Narrow Results

In this paper, we present a method for calculation of spin groups elements for known pseudo-orthogonal group elements with respect to the corresponding two-sheeted coverings. We present our results using the Clifford algebra formalism in the case of arbitrary dimension and signature, and then explicitly using matrices, quaternions, and split-quaternions in the cases of all possible signatures $(p,q)$ of space up to dimension $n=p+q=3$. The different formalisms are convenient for different possible applications in physics, engineering, and computer science.

In this paper, we first construct a symplectic Schur function solution to a newly defined two-component symplectic Kadomtsev–Petviashvili hierarchy. As a generalization of a two-component symplectic Schur function, we construct two-component symplectic universal characters which satisfy quadratic equations in an infinite-dimensional integrable dynamic system called a two-component symplectic universal character hierarchy. Then, we define a modified symplectic universal character hierarchy whose tau function can be represented by free fermions in Clifford algebras.

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.

We demonstrate a generalization of relativistic quantum mechanics using eight-component octonic wave function and octonic spatial operators. It is shown that the second-order equation for octonic wave function describing particles with spin 1/2 can be reformulated in the form of a system of first-order equations for quantum fields, which is analogous to the system of Maxwell equations for the electromagnetic field. It is established that for the special types of wave functions the second-order equation can be reduced to the single first-order equation analogous to the Dirac equation. At the same time it is shown that this first-order equation describes particles, which do not have quantum fields.

Generalized field equations of linear gravity are formulated on the basis of octons. When compared to the other eight-component noncommutative hypercomplex number systems, it is demonstrated that associative octons with scalar, pseudoscalar, pseudovector and vector values present a convenient and capable tool to describe the Maxwell–Proca-like field equations of gravitoelectromagnetism in a compact and simple way. Introducing massive graviton and gravitomagnetic monopole terms, the generalized gravitational wave equation and Klein–Gordon equation for linear gravity are also developed.

We analyze the group of maximal automorphisms of the N-extended worldline supersymmetry algebra, and its action on off-shell supermultiplets. This defines a concept of "holoraumy" that extends the notions of holonomy and curvature in a novel way and provides information about the geometry of the supermultiplet field-space. In turn, the "holoraumy" transformations of 0-brane dimensionally reduced supermultiplets provide information about Lorentz transformations in the higher-dimensional space–time from which the 0-brane supermultiplets are descended. Specifically, Spin(3) generators are encoded within 0-brane "holoraumy" tensors. Worldline supermultiplets are thus able to holographically encrypt information about higher-dimensional space–time geometry.

In this paper, it is proven that the associative octons including scalar, pseudoscalar, pseudovector and vector values are convenient and capable tools to generalize the Maxwell–Dirac like field equations of electromagnetism and linear gravity in a compact and simple way. Although an attempt to describe the massless field equations of electromagnetism and linear gravity needs the sixteen real component mathematical structures, it is proved that these equations can be formulated in terms of eight components of octons. Furthermore, the generalized wave equation in terms of potentials is derived in the presence of electromagnetic and gravitational charges (masses). Finally, conservation of energy concept has also been investigated for massless fields.

We propose a Clifford algebra approach to chiral symmetry breaking and fermion mass hierarchies in the context of composite Higgs bosons. Standard model fermions are represented by algebraic spinors of six-dimensional binary Clifford algebra, while ternary Clifford algebra-related flavor projection operators control allowable flavor-mixing interactions. There are three composite electroweak Higgs bosons resulted from top quark, tau neutrino, and tau lepton condensations. Each of the three condensations gives rise to masses of four different fermions. The fermion mass hierarchies within these three groups are determined by four-fermion condensations, which break two global chiral symmetries. The four-fermion condensations induce axion-like pseudo-Nambu–Goldstone bosons and can be dark matter candidates. In addition to the 125 GeV Higgs boson observed at the Large Hadron Collider, we anticipate detection of tau neutrino composite Higgs boson via the charm quark decay channel.

We bring to light an electroweak model which has been reappearing in the literature under various guises.${}^{1-5}$ In this model, weak isospin is shown to act automatically on states of only a single chirality (left). This is achieved by building the model exclusively from the raising and lowering operators of the Clifford algebra $ℂl(4)$. That is, states constructed from these ladder operators mimic the behaviour of left- and right-handed electrons and neutrinos under unitary ladder operator symmetry. This ladder operator symmetry is found to be generated uniquely by $su{(2)}_L$ and $u{(1)}_Y$. Crucially, the model demonstrates how parity can be maximally violated, without the usual step of introducing extra gauge and extra Higgs bosons, or ad hoc projectors.

We construct the Rarita–Schwinger basis vectors, $U^{\mu }$, spanning the direct product space, $U^{\mu }:=A^{\mu }\otimes u_M$, of a massless four-vector, $A^{\mu }$, with massless Majorana spinors, $u_M$, together with the associated field-strength tensor, ${{𝒯}}^{\mu \nu }:=p^{\mu }{U}^{\nu }-p^{\nu }{U}^{\mu }$. The ${{𝒯}}^{\mu \nu }$ space is reducible and contains one massless subspace of a pure spin-$3/2\in (3/2,0)\oplus (0,3/2)$. We show how to single out the latter in a unique way by acting on ${{𝒯}}^{\mu \nu }$ with an earlier derived momentum independent projector, ${{𝒫}}^{(3/2,0)}$, properly constructed from one of the Casimir operators of the algebra $so(1,3)$ of the homogeneous Lorentz group. In this way, it becomes possible to describe the irreducible massless $(3/2,0)\oplus (0,3/2)$ carrier space by means of the antisymmetric tensor of second rank with Majorana spinor components, defined as ${[w^{(3/2,0)}]}^{\mu \nu }:={[{{𝒫}}^{(3/2,0)}]}^{\mu \nu }{}_{\gamma \delta }{{𝒯}}^{\gamma \delta }$. The conclusion is that the $(3/2,0)\oplus (0,3/2)$ bi-vector spinor field can play the same role with respect to a $U^{\mu }$ gauge field as the bi-vector, $(1,0)\oplus (0,1)$, associated with the electromagnetic field-strength tensor, $F_{\mu \nu }$, plays for the Maxwell gauge field, $A_{\mu }$. Correspondingly, we find the free electromagnetic field equation, $p^{\mu }{F}_{\mu \nu }=0$, is paralleled by the free massless Rarita–Schwinger field equation, $p^{\mu }{[w^{(3/2,0)}]}_{\mu \nu }=0$, supplemented by the additional condition, ${\gamma }^{\mu }{\gamma }^{\nu }{[w^{(3/2,0)}]}_{\mu \nu }=0$, a constraint that invokes the Majorana sector.

In this paper, we show that a quadratic Lagrangian, with no constraints, containing ordinary time derivatives up to the order $m$ of $N$ dynamical variables, has $2mN$ symmetries consisting in the translation of the variables with solutions of the equations of motion. We construct explicitly the generators of these transformations and prove that they satisfy the Heisenberg algebra. We also analyze other specific cases which are not included in our previous statement: the Klein–Gordon Lagrangian, $N$ Fermi oscillators and the Dirac Lagrangian. In the first case, the system is described by an equation involving partial derivatives, the second case is described by Grassmann variables and the third shows both features. Furthermore, the Fermi oscillator and the Dirac field Lagrangians lead to second class constraints. We prove that also in these last two cases there are translational symmetries and we construct the algebra of the generators. For the Klein–Gordon case we find a continuum version of the Heisenberg algebra, whereas in the other cases, the Grassmann generators satisfy, after quantization, the algebra of the Fermi creation and annihilation operators.

The stabilized Poincare–Heisenberg algebra (SPHA) is a Lie algebra of quantum relativistic kinematics generated by fifteen generators. It is obtained from imposing stability conditions after combining the Lie algebras of quantum mechanics and relativity. In this paper, we show how the sixteen-dimensional real Clifford algebras Cℓ(1,3) and Cℓ(3,1) can both be used to generate the SPHA. The Clifford algebra path to the SPHA avoids the traditional stability considerations. It is conceptually easier and more straightforward to work with a Clifford algebra. The Clifford algebra path suggests that the next evolutionary step toward a theory of physics at the interface of GR and QM might be to depart from working in spacetime and instead to work in spacetime–momentum.

An $F$-algebra $A$ with unit $1$ is said to be a locally matrix algebra if an arbitrary finite collection of elements $a_1,\dots ,a_s$ from $A$ lies in a subalgebra $B$ with $1$ of the algebra $A$, that is isomorphic to a matrix algebra $M_n(F)$, $n\ge 1$. To an arbitrary unital locally matrix algebra $A$, we assign a Steinitz number $st(A)$ and study a relationship between $st(A)$ and $A$.

The Heisenberg uncertainty principle and the uncertainty principle for self-adjoint operators have been known and applied for decades. In this paper, in the framework of Clifford algebra, we establish a stronger Heisenberg–Pauli–Wely type uncertainty principle for the Fourier transform of multivector-valued functions, which generalizes the recent results about uncertainty principles of Clifford–Fourier transform. At the end, we consider another stronger uncertainty principle for the Dunkl transform of multivector-valued functions.

Several classes of classical cardinal B-splines can be obtained as solutions of operator equations of the form $Ly=0$ where $L$ is a linear differential operator of integral order. In this paper, we consider classes of generalized B-splines consisting of cardinal polynomial B-splines of complex and hypercomplex orders and cardinal exponential B-splines of complex order and derive the fractional linear differential operators that are naturally associated with them. For this purpose, we also present the spaces of distributions onto which these fractional differential operators act.

We study the regular representation ρ_ζ of the single-fermion algebra , i.e., c² = c⁺² = 0, cc⁺ + c⁺c = ζ1, for ζ ∈ [0,1]. We show that ρ₀ is a four-dimensional nonunitary representation of which is faithfully irreducible (it does not admit a proper faithful subrepresentation). Moreover, ρ₀ is the minimal faithfully irreducible representation of in the sense that every faithful representation of has a subrepresentation that is equivalent to ρ₀. We therefore identify a classical fermion with ρ₀ and view its quantization as the deformation: ζ : 0 → 1 of ρ_ζ. The latter has the effect of mapping ρ₀ into the four-dimensional, unitary, (faithfully) reducible representation ρ₁ of that is reminiscent of a Dirac fermion.

It is well known that starting with real structure, the Cayley–Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions p = 2, 4 and 8, respectively, but the procedure fails for p = 16 in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are n = 2⁷. Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower type related to complex and Pauli structures and, in relation to the iteraton process p → p + 2 → p + 4 → ⋯, they have constructed 2⁴-dimensional "bipetals" for p = 9 and 2⁷-dimensional "bisepals" for p = 13. The objects constructed appear to have an interesting property of periodicity related to the gradating function on the fractal diagonal interpreted as the "pistil" and a family of pairs of segments parallel to the diagonal and equidistant from it, interpreted as the "stamens." The present paper aims at an effective, explicit determination of the periods and expressing them in terms of complex and quaternionic structures, thus showing the quaternionic background of that periodicity. The proof of the Periodicity Theorem is given in the case where the index of the generator of the algebra in question exceeds the order of the initial algebra. In contrast to earlier results, the fractal bundle flower structure, in particular sepals, bisepals, perianths, and calyces are not introduced ab initio; they are quoted a posteriori, when they are fully motivated. The same concerns canonical two-layer pairing of prianth sepals and the relationship of fractal bundles of algebraic structure with the Hurwitz problem.

Gravitation theory is formulated as gauge theory on natural bundles with spontaneous symmetry breaking, where gauge symmetries are general covariant transformations, gauge fields are general linear connections, and Higgs fields are pseudo-Riemannian metrics.

The Dirac equation in four time and four space dimensions (or ($4+4$)-dimensions) is considered. Step by step, we show that such an equation admits Majorana and Weyl solutions. In order to obtain the Majorana or Weyl spinors, we used a method based on the construction of Clifford algebra in terms of $2\times 2$-matrices. We argue that our approach can be useful in supergravity, superstrings and qubit theory.

In the light of the analogy between electromagnetism and fluid dynamics, the Maxwell-type equations of compressible fluids are reformulated on the basis of spacetime algebra. In this paper, it is proved that this algebra provides an efficient mathematical tool for describing fluid fields in a compact and elegant way. Moreover, the fluid wave equation in terms of potentials are derived in a form similar to electromagnetic and gravitational counterparts previously derived using spacetime algebra.