Clifford Algebra Research Articles

Global exponential stability for delayed Clifford-valued coupled neural networks with impulsive effects

In the paper, the global exponential stability for impulsive delayed Clifford-valued coupled neural networks is studied. Firstly, a delayed Clifford-valued coupled neural network model is established. So as to avoid the non-commutativity issue of Clifford numbers multiplication, the original n-dimensional Clifford-valued models are decomposed into 2mn-dimensional real-valued models. Then, by using Lyapunov-Krasovskii (L-K) functional approach and the technique about linear matrix inequality, some new sufficient conditions are presented for the global exponential stability of the considered neural network models. Finally, the effectiveness of the results are verified by two numerical simulations.

On the Representations of Clifford and SO(1,9) Algebras for 8-Component Dirac Equation

On the Representations of Clifford and SO(1,9) Algebras for 8-Component Dirac Equation

Matrix formulas for multiplicities in the spin module

We obtain inductive and enumerative formulas for the multiplicities of the weights of the spin module for the Clifford algebra of a Levi subalgebra in a complex semisimple Lie algebra. Our formulas involve only matrices and tableaux, and our techniques combine linear algebra, Lie theory, and combinatorics. Moreover, this suggests a relationship with complex nilpotent orbits. The case of the special linear Lie algebra $\mathfrak{sl}(n,{\mathbb C})$ is emphasized.

Development of the Method of Averaging in Clifford Geometric Algebras

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.

Scalar fields described by Dirac quaternion wave equation

In this paper, we discuss the fields described by Dirac wave equation written in Clifford algebra based on Macfalane quaternions. It is shown that the strengths of these fields are nonzero only in the area of sources and the interaction of such fields occurs by overlapping. We consider both the simple spherically symmetric models of sources, which demonstrate attractive and repulsive interaction, and more complicated core-shell systems, which provide the bound states formation.

Properties of the fractional Clifford–Fourier transform

ABSTRACT In this paper, we establish a Riemann–Lebesgue theorem and some real Paley–Wiener-type theorems for the fractional Clifford–Fourier transform (FrCFT). Furthermore, because of the non-commutativity of Clifford algebra, we study some basic properties from linearity to modulation of the FrCFT about the left-multiplied functions and the right-multiplied functions.

Murnaghan–Nakayama Rule and Spin Bitrace for the Hecke–Clifford Algebra

AbstractA Pfaffian-type Murnaghan–Nakayama rule is derived for the Hecke–Clifford algebra $\mathcal{H}^{c}_{n}$ based on the Frobenius formula and vertex operators, and this leads to a combinatorial version via the tableaux realization of Schur’s $Q$-functions. As a consequence, a general formula for the irreducible characters $\zeta ^{\lambda }_{\mu }(q)$ using partition-valued functions is derived. Meanwhile, an iterative formula on the indexing partition $\lambda $ via the Pieri rule is also deduced. As applications, some compact formulae of the irreducible characters are given for special partitions and a symmetric property of the irreducible character is found. We also introduce the spin bitrace as the analogue of the bitrace for the Hecke algebra and derive its general combinatorial formula. Tables of irreducible characters are listed for $n\leq 7.$

Revisiting Takahashi's inversion theorem in discrete symmetry-based dual frameworks

The so-called Takahashi's Inversion Theorem, the reconstruction of a given spinor based on its bilinear covariants [1], are re-examined, considering alternative dual structures. In contrast to the classical results, where the Dirac dual structure plays the central role, new duals are built using the discrete symmetries C,P,T. Their combinations are also taken into account. Furthermore, the imposition of a new adjoint structure led us also to re-examine the representation of the Clifford algebra basis elements, uncovering new bilinear structures and a new Fierz aggregate. Those results might help construct theories for new beyond standard model fields.

Various Characteristic Properties of Lipschitzian Elements in Clifford Algebras

Various Characteristic Properties of Lipschitzian Elements in Clifford Algebras

The characteristic polynomial in calculation of exponential and elementary functions in Clifford algebras

Formulas to calculate multivector exponentials in a basis‐free representation and orthonormal basis are presented for an arbitrary Clifford geometric algebra . The formulas are based on the analysis of roots of characteristic polynomial of a multivector. Elaborate examples how to use the formulas in practice are presented. The results are generalized to arbitrary functions of multivector and may be useful in the quantum circuits or in the problems ofanalysis of evolution of the entangled quantum states.

Moduli spaces of modules over even Clifford algebras and Prym varieties

Moduli spaces of modules over even Clifford algebras and Prym varieties

Generalized Quaternions and Matrix Algebra

In this paper, we established the connection between generalized quaternion algebra and real (complex) matrix algebras by using Hamilton operators. We obtained real and complex matrices corresponding to the real and complex basis of the generalized quaternions. Also, we investigated the basis features of real and complex matrices. We get Pauli matrices corresponding to generalized quaternions. Then, we have shown that the algebra produced by these matrices is isomorphic to the Clifford algebra Cl(E_αβ^3) produced by generalized space E_αβ^3.
 Finally, we studied the relations among the symplectic matrices group corresponding to generalized unit quaternions, generalized unitary matrices group, and generalized orthogonal matrices group.

Clifford Algebras, Quantum Neural Networks and Generalized Quantum Fourier Transform

Clifford Algebras, Quantum Neural Networks and Generalized Quantum Fourier Transform

Experimental Test of High-Dimensional Quantum Contextuality Based on Contextuality Concentration.

Contextuality is a distinctive feature of quantum theory and a fundamental resource for quantum computation. However, existing examples of contextuality in high-dimensional systems lack the necessary robustness required in experiments. Here, we address this problem by identifying a family of noncontextuality inequalities whose maximum quantum violation grows with the dimension of the system. At first glance, this contextuality is the single-system version of multipartite Bell nonlocality taken to an extreme form. What is interesting is that the single-system version achieves the same degree of contextuality but uses a Hilbert space of lower dimension. That is, contextuality "concentrates" as the degree of contextuality per dimension increases. We show the practicality of this result by presenting an experimental test of contextuality in a seven-dimensional system. By simulating sequences of quantum ideal measurements with destructive measurements and repreparation in an all-optical setup, we report a violation of 68.7 standard deviations of the simplest case of the noncontextuality inequalities identified. Our results advance the investigation of high-dimensional contextuality, its connection to the Clifford algebra, and its role in quantum computation.

A note on the wedge reversion antisymmetry operation and 51 types of physical quantities in arbitrary dimensions.

The paper by Gopalan [(2020). Acta Cryst. A76, 318-327] presented an enumeration of the 41 physical quantity types in non-relativistic physics, in arbitrary dimensions, based on the formalism of Clifford algebra. Gopalan considered three antisymmetries: spatial inversion, 1, time reversal, 1', and wedge reversion, 1†. A consideration of the set of all seven antisymmetries (1, 1', 1†, 1'†, 1†, 1', 1'†) leads to an extension of the results obtained by Gopalan. It is shown that there are 51 types of physical quantities with distinct symmetry properties in total.

Boundary value problems for a second‐order elliptic partial differential equation system in Euclidean space

Let be a bounded regular domain, let be the standard Dirac operator in , and let be the Clifford algebra constructed over the quadratic space . For fixed, denotes the space of ‐vectors in . In the framework of Clifford analysis, we consider two boundary value problems for a second‐order elliptic system of partial differential equations of the form in , where is a smooth ‐vector valued function. The boundary conditions of the problems contain the inner and outer products of the ‐vector solution with both the Dirac operator and the normal vector to , ensuring the well‐posedness for the problems. Investigation of the spectral properties of the sandwich operator is considered by using the Fredholm theory. Finally, it is shown that satisfactory problem‐solving properties, in general, fail when we replace the standard Dirac operator by those, obtained via unusual orthogonal bases of .

Gravitation, and quantum theory, as emergent phenomena

There must exist a reformulation of quantum field theory, even at low energies, which does not depend on classical time. The octonionic theory proposes such a reformulation, leading to a pre-quantum pre-spacetime theory. The ingredients for constructing such a theory, which is also a unification of the standard model with gravitation, are : (i) the pre-quantum theory of trace dynamics – a matrix-valued Lagrangian dynamics, (ii) the spectral action principle of non-commutative geometry, (iii) the number system known as the octonions, for constructing a non-commutative manifold and for defining elementary particles via Clifford algebras, (iv) a Lagrangian with E 8 × E 8 symmetry. The split bioctonions define a sixteen dimensional space (with left-right symmetry) whose geometry (evolving in Connes time) relates to the four known fundamental forces, while predicting two new forces, SU(3) grav and U(1) grav . This latter interaction is possibly the theoretical origin of MOND. Coupling constants of the standard model result from left-right symmetry breaking, and their values are theoretically determined by the characteristic equation of the exceptional Jordan algebra of the octonions. The quantum-to-classical transition, precipitated by the entanglement of a critical number of fermions, is responsible for the emergence of classical spacetime, and also for the familiar formulation of quantum theory on a spacetime background.

Dynamical characterization of topological phases beyond the minimal models

Dynamical characterization of topological phases under quantum quench dynamics has been demonstrated as a powerful and efficient tool. Previous studies have been focused on systems of which the Hamiltonian consists of matrices that commute with each other and satisfy Clifford algebra. In this work, we consider the characterization of topological phases with Hamiltonians that are beyond the minimal model. Specifically, the quantum quench dynamics of two types of layered systems is studied, of which the consisting matrices of Hamiltonians do not all satisfy Clifford algebra. We find that the terms which anti-commute with others can hold common band-inversion surfaces, which controls the topology of all the bands, but for other terms, there is no universal behavior and need to be treated case by case.

Octonion Internal Space Algebra for the Standard Model

This paper surveys recent progress in our search for an appropriate internal space algebra for the standard model (SM) of particle physics. After a brief review of the existing approaches, we start with the Clifford algebras involving operators of left multiplication by octonions. A central role is played by a distinguished complex structure that implements the splitting of the octonions O=C⊕C3, which reflect the lepton-quark symmetry. Such a complex structure on the 32-dimensional space S of Cℓ10 Majorana spinors is generated by the Cℓ6(⊂Cℓ10) volume form, ω6=γ1⋯γ6, and is left invariant by the Pati–Salam subgroup of Spin(10), GPS=Spin(4)×Spin(6)/Z2. While the Spin(10) invariant volume form ω10=γ1…γ10 of Cℓ10 is known to split S on a complex basis into left and right chiral (semi)spinors, P=12(1−iω6) is interpreted as the projector on the 16-dimensional particle subspace (which annihilates the antiparticles).The standard model gauge group appears as the subgroup of GPS that preserves the sterile neutrino (which is identified with the Fock vacuum). The Z2-graded internal space algebra A is then included in the projected tensor product A⊂PCℓ10P=Cℓ4⊗Cℓ60. The Higgs field appears as the scalar term of a superconnection, an element of the odd part Cℓ41 of the first factor. The fact that the projection of Cℓ10 only involves the even part Cℓ60 of the second factor guarantees that the color symmetry remains unbroken. As an application, we express the ratio mHmW of the Higgs to the W boson masses in terms of the cosine of the theoretical Weinberg angle.

On the electron spin and spectrum energy problems within the Fock many temporal and Feynman proper time paradigms

The quantum electron spin structure and the photon dressed electron energy spectrum problem are reanalyzed in the framework of the Fock many-temporal parameter approach. There is analyzed the spin structure and its dependence on the symmetry properties of the related representations of the basic Clifford algebra, generated by creation-annihilation operators. The self-interaction phenomenon is discussed within the quantum renormalized Lorenz constraint on a suitably reduced Fock space. The electron energy spectrum is analyzed within the many-temporal Fock and Feynman proper time paradigms.