Complex Clifford Algebra Research Articles

A new approach to matrix isomorphisms of complex Clifford algebras via Cantor set

We give a new way to obtain the standard isomorphisms of complex Clifford algebras, known as the tensor product of Pauli matrices, by representing the complex Clifford algebras on the space of complex valued functions defined over a finite subset of the Cantor set.

Quantum computing based on complex Clifford algebras

We propose to represent both n-qubits and quantum gates acting on them as elements in the complex Clifford algebra defined on a complex vector space of dimension 2n. In this framework, the Dirac formalism can be realized in straightforward way. We demonstrate its functionality by performing quantum computations with several well known examples of quantum gates. We also compare our approach with representations that use real geometric algebras.

Complex Clifford algebra in repeated quantum prisoner's dilemma

This paper introduces an application of complex Clifford algebra in a representation of the quantum prisoner's dilemma. The authors propose a novel modification of the Eisert–Lewenstein–Wilkens protocol to represent a repeated version of the quantum game. This repeated modification allows to embed entanglement into players' strategy sets and to see how players will operate with it. The apparatus of complex Clifford algebra enables an intuitive representation of the suggested protocol and efficient computation of the resulting payoff functions. The presented findings provide a new point of view on the interpretation of entanglement as a measure of information transition between rounds of the game.

Granular sieving algorithm for selecting best n$$ n $$ parameters

A common type problem of optimization is to find simultaneously parameters that globally minimize an objective function of variables. Such problems are seen in signal and image processing and in various applications of mathematical analysis of several complex variables and Clifford algebras. Objective functions are usually assumed to be Lipschitzian with maybe unknown Lipschitz constants. A number of methods have been established to discard the sets called “bad sets” in a partition that is impossible to contain any optimal point, as well as to treat the unknown Lipschitz bound problem along with the algorithm. In the present paper, a simple criterion of eliminating bad sets is proposed for the first time. The elimination method leads to a concise and rigorous proof of convergence. The algorithm, on the range space side, converges to the global minimum with an exponential rate, while on the domain space side, converges with equal accuracy to the set of all the global minimizers. To treat the unknown Lipschitz constant dilemma, we propose a practical pseudo‐Lipshitz bound process. The methodology is of fundamental nature with straightforward mathematical formulation applicable to multivariate objective functions defined on any compactly connected manifolds in higher dimensions. The method is tested against an extensive number of benchmark functions in the literature. The experimental results exhibit considerable effectiveness and applicability of the algorithm.

Iterants, Majorana Fermions and the Majorana-Dirac Equation

This paper explains a method of constructing algebras, starting with the properties of discrimination in elementary discrete systems. We show how to use points of view about these systems to construct what we call iterant algebras and how these algebras naturally give rise to the complex numbers, Clifford algebras and matrix algebras. The paper discusses the structure of the Schrödinger equation, the Dirac equation and the Majorana Dirac equations, finding solutions via the nilpotent method initiated by Peter Rowlands.

Topological preons from algebraic spinors

It is demonstrated that many of the assumed rules that govern the structure of a previously proposed topological preon model, in which simple non-trivial braids consisting of three twisted ribbons are mapped to the first generation of leptons and quarks, are automatically adhered to when the algebraic spinors of two complex Clifford algebras are identified with braids via a suitable map. Much of the assumed topological architecture of this model can therefore be interpreted as a direct consequence of the deeper algebraic structures upon which the minimal ideals of these Clifford algebras are constructed. This result deepens the understanding of how these two complementary descriptions, one topological and one algebraic, of Standard Model symmetries are intimately connected despite originating from very different perspectives.

On Inner Automorphisms Preserving Fixed Subspaces of Clifford Algebras

In this paper, we consider inner automorphisms that leave invariant fixed subspaces of real and complex Clifford algebras -- subspaces of fixed grades and subspaces determined by the reversion and the grade involution. We present groups of elements that define such inner automorphisms and study their properties. Some of these Lie groups can be interpreted as generalizations of Clifford, Lipschitz, and spin groups. We study the corresponding Lie algebras. Some of the results can be reformulated for the case of more general algebras -- graded central simple algebras or graded central simple algebras with involution.

The Standard Model particle content with complete gauge symmetries from the minimal ideals of two Clifford algebras

Building upon previous works, it is shown that two minimal left ideals of the complex Clifford algebra mathbb {C}ell (6) and two minimal right ideals of mathbb {C}ell (4) transform as one generation of leptons and quarks under the gauge symmetry SU(3)_Ctimes U(1)_{EM} and SU(2)_L respectively. The SU(2)_L weak symmetries are naturally chiral. Combining the mathbb {C}ell (6) and mathbb {C}ell (4) ideals, all the gauge symmetries of the Standard Model, together with its lepton and quark content for a single generation are represented. The combined ideals can be written as minimal left ideals of mathbb {C}ell (6)otimes mathbb {C}ell (4)cong mathbb {C}ell (10) in a way that preserves individually the mathbb {C}ell (6) structure and mathbb {C}ell (4) structure of physical states. This resulting model captures many of the attractive features of the Georgi and Glashow SU(5) Grand Unified Theory without introducing proton decay or other unobserved processes. Such processes are naturally excluded because they do not preserve the underlying algebraic structure.

A note on Jordan algebras, three generations and exceptional periodicity

It is shown that the algebra [Formula: see text] based on the complexified Exceptional Jordan, and the complex Clifford algebra in 4D, is rich enough to describe all the spinorial degrees of freedom of three generations of fermions in 4D, and include additional fermionic dark matter candidates. Furthermore, the model described in this paper can account also for the Standard Model gauge symmetries. We extend these results to the Magic Star algebras of Exceptional Periodicity developed by Marrani–Rios–Truini and based on the Vinberg cubic [Formula: see text] algebras which are generalizations of exceptional Jordan algebras. It is found that there is a one-to-one correspondence among the real spinorial degrees of freedom of four generations of fermions in 4D with the off-diagonal entries of the spinorial elements of the [Formula: see text] [Formula: see text] of Vinberg matrices at level [Formula: see text]. These results can be generalized to higher levels [Formula: see text] leading to a higher number of generations beyond 4. Three [Formula: see text] of [Formula: see text] algebras and their conjugates [Formula: see text] were essential in the Magic Star construction of Exceptional Periodicity that extends the [Formula: see text] algebra to [Formula: see text] with [Formula: see text] integer.

Fermion condensation and super pivotal categories

We study fermionic topological phases using the technique of fermion condensation. We give a prescription for performing fermion condensation in bosonic topological phases that contain a fermion. Our approach to fermion condensation can roughly be understood as coupling the parent bosonic topological phase to a phase of physical fermions and condensing pairs of physical and emergent fermions. There are two distinct types of objects in the resulting fermionic fusion categories, which we call “m-type” and “q-type” objects. The endomorphism algebras of q-type objects are complex Clifford algebras, and they have no analogs in bosonic theories. We construct a fermionic generalization of the tube category, which allows us to compute the quasiparticle excitations arising from the condensed theories. We prove a series of results relating data in fermionic theories to data in their parent bosonic theories; for example, if C is a modular tensor category containing a fermion, then the tube category constructed from the condensed theory satisfies Tube(C/ψ)≅C×(C/ψ). We also study how modular transformations, fusion rules, and coherence relations are modified in the fermionic setting, prove a fermionic version of the Verlinde dimension formula, construct a commuting projector lattice Hamiltonian for fermionic theories, and write down a fermionic version of the Turaev-Viro-Barrett-Westbury state sum. A large portion of this work is devoted to three detailed examples of performing fermion condensation to produce fermionic topological phases: we condense fermions in the Ising theory, the SO(3)6 theory, and the 12E6 theory and compute the quasiparticle excitation spectrum in each of the condensed theories.

Using Raising and Lowering Operators from Geometric Algebra for Electroweak Theory in Particle Physics

This paper has two objectives. The first is to explore the form and action of raising and lowering operators expressed in geometric algebra (GA). The second is to show how increasing the number of dimensions of Euclidean space from three to four opens a new avenue for understanding the chiral asymmetry of electroweak interactions. These explorations are guided by isomorphisms among groups represented in complex Clifford algebra, matrix algebra, and GA. With these isomorphisms, expressions for raising and lowering operators for electron and neutrino states in complex Clifford algebra are translated into GA and elaborated to include positrons and antineutrinos. This paper addresses such operators in the context of the electroweak sector of the SM utilizing (1) the GA $$\mathcal {G}_3$$ for the Hestenes–Dirac equation in a Euclidean lab frame, (2) $$\mathcal {G}_4$$ to introduce chiral asymmetry, and (3) $$\mathcal {G}_{4,1}$$ to express the electroweak fermion states of the first generation of the SM and demonstrate their SU(2) relationships.

On C $$\varvec{\otimes }$$ H $$\varvec{\otimes }$$ O-Valued Gravity, Sedenions, Hermitian Matrix Geometry and Nonsymmetric Kaluza–Klein Theory

We review briefly how R $$\otimes $$ C $$\otimes $$ H $$\otimes $$ O-valued gravity (real-complex-quaterno-octonionic gravity) naturally can describe a grand unified field theory of Einstein’s gravity with a Yang–Mills theory containing the Standard Model group $$SU(3) \times SU(2) \times U(1)$$ . The algebra of left actions associated with the composite algebras involving the Division algebras, and the Sedenions $$\mathbf{S}$$ , and acting on themselves, all lead to complex Clifford algebras (complex matrix algebras). The complex Cl(16) algebra is the most appealing one since it is the one corresponding to the algebra of left actions of $$\mathbf{C} \otimes \mathbf{H} \otimes \mathbf{O} \otimes \mathbf{S}$$ acting on itself, and containing the $$\mathbf{e_8 \oplus e_8}$$ algebra of the anomaly free 10D Heterotic String. An analysis of $$ C \otimes H \otimes O$$ -valued gravity reveals that it bears a connection to Nonsymmetric Kaluza–Klein theories and complex Hermitian Matrix Geometry. The key behind these connections is in finding the relation between $$ C \otimes H \otimes O$$ -valued metrics in two complex dimensions with complex metrics in higher dimensional real manifolds ( $$ D = 32 $$ real dimensions in particular). It is desirable to extend these results to hypercomplex, quaternionic manifolds and Exceptional Jordan Matrix Models.

A New Cauchy Integral Formula in the Complex Clifford Analysis

In this paper, we construct an analogue of Bochner–Martinelli kernel based on theory of functions of several complex variables in complex Clifford analysis, which has generalized complex differential forms with Clifford basis vectors. Using these complex differential forms, we obtain the Stoke’s formula of complex Clifford functions which are defined on a domain \(\Omega \subset C^{n+1}\) and take values in a complex Clifford algebra \(Cl_{0,n}(C)\). Then, we give a Stoke’s formula which has a classical form and an analogue of Cauchy–Pompeiu formula which is represented by Bochner–Martinelli kernel, and establish an analogue of Cauchy integral formula in complex Clifford analysis. It is possible to promote these results to complex manifold’s corresponding results in the Clifford analysis using the representation by generalized complex differential forms.

Superalgebraic structure of Lorentz transformations

We constructed superalgebraic representation of the algebra of second quantization of spinors. It is a fermionic analog of the Bargmann-Fock representation and is based on the use of the density of Grassmann variables in impulse space and their derivatives. We showed that Dirac matrices and Lorentz transformation generators can be expressed in terms of such densities. Lorentz transformations are considered as Bogolyubov transformations of creation and annihilation operators. We constructed superalgebraic form of the Dirac equation and the vacuum state vector. We showed that in the superalgebraic form of the complex Clifford algebra generators corresponding to Dirac gamma matrices are not equivalent. Clifford vector corresponding to diagonal matrix γ0 annihilates the vacuum, and the remaining ones give nonzero values. This means that there is asymmetric direction corresponding to the time axis.

Leptons, Quarks, and Gauge from the Complex Clifford Algebra $$\mathbb {C}\ell _6$$ C ℓ 6

A simple geometric algebra is shown to contain automatically the leptons and quarks of a family of the Standard Model, and the electroweak and color gauge symmetries, without predicting extra particles and symmetries. The algebra is already naturally present in the Standard Model, in two instances of the Clifford algebra $\mathbb{C}\ell_6$, one being algebraically generated by the Dirac algebra and the weak symmetry generators, and the other by a complex three-dimensional representation of the color symmetry, which generates a Witt decomposition which leads to the decomposition of the algebra into ideals representing leptons and quarks. The two instances being isomorphic, the minimal approach is to identify them, resulting in the model proposed here. The Dirac and Lorentz algebras appear naturally as subalgebras acting on the ideals representing leptons and quarks. The resulting representations on the ideals are invariant to the electromagnetic and color symmetries, which are generated by the bivectors of the algebra. The electroweak symmetry is also present, and it is already broken by the geometry of the algebra. The model predicts a bare Weinberg angle $\theta_W$ given by $\sin^2\theta_W=0.25$. The model shares common ideas with previously known models, particularly with Chisholm and Farwell, 1996, Trayling and Baylis, 2004, and Furey, 2016.

Plethystic vertex operators and boson-fermion correspondences

We study the algebraic properties of plethystic vertex operators, introduced in (2010 J. Phys. A: Math. Theor. 43 405202), underlying the structure of symmetric functions associated with certain generalized universal character rings of subgroups of the general linear group, defined to stabilize tensors of Young symmetry type characterized by a partition of arbitrary shape π. Here we establish an extension of the well-known boson-fermion correspondence involving Schur functions and their associated (Bernstein) vertex operators: for each π, the modes generated by the plethystic vertex operators and their suitably constructed duals, satisfy the anticommutation relations of a complex Clifford algebra. The combinatorial manipulations underlying the results involve exchange identities exploiting the Hopf-algebraic structure of certain symmetric function series and their plethysms.

Weighted solution of the Dirac Beltrami equation with coefficient in VMO

We study the generalized Beltrami equation , where is the left Dirac operator in acting on functions in and with values in the complex Clifford algebra , is its conjugate, and is a -valued function with compact support, with vanishing mean oscillation, satisfying , where are the coordinates of in . Let be a weight function in . We prove that if belongs to the Muckenhoupt class with , that makes continuous the Riesz potential , then for every , there exists a solution of the equation above satisfying for every distributional partial derivative. If , we prove the same result for any .

Demonstration representation and tensor products of Clifford algebras

It is proved that the tensor product of any Clifford algebras is isomorphic to a single Clifford algebra over some commutative algebra. It is also proved that any complex or real Clifford algebra C(p, q) can be represented as a tensor product of Clifford algebras of the second and first orders. A canonical form of such a representation is proposed.

Semigroups over real alternative *-algebras: Generation theorems and spherical sectorial operators

The aim of this paper is twofold. On one hand, generalizing some recent results obtained in the quaternionic setting, but using simpler techniques, we prove the generation theorems for semigroups in Banach spaces whose set of scalars belongs to the class of real alternative *-algebras, which includes, besides real and complex numbers, quaternions, octonions and Clifford algebras. On the other hand, in this new general framework, we introduce the notion of spherical sectorial operator and we prove that a spherical sectorial operator generates a semigroup that can be represented by a Cauchy integral formula. It follows that such a semigroup is analytic in time.

Spinor Structure and Internal Symmetries

Spinor structure and internal symmetries are considered within one theoretical framework based on the generalized spin and abstract Hilbert space. Complex momentum is understood as a generating kernel of the underlying spinor structure. It is shown that tensor products of biquaternion algebras are associated with the each irreducible representation of the Lorentz group. Space-time discrete symmetries $P$, $T$ and their combination $PT$ are generated by the fundamental automorphisms of this algebraic background (Clifford algebras). Charge conjugation $C$ is presented by a pseudoautomorphism of the complex Clifford algebra. This description of the operation $C$ allows one to distinguish charged and neutral particles including particle-antiparticle interchange and truly neutral particles. Spin and charge multiplets, based on the interlocking representations of the Lorentz group, are introduced. A central point of the work is a correspondence between Wigner definition of elementary particle as an irreducible representation of the Poincar\'{e} group and $SU(3)$-description (quark scheme) of the particle as a vector of the supermultiplet (irreducible representation of $SU(3)$). This correspondence is realized on the ground of a spin-charge Hilbert space. Basic hadron supermultiplets of $SU(3)$-theory (baryon octet and two meson octets) are studied in this framework. It is shown that quark phenomenologies are naturally incorporated into presented scheme. The relationship between mass and spin is established. The introduced spin-mass formula and its combination with Gell-Mann--Okubo mass formula allows one to take a new look at the problem of mass spectrum of elementary particles.