Clifford algebras and geometric algebra

1997 ◽  
Vol 7 (2) ◽  
pp. 91-102 ◽  
Author(s):  
G. Aragón ◽  
J. L. Aragón ◽  
M. A. Rodríguez
Keyword(s):  
Geometric Algebra ◽  
Clifford Algebras

scholarly journals Explicit Baker–Campbell–Hausdorff–Dynkin formula for spacetime via geometric algebra

2021 ◽  
Author(s):  
Joseph Wilson ◽  
Matt Visser
Keyword(s):  
Efficient Method ◽  
Geometric Algebra ◽  
Clifford Algebras ◽  
Matrix Representation ◽  
Spin Representation ◽  
Lorentz Transformations ◽  
Rodrigues Formula ◽  
Dimension Formula ◽  
Spacetime Algebra ◽  
Geometric Algebras

We present a compact Baker–Campbell–Hausdorff–Dynkin formula for the composition of Lorentz transformations [Formula: see text] in the spin representation (a.k.a. Lorentz rotors) in terms of their generators [Formula: see text]: [Formula: see text] This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension [Formula: see text], naturally generalizing Rodrigues’ formula for rotations in [Formula: see text]. In particular, it applies to Lorentz rotors within the framework of Hestenes’ spacetime algebra, and provides an efficient method for composing Lorentz generators. Computer implementations are possible with a complex [Formula: see text] matrix representation realized by the Pauli spin matrices. The formula is applied to the composition of relativistic 3-velocities yielding simple expressions for the resulting boost and the concomitant Wigner angle.

Clifford Geometric Algebra

2021 ◽  
Author(s):  
Gennadiy Kondrat'ev
Keyword(s):  
Differential Geometry ◽  
Geometric Algebra ◽  
Clifford Algebras ◽  
Matrix Representation ◽  
Geometric Interpretation ◽  
Vector Spaces ◽  
Associative Algebras ◽  
Euclidean Manifold ◽  
Clifford Geometric Algebra ◽  
Algebraic Operations

The monograph is devoted to the fundamental aspects of geometric algebra and closely related issues. The category of Clifford algebras is considered as the conjugate category of vector spaces with a quadratic form. Possible constructions in this category and internal algebraic operations of an algebra with a geometric interpretation are studied. An application to the differential geometry of a Euclidean manifold based on a shape tensor is included. We consider products, coproducts and tensor products in the category of associative algebras with application to the decomposition of Clifford algebras into simple components. Spinors are introduced. Methods of matrix representation of the Clifford algebra are studied. It may be of interest to students, postgraduates and specialists in the field of mathematics, physics and cybernetics.

Series expansions for monogenic functions in Clifford algebras and their application

2020 ◽  
Vol 17 (3) ◽  
pp. 365-371
Author(s):  
Anatoliy Pogorui ◽  
Tamila Kolomiiets
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Vector Space ◽  
Clifford Algebra ◽  
Clifford Algebras ◽  
Monogenic Function ◽  
Second Order ◽  
Series Expansions ◽  
Monogenic Functions ◽  
Partial Differential

This paper deals with studying some properties of a monogenic function defined on a vector space with values in the Clifford algebra generated by the space. We provide some expansions of a monogenic function and consider its application to study solutions of second-order partial differential equations.

Isomorphisms of graded skew Clifford algebras

2020 ◽  
Vol 13 (5) ◽  
pp. 871-878
Author(s):  
Richard G. Chandler ◽  
Nicholas Engel
Keyword(s):  
Clifford Algebras

scholarly journals Geometric Algebra Framework Applied to Symmetrical Balanced Three-Phase Systems for Sinusoidal and Non-Sinusoidal Voltage Supply

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1259
Author(s):  
Francisco G. Montoya ◽  
Raúl Baños ◽  
Alfredo Alcayde ◽  
Francisco Manuel Arrabal-Campos ◽  
Javier Roldán Roldán Pérez
Keyword(s):  
Geometric Algebra ◽  
Dimensional Euclidean Space ◽  
Active Components ◽  
Electrical Circuits ◽  
Current Harmonics ◽  
Operation Conditions ◽  
Multiple Phase ◽  
Three Phase ◽  
Phase Systems ◽  
Definition Of

This paper presents a new framework based on geometric algebra (GA) to solve and analyse three-phase balanced electrical circuits under sinusoidal and non-sinusoidal conditions. The proposed approach is an exploratory application of the geometric algebra power theory (GAPoT) to multiple-phase systems. A definition of geometric apparent power for three-phase systems, that complies with the energy conservation principle, is also introduced. Power calculations are performed in a multi-dimensional Euclidean space where cross effects between voltage and current harmonics are taken into consideration. By using the proposed framework, the current can be easily geometrically decomposed into active- and non-active components for current compensation purposes. The paper includes detailed examples in which electrical circuits are solved and the results are analysed. This work is a first step towards a more advanced polyphase proposal that can be applied to systems under real operation conditions, where unbalance and asymmetry is considered.

scholarly journals Iterants, Majorana Fermions and the Majorana-Dirac Equation

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1373
Author(s):  
Louis H. Kauffman
Keyword(s):  
Dirac Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Clifford Algebras ◽  
Discrete Systems ◽  
Complex Numbers ◽  
Majorana Fermions ◽  
Matrix Algebras ◽  
Dirac Equations ◽  
Points Of View

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.

scholarly journals Invariants of Space Line Element Structure Based on Projective Geometric Algebra

IEEE Access ◽  
2020 ◽  
Vol 8 ◽  
pp. 62610-62618 ◽  
Author(s):  
Zhang Youzheng ◽  
Mui Yanping
Keyword(s):  
Geometric Algebra ◽  
Line Element ◽  
Space Line
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