pinecone/core-2020-05-10-deduplication huggingface dataset:
Extended Grassmann and Clifford algebras

This paper is intended to investigate Grassmann and Clifford algebras over
Peano spaces, introducing their respective associated extended algebras, and to
explore these concepts also from the counterspace viewpoint. The exterior
(regressive) algebra is shown to share the exterior (progressive) algebra in
the direct sum of chiral and achiral subspaces. The duality between scalars and
volume elements, respectively under the progressive and the regressive products
is shown to have chirality, in the case when the dimension n of the Peano space
is even. In other words, the counterspace volume element is shown to be a
scalar or a pseudoscalar, depending on the dimension of the vector space to be
respectively odd or even. The de Rham cochain associated with the differential
operator is constituted by a sequence of exterior algebra homogeneous subspaces
subsequently chiral and achiral. Thus we prove that the exterior algebra over
the space and the exterior algebra constructed on the counterspace are only
pseudoduals each other, when we introduce chirality. The extended Clifford
algebra is introduced in the light of the periodicity theorem of Clifford
algebras context, wherein the Clifford and extended Clifford algebras Cl(p,q)
can be embedded in Cl(p+1,q+1), which is shown to be exactly the extended
Clifford algebra. Clifford algebras are constructed over the counterspace, and
the duality between progressive and regressive products is presented using the
dual Hodge star operator. The differential and codifferential operators are
also defined for the extended exterior algebras from the regressive product
viewpoint, and it is shown they uniquely tumble right out progressive and
regressive exterior products of 1-forms.Comment: 17 pages, to appear in Adv. Appl. Clifford Algebras 16 (3) (2006
Bicomplex Quantum Mechanics: II. The Hilbert Space

Using the bicomplex numbers $\mathbb{T}$ which is a commutative ring with
zero divisors defined by $\mathbb{T}=\{w_0 + w_1 i_1 + w_2 i_2 + w_3 j | w_0,
w_1, w_2, w_3 \in \mathbb{R}\}$ where $i_{1}^{2} = -1, i_{2}^{2} = -1, j^2 = 1,
i_1 i_2 = j = i_2 i_1$, we construct hyperbolic and bicomplex Hilbert spaces.
Linear functionals and dual spaces are considered and properties of linear
operators are obtained; in particular it is established that the eigenvalues of
a bicomplex self-adjoint operator are in the set of hyperbolic numbers.Comment: 25 pages, no figur
The Spacetime Algebra Approach to Massive Classical Electrodynamics with
  Magnetic Monopoles

Maxwell's equations with massive photons and magnetic monopoles are
formulated using spacetime algebra. It is demonstrated that a single
non-homogeneous multi-vectorial equation describes the theory. Two limiting
cases are considered and their symmetries highlighted: massless photons with
magnetic monopoles and finite photon mass in the absence of monopoles. Finally,
it is shown that the EM-duality invariance is a symmetry of the Hamiltonian
density (for Minkowskian spacetime) and Lagrangian density (for Euclidean
4-space) that reflects the signature of the respective metric manifold.Comment: 10 pages, no figure
Quantum fractals on n-spheres. Clifford Algebra approach

Using the Clifford algebra formalism we extend the quantum jumps algorithm of
the Event Enhanced Quantum Theory (EEQT) to convex state figures other than
those stemming from convex hulls of complex projective spaces that form the
basis for the standard quantum theory. We study quantum jumps on n-dimensional
spheres, jumps that are induced by symmetric configurations of non-commuting
state monitoring detectors. The detectors cause quantum jumps via geometrically
induced conformal maps (Mobius transformations) and realize iterated function
systems (IFS) with fractal attractors located on n-dimensional spheres. We also
extend the formalism to mixed states, represented by "density matrices". As a
numerical illustration we study quantum fractals on the circle, two--sphere
(octahedron), and on three-dimensional sphere (hypercube-tesseract, 24 cell,
600 cell,and 120 cell). The invariant measure on the attractor is approximated
by the powers of the Markov operator. In the appendices we calculate the
Radon-Nikodym derivative of the SO(n+1) invariant measure on S^n under
SO(1,n+1) transformations and discuss the Hamilton's "icossian calculus" as
well as its application to quaternionic realization of the binary icosahedral
group that is at the basis of the 600 cell and its dual, the 120 cell. As a
by-product of this work we obtain several Clifford algebraic results, such as a
characterization of positive elements in a Clifford algebra Cl(n+1) as
generalized Lorentz boosts, and their action as Moebius transformation on
n-sphere, and a decomposition of any element of Spin^+(1,n+1) into a boost and
a rotation, including the explicit formula for the pullback of the O(n+1)
invariant Riemannian metric with respect to the associated Mobius
transformation.Comment: Latex. 49 pages. 9 figures. Corrected two misprints in Sec. 3. R(2,C)
  at the beginning has been replaced by C(V,Q), and `isomorphism' just before
  Theorem 1 has been replaced by `homomorphism
Semi-harmonicity, Integral Means and Euler Type Vector Fields

The Dirichlet product of functions on a semi-Riemann domain and generalized
Euler vector fields, which include the radial, $\bar \partial$-Euler, and the
$\bar \partial$-Neumann vector fields, are introduced. The integral means and
the harmonic residues of functions on a Riemann domain are studied. The notion
of semi-harmonicity of functions on a complex space is introduced. It is shown
that, on a Riemann domain, the semi-harmonicity of a locally forwardly
$L^2$-function is characterized by local mean-value properties as well as by
weak-harmonicity. In particular, the Weyl's Lemma is extended to a Riemann
domain.Comment: This paper is a corrected version of "Semi-Harmonicity, Integral
  Means and Euler Type Vector Fields, Adv. Appl. Clifford Alg. 17 (2007),
  555-573". This publication is available at Springer via http://dx.doi.org/
  DOI 10.1007/s00006-007-0036-
The Barut Second-Order Equation: Lagrangian, Dynamical Invariants and
  Interactions

The second-order equation in the (1/2,0)+(0,1/2) representation of the
Lorentz group has been proposed by A. Barut in the 70s. It permits to explain
the mass splitting of leptons (e,mu,tau). Recently, the interest has grown to
this model (see, for instance, the papers by S. Kruglov and J. P. Vigier et
al). We continue the research deriving the equation from the first principles,
finding dynamical invariants for this model, investigating the influence of
potential interactions.Comment: 8 pages, no figures. The talk given at the Int. Conf. on Clifford
  Algebras and Applications (ICCA7), Toulouse, France, May 19-29, 2005. To be
  published in the Proceeding
Diffeomorphism Invariance and Local Lorentz Invariance

We show that diffeomorphism invariance of the Maxwell and the Dirac-Hestenes
equations implies the equivalence among different universe models such that if
one has a linear connection with non-null torsion and/or curvature the others
have also. On the other hand local Lorentz invariance implies the surprising
equivalence among different universe models that have in general different
G-connections with different curvature and torsion tensors.Comment: 19 pages, Revtex, Plenary Talk presented at VII International
  Conference on Clifford Algebras and their Applications, Universite Paul
  Sabatier UFR MIG, Toulouse (FRANCE), to appear in "Clifford Algebras,
  Applications to Mathematics, Physics and Engineering", Progress in Math.
  Phys., Birkhauser, Berlin 200
Quaternion polar representation with a complex modulus and complex argument inspired by the Cayley-Dickson form

We present a new polar representation of quaternions inspired by the Cayley-Dickson representation. In this new polar representation, a quaternion is represented by a pair of complex numbers as in the Cayley-Dickson form, but here these two complex numbers are a complex 'modulus' and a complex 'argument'. As in the Cayley-Dickson form, the two complex numbers are in the same complex plane (using the same complex root of - 1), but the complex phase is multiplied by a different complex root of - 1 in the exponential function. We show how to calculate the 'modulus' and 'argument' from an arbitrary quaternion in Cartesian form. © 2008 Birkhäuser Verlag Basel/Switzerland
Isometric action of SL(2,R) on homogeneous spaces

We investigate the SL(2,R) invariant geodesic curves with the as- sociated
invariant distance function in parabolic geometry. Parabolic geom- etry
naturally occurs in the study of SL(2,R) and is placed in between the elliptic
and the hyperbolic (also known as the Lobachevsky half-plane and 2- dimensional
Minkowski half-plane space-time) geometries. Initially we attempt to use
standard methods of finding geodesics but they lead to degeneracy in this
setup. Instead, by studying closely the two related elliptic and hyperbolic
geometries we discover a unified approach to a more exotic and less obvious
parabolic case. With aid of common invariants we describe the possible dis-
tance functions that turn out to have some unexpected, interesting properties.Comment: LaTeX, 10 pages, 9 EPS figure
Yang-Mills Interactions and Gravity in Terms of Clifford Algebra

A model of Yang-Mills interactions and gravity in terms of the Clifford
algebra Cl(0,6) is presented. The gravity and Yang-Mills actions are formulated
as different order terms in a generalized action. The feebleness of gravity as
well as the smallness of the cosmological constant and theta terms are
discussed at the classical level. The invariance groups, including the de
Sitter and the Pati-Salam SU(4) subgroups, consist of gauge transformations
from either side of an algebraic spinor. Upon symmetry breaking via the Higgs
fields, the remaining symmetries are the Lorentz SO(1,3), color SU(3),
electromagnetic U(1)_EM, and an additional U(1). The first generation leptons
and quarks are identified with even and odd parts of spinor idempotent
projections. There are still several shortcomings with the current model.
Further research is needed to fully recover the standard model results.Comment: 20 pages, to appear in Advances in Applied Clifford Algebra
Exact Description of Rotational Waves in an Elastic Solid

Conventional descriptions of transverse waves in an elastic solid are limited
by an assumption of infinitesimally small gradients of rotation. By assuming a
linear response to variations in orientation, we derive an exact description of
a restricted class of rotational waves in an ideal isotropic elastic solid. The
result is a nonlinear equation expressed in terms of Dirac bispinors. This
result provides a simple classical interpretation of relativistic quantum
mechanical dynamics. We construct a Lagrangian of the form L=-E+U+K=0, where E
is the total energy, U is the potential energy, and K is the kinetic energy.Comment: 9 pages; Added references in revisio
One-Parameter Homothetic Motion in the Hyperbolic Plane and Euler-Savary
  Formula

In \cite{Mul} one-parameter planar motion was first introduced and the
relations between absolute, relative, sliding velocities (and accelerations) in
the Euclidean plane $\mathbb{E}^2$ were obtained. Moreover, the relations
between the Complex velocities one-parameter motion in the Complex plane were
provided by \cite{Mul}. One-parameter planar homothetic motion was defined in
the Complex plane, \cite{Kur}. In this paper, analogous to homothetic motion in
the Complex plane given by \cite{Kur}, one-parameter planar homothetic motion
is defined in the Hyperbolic plane. Some characteristic properties about the
velocity vectors, the acceleration vectors and the pole curves are given.
Moreover, in the case of homothetic scale $h$ identically equal to 1, the
results given in \cite{Yuc} are obtained as a special case. In addition, three
hyperbolic planes, of which two are moving and the other one is fixed, are
taken into consideration and a canonical relative system for one-parameter
planar hyperbolic homothetic motion is defined. Euler-Savary formula, which
gives the relationship between the curvatures of trajectory curves, is obtained
with the help of this relative system
Fundamental Representations and Algebraic Properties of Biquaternions or Complexified Quaternions

The fundamental properties of biquaternions (complexified quaternions) are presented including several different representations, some of them new, and definitions of fundamental operations such as the scalar and vector parts, conjugates, semi-norms, polar forms, and inner products. The notation is consistent throughout, even between representations, providing a clear account of the many ways in which the component parts of a biquaternion may be manipulated algebraically. © 2010 Springer Basel AG
A Geometric Algebra Perspective On Quantum Computational Gates And
  Universality In Quantum Computing

We investigate the utility of geometric (Clifford) algebras (GA) methods in
two specific applications to quantum information science. First, using the
multiparticle spacetime algebra (MSTA, the geometric algebra of a relativistic
configuration space), we present an explicit algebraic description of one and
two-qubit quantum states together with a MSTA characterization of one and
two-qubit quantum computational gates. Second, using the above mentioned
characterization and the GA description of the Lie algebras SO(3) and SU(2)
based on the rotor group Spin+(3, 0) formalism, we reexamine Boykin's proof of
universality of quantum gates. We conclude that the MSTA approach does lead to
a useful conceptual unification where the complex qubit space and the complex
space of unitary operators acting on them become united, with both being made
just by multivectors in real space. Finally, the GA approach to rotations based
on the rotor group does bring conceptual and computational advantages compared
to standard vectorial and matricial approaches.Comment: 18 pages; accepted for publication in Adv. Appl. Clifford Alg. (2010
On Conformal Infinity and Compactifications of the Minkowski Space

Using the standard Cayley transform and elementary tools it is reiterated
that the conformal compactification of the Minkowski space involves not only
the "cone at infinity" but also the 2-sphere that is at the base of this cone.
We represent this 2-sphere by two additionally marked points on the Penrose
diagram for the compactified Minkowski space. Lacks and omissions in the
existing literature are described, Penrose diagrams are derived for both,
simple compactification and its double covering space, which is discussed in
some detail using both the U(2) approach and the exterior and Clifford algebra
methods. Using the Hodge * operator twistors (i.e. vectors of the
pseudo-Hermitian space H_{2,2}) are realized as spinors (i.e., vectors of a
faithful irreducible representation of the even Clifford algebra) for the
conformal group SO(4,2)/Z_2. Killing vector fields corresponding to the left
action of U(2) on itself are explicitly calculated. Isotropic cones and
corresponding projective quadrics in H_{p,q} are also discussed. Applications
to flat conformal structures, including the normal Cartan connection and
conformal development has been discussed in some detail.Comment: 38 pages, 8 figures, late
A generalization of the Lorentz ether to gravity with
  general-relativistic limit

Does relativistic gravity provide arguments against the existence of a
preferred frame? Our answer is negative. We define a viable theory of gravity
with preferred frame. In this theory, the EEP holds exactly, and the Einstein
equations of GR limit are obtained in a natural limit. Despite some remarkable
differences (stable "frozen stars" instead of black holes, a "big bounce"
instead of the big bang, exclusion of nontrivial topologies and closed causal
loops, and a preference for a flat universe) the theory is viable.
  The equations of the theory are derived from simple axioms about some
fundamental condensed matter (the generalized Lorentz ether), so that, in
particular, the EEP is not postulated but derived.
  The theory is compatible with the condensed matter interpretation for the
fermions and gauge fields of the standard model.Comment: Some changes in the presentatio
The Extended Fock Basis of Clifford Algebra

We investigate the properties of the Extended Fock Basis (EFB) of Clifford
algebras introduced in [1]. We show that a Clifford algebra can be seen as a
direct sum of multiple spinor subspaces that are characterized as being left
eigenvectors of \Gamma. We also show that a simple spinor, expressed in Fock
basis, can have a maximum number of non zero coordinates that equals the size
of the maximal totally null plane (with the notable exception of vectorial
spaces with 6 dimensions).Comment: Minimal corrections to the published versio
Clifford A-algebras of quadratic A-modules

A Clifford A-algebra of a quadratic A-module (E, q) is an associative
and unital A-algebra (i.e. sheaf of A-algebras) associated with
the quadratic ShSetX-morphism q, and satisfying a certain universal
property. By introducing sheaves of sets of orthogonal bases (or simply
sheaves of orthogonal bases), we show that with every Riemannian quadratic
free A-module of finite rank, say, n, one can associate a Clifford
free A-algebra of rank 2n. This “main” result is stated in Theorem 3.2.http://link.springer.com/journal/6hb2016Mathematics and Applied Mathematic
Non-Associativity in the Clifford Bundle on the Parallelizable Torsion
  7-Sphere

In this paper we discuss generalized properties of non-associativity in
Clifford bundles on the 7-sphere S7. Novel and prominent properties inherited
from the non-associative structure of the Clifford bundle on S7 are
demonstrated. They naturally lead to general transformations of the spinor
fields on S7 and have dramatic consequences for the associated Kac-Moody
current algebras. All additional properties concerning the non-associative
structure in the Clifford bundle on S7 are considered. We further discuss and
explore their applications.Comment: 16 page
Clifford algebra unveils a surprising geometric significance of quaternionic root systems of Coxeter groups.

Quaternionic representations of Coxeter (reflection) groups of ranks 3 and 4, as well as those of E8, have been used extensively in the literature. The present paper analyses such Coxeter groups in the Clifford Geometric Algebra framework, which affords a simple way of performing reflections and rotations whilst exposing more clearly the underlying geometry. The Clifford approach shows that the quaternionic representations in fact have very simple geometric interpretations. The representations of the groups A1 × A1 × A1, A3, B3 and H3 of rank 3 in terms of pure quaternions are shown to be simply the Hodge dualised root vectors, which determine the reflection planes of the Coxeter groups. Two successive reflections result in a rotation, described by the geometric product of the two reflection vectors, giving a Clifford spinor. The spinors for the rank-3 groups A1 × A1 × A1, A3, B3 and H3 yield a new simple construction of binary polyhedral groups. These in turn generate the groups A1 × A1 × A1 × A1, D4, F4 and H4 of rank 4 and their widely used quaternionic representations are shown to be spinors in disguise. Therefore, the Clifford geometric product in fact induces the rank-4 groups from the rank-3 groups. In particular, the groups D4, F4 and H4 are exceptional structures, which our study sheds new light on. IPPP/12/26, DCPT/12/5
Applications of Clifford's Geometric Algebra

We survey the development of Clifford's geometric algebra and some of its
engineering applications during the last 15 years. Several recently developed
applications and their merits are discussed in some detail. We thus hope to
clearly demonstrate the benefit of developing problem solutions in a unified
framework for algebra and geometry with the widest possible scope: from quantum
computing and electromagnetism to satellite navigation, from neural computing
to camera geometry, image processing, robotics and beyond.Comment: 26 pages, 91 reference
Representation of Crystallographic Subperiodic Groups in Clifford's
  Geometric Algebra

This paper explains how, following the representation of 3D crystallographic
space groups in Clifford's geometric algebra, it is further possible to
similarly represent the 162 so called subperiodic groups of crystallography in
Clifford's geometric algebra. A new compact geometric algebra group
representation symbol is constructed, which allows to read off the complete set
of geometric algebra generators. For clarity moreover the chosen generators are
stated explicitly. The group symbols are based on the representation of point
groups in geometric algebra by versors (Clifford monomials, Lipschitz
elements).
  Keywords: Subperiodic groups, Clifford's geometric algebra, versor
representation, frieze groups, rod groups, layer groups .Comment: 17 pages, 6 figures, 11 tables. arXiv admin note: substantial text
  overlap with arXiv:1306.128
Chisholm-Caianiello-Fubini Identities for S=1 Barut-Muzinich-Williams
  Matrices

The formulae of the relativistic products are found S=1
Barut-Muzinich-Williams matrices. They are analogs of the well-known
Chisholm-Caianiello-Fubini identities. The obtained results can be useful in
the higher-order calculations of the high-energy processes with S=1 particles
in the framework of the 2(2S+1) Weinberg formalism, which recently attracted
attention again. PACS numbers: 02.90.+p, 11.90.+t, 12.20.DsComment: 5pp. This is the modernized version of the EFUAZ FT-95-14 (never
  submitted and published) preprint of the second author. The modifications are
  due to the Thesis of the first autho
A Clifford algebraic framework for Coxeter group theoretic computations.

Real physical systems with reflective and rotational symmetries such as viruses, fullerenes and quasicrystals have recently been modeled successfully in terms of three-dimensional (affine) Coxeter groups. Motivated by this progress, we explore here the benefits of performing the relevant computations in a Geometric Algebra framework, which is particularly suited to describing reflections. Starting from the Coxeter generators of the reflections, we describe how the relevant chiral (rotational), full (Coxeter) and binary polyhedral groups can be easily generated and treated in a unified way in a versor formalism. In particular, this yields a simple construction of the binary polyhedral groups as discrete spinor groups. These in turn are known to generate Lie and Coxeter groups in dimension four, notably the exceptional groups D4, F4 and H4. A Clifford algebra approach thus reveals an unexpected connection between Coxeter groups of ranks 3 and 4. We discuss how to extend these considerations and computations to the Conformal Geometric Algebra setup, in particular for the non-crystallographic groups, and construct root systems and quasicrystalline point arrays. We finally show how a Clifford versor framework sheds light on the geometry of the Coxeter element and the Coxeter plane for the examples of the twodimensional non-crystallographic Coxeter groups I2(n) and the threedimensional groups A3, B3, as well as the icosahedral group H3. IPPP/12/49, DCPT/12/9