# Clifford algebras and geometric algebra

@article{Aragon1997CliffordAA, title={Clifford algebras and geometric algebra}, author={G. Aragon and Jos{\'e} Luis Arag{\'o}n and M. A. Rodŕıguez}, journal={Advances in Applied Clifford Algebras}, year={1997}, volume={7}, pages={91-102} }

The geometric algebra as defined by D. Hestenes is compared with a constructive definition of Clifford algebras. Both approaches are discussed and the equivalence between a finite geometric algebra and the universal Clifford algebra Rp, q is shown. Also an intermediate way to construct Clifford algebras is sketched. This attempt to conciliate two separated approaches may be useful taking into account the recognized importance of Clifford algebras in theoretical and applied physics.

## 10 Citations

### Geometry of complex data

- MathematicsIEEE Aerospace and Electronic Systems Magazine
- 2016

This tutorial provides a basic introduction to geometric algebra and presents formulations of known electrical engineering and signal processing concepts to illustrate some inherent advantages of geometric algebra for formulating and solving problems involving vectors.

### Constructing a “string theory” through the use of geometric algebra

- Physics
- 2009

String theory has allowed us to make advances in theoretical physics by exchanging point particles for one-dimensionally extended ‘strings’ as elementary object; for example string theories are the…

### Fourvector algebra

- Physics
- 2007

The algebra of fourvectors is described. The fourvectors are more appropriate than the Hamilton quaternions for its use in Physics and the sciences in general. The fourvectors embrace the 3D vectors…

### A Proposal of an Algebra for Vectors and an Application to Electromagnetism

- Mathematics, Physics
- 2014

A new mathematical structure intended to formalize the classical 3D and 4D vectors is briefly described. This structure is offered to the investigators as a tool that bears the potential of being…

### A pr 2 02 1 Gravity as a ‘ Third ’ Quantization Gauge Theory with Minkowski Metric , Non-Zero Torsion and Curvature

- Physics
- 2021

It is understood that, in general, a global and unique vacuum state cannot be considered in a curved space. Hence, we have introduced a geometry that, in addition of having non-zero local curvature…

### A Different Polar Representation for Generalized and Generalized Dual Quaternions

- MathematicsAdvances in Applied Clifford Algebras
- 2018

In this paper, we present a new polar representation for generalized quaternions and generalized dual quaternions. In this different polar form representation, as in the Cayley–Dickson form, a…

### A `Third' Quantization Constructed for Gauge Theory of Gravity

- Physics
- 2021

In general, a global and unique vacuum state cannot be constructed for a curved space. As a remedy, we introduce a curved space background geometry with a Minkowski metric tensor and locally nonzero…

### Visual Servoing on the Sphere Using Conformal Geometric Algebra

- Mathematics
- 2013

This paper deals with a visual servoing scheme, which uses a paracatadioptric sensor as visual input. The paracatadioptric sensor provides a wide field of view while maintaining the single center of…

### Pattern Classification Approaches for Breast Cancer Identification via MRI: State-Of-The-Art and Vision for the Future

- Computer ScienceApplied Sciences
- 2020

The proposed data fusion approaches provide a novel Artificial Intelligence (AI) based framework for more robust image registration that can potentially advance the early identification of heterogeneous tumour types, even when the associated imaged organs are registered as separate entities embedded in more complex geometric spaces.

## References

SHOWING 1-8 OF 8 REFERENCES

### Dirac equation and Clifford algebra

- Mathematics
- 1994

The Dirac equation is rewritten in the framework of a Clifford algebra, revealing some new features, of which the most interesting is the U(1)×SU(2) symmetry of the corresponding Lagrangian.

### Imaginary numbers are not real—The geometric algebra of spacetime

- Mathematics
- 1993

This paper contains a tutorial introduction to the ideas of geometric algebra, concentrating on its physical applications. We show how the definition of a “geometric product” of vectors in 2-and…

### Space-time algebra

- Mathematics, Physics
- 1966

Preface to the Second Edition.- Introduction.- Part I:Geometric Algebra.- 1.Intrepretation of Clifford Algebra.- 2.Definition of Clifford Algebra.- 3.Inner and Outer Products.- 4.Structure of…

### New Foundations for Classical Mechanics

- Physics
- 1986

1: Origins of Geometric Algebra.- 1-1. Geometry as Physics.- 1-2. Number and Magnitude.- 1-3. Directed Numbers.- 1-4. The Inner Product.- 1-5. The Outer Product.- 1-6. Synthesis and Simplification.-…

### The applications of Clifford algebras to crystallography using Mathematica

- Materials Science
- 1996

The discovery of materials such as incommensurate structures and quasicrystals makes necessary to extend crystallography to more than three dimensions. In this work, we explore the use of Clifford…