach for trajectory shape analysis) were extracted. The resulting data were used to train various machine learning algorithms (k-NN, SVM, and Naive Bayes) to distinguish healthy subjects from PD patients. The integration of traditional kinematic and geometric approaches improves objective hand movement analysis, providing new diagnostic opportunities. In particular, geometric trajectory analysis provides more interpretable information than conventional signal processing methods. This study highlights the value of wearable technologies and Clifford algebra-based algorithms as tools that can complement clinical assessment. They are capable of reducing inter-rater variability and enabling more continuous and precise monitoring of hand motor movements in patients with PD. Full article

Elasticity is a well-established field within mathematical physics, yet new formulations can provide deeper insight and computational advantages. This study explores the geometry of two- and three-dimensional elastic curves using the formalism of geometric algebra, offering a unified and coordinate-free approach. This work systematically derives the Frenet, Darboux, and Bishop frames within the three-dimensional geometric algebra and employs them to integrate the elastica equation. A concise Lagrangian formulation of the problem is introduced, enabling the identification of Noetherian, conserved, multi-vector moments associated with the elastic system. A particularly compact form of the elastica equation emerges when expressed in the Bishop frame, revealing structural simplifications and making the equations more amenable to analysis. Ultimately, the geometric algebra perspective uncovers a natural correspondence between the theory of free elastic curves and classical beam models, showing how constrained theories, such as Euler–Bernoulli and Kirchhoff beam formulations, arise as special cases. These results not only clarify foundational aspects of elasticity theory but also provide a framework for future applications in continuum mechanics and geometric modeling. Full article

The unital associative algebra structure $\mathbb{A}$ on ${\mathbb{R}}^n$ allows for defining elementary functions and functions defined by convergent power series. For these, the usual derivative has a simple form even for higher-order derivatives, which allows us to have the $\mathbb{A}$-calculus. Thus, we introduce $\mathbb{A}$-differentiability. Rules for $\mathbb{A}$-differentiation are obtained: a product rule, left and right quotients, and a chain rule. Convergent power series are $\mathbb{A}$-differentiable, and their $\mathbb{A}$-derivatives are the power series defined by their $\mathbb{A}$-derivatives. Therefore, we use associative algebra structures to calculate the usual derivatives. These calculations are carried out without using partial derivatives, but only by performing operations in the corresponding algebras. For $f\left(x\right)=x^2$, we obtain $d{f}_x\left(v\right)=vx+xv$, and for $f\left(x\right)=x^{-1}$, $d{f}_x\left(v\right)=-x^{-1}v{x}^{-1}$. Taylor approximations of order k and expansion by the Taylor series are performed. The pre-twisted differentiability for the case of non-commutative algebras is introduced and used to solve families of quadratic ordinary differential equations. Full article

Clifford algebras are an active area of mathematical research having numerous applications in mathematical physics and computer graphics, among many others. This paper demonstrates algorithms for the computation of characteristic polynomials, inverses, and minimal polynomials of general multivectors residing in a non-degenerate Clifford algebra of an arbitrary dimension. The characteristic polynomial and inverse computation are achieved by a translation of the classical Faddeev–LeVerrier–Souriau (FVS) algorithm in the language of Clifford algebra. The demonstrated algorithms are implemented in the Clifford package of the open source computer algebra system Maxima. Symbolic and numerical examples residing in different Clifford algebras are presented. Full article

This article is a shortened review of previous results obtained by the author. The advantages of describing the geometric nature of the physical properties of the early universe using the Clifford algebra approach are demonstrated. A geometric representation of the wave function of the early universe is used, and a new mechanism of spontaneous symmetry breaking with different degrees of freedom is proposed. A possible supersymmetry is revealed, and it is shown that the energy of the initial vacuum can be considered equal to zero. The origin of baryonic asymmetry and the nature of dark matter can be explained using a geometric representation of the wave function of the early universe. Full article

An influence network of events is a view of the universe based on events that may be related to one another via influence. The network of events forms a partially ordered set which, when quantified consistently via a technique called chain projection, results in the emergence of spacetime and the Minkowski metric as well as the Lorentz transformation through changing an observer from one frame to another. Interestingly, using this approach, the motion of a free electron as well as the Dirac equation can be described. Indeed, the same approach can be employed to show how a discrete version of some of the features of Euclidean geometry including directions, dimensions, subspaces, Pythagorean theorem, and geometric shapes can emerge. In this paper, after reviewing the essentials of the influence network formalism, we build on some of our previous works to further develop aspects of Euclidean geometry. Specifically, we present the emergence of geometric shapes, a discrete version of the parallel postulate, the dot product, and the outer (wedge product) in $2+1$ dimensions. Finally, we show that the scalar quantification of two concatenated orthogonal intervals exhibits features that are similar to those of the well-known concept of a geometric product in geometric Clifford algebras. Full article