An Introduction to Clifford Algebras and Spinors

  title={An Introduction to Clifford Algebras and Spinors},
  author={Jayme Vaz and Rold{\~a}o da Rocha},
A primer on the differential geometry of quaternionic curves
This paper describes the foundations of a differential geometry of a quaternionic curves. The Frenet–Serret equations and the evolutes and evolvents of a particular quaternionic curve are accordingly
Complexifying the Spacetime Algebra by Means of an Extra Timelike Dimension: Pin, Spin and Algebraic Spinors
Because of the isomorphism $\operatorname{Cl}_{1,3}(\Bbb{C})\cong\operatorname{Cl}_{2,3}(\Bbb{R})$, it is possible to complexify the spacetime Clifford algebra $\operatorname{Cl}_{1,3}(\Bbb{R})$ by
On the Clifford Algebraic Description of the Geometry of a 3D Euclidean Space
We discuss how transformations in a three dimensional euclidean space can be described in terms of the Clifford algebra $\mathcal{C}\ell_{3,3}$ of the quadratic space $\mathbb{R}^{3,3}$. We show that
\'Algebras y grupos de Clifford, espinores algebraicos y aplicaciones a la f\'isica.
In these notes we introduce the Clifford algebra of a quadratic space using techniques from universal algebra and algebraic theory of quadratic forms. We also define the Clifford, Pin and Spin groups
Revealing how different spinors can be: the Lounesto spinor classification
This paper aims to give a coordinate based introduction to the so-called Lounesto spinorial classification scheme. We introduce the main ideas and aspects of this spinorial categorization in an
Compendium on Multivector Contractions
We reorganize and expand the theory of contractions or interior products of multivectors, and related topics like Hodge duality. Results are generalized and new ones are presented, like a
Graded Symmetry Groups: Plane and Simple
The current work shows how an analysis using geometric algebra provides a picture complementary to that of the classic matrix Lie algebra approach, while retaining information about the number of reflections in a given transformation, by presenting novel matrix/vector representations for geometric algebras Rpqr.
Further investigation of mass dimension one fermionic duals
Geometric-Algebra Adaptive Filters
This paper reformulates adaptive filters in the framework of geometric algebra in a complete study of the resulting geometric-algebra adaptive filters (GAAFs), providing a generalization of regular AFs to subalgebras of GA.
New fermions in the bulk
Spinor fields on 5-dimensional Lorentzian manifolds are classified, according to the geometric Fierz identities that involve their bilinear covariants. Based upon this classification that generalises


Twistor algebra
  • Journal of Mathematical Physics
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Topological Geometry
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On the algebra of Dirac bispinor densities: Factorization and inversion theorems
The algebraic system formed by Dirac bispinor densities ρi≡ψΓiψ is discussed. The inverse problem—given a set of 16 real functions ρi, which satisfy the bispinor algebra, find the spinor ψ to which
An Introduction to Spinors and Geometry with Applications in Physics
There is now a greater range of mathematics used in theoretical physics than ever. The aim of this book is to introduce theoretical physicists, of graduate student level upwards, to the methods of
Quadratic Mappings and Clifford Algebras
Algebraic Preliminaries.- Quadratic Mappings.- Clifford Algebras.- Comultiplications. Exponentials. Deformations.- Orthogonal Groups and Lipschitz Groups.- Further Algebraic Developments.- Hyperbolic
Octonions and triality
In this chapter, we explore another generalization of C and IE, a non-associative real algebra, the Cayley algebra of octonions, (D. Like complex numbers and quaternions, octonions f o r m a real
Modulo (1,1) periodicity of Clifford algebras and generalized (anti-)Möbius transformations
Cli ord Algebras and the Classical Groups
The Clifford algebras of real quadratic forms and their complexifications are studied here in detail, and those parts which are immediately relevant to theoretical physics are seen in the proper
Clifford algebras and spinors
1. Quadratic spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Quaternion algebras . . . . . . . . . .
The design of linear algebra and geometry
Conventional formulations of linear algebra do not do justice to the fundamental concepts of meet, join, and duality in projective geometry. This defect is corrected by introducing Clifford algebra