Clifford Algebra

Clifford Algebra

In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry and theoretical physics. They are named after the English geometer William Kingdon Clifford.

The most familiar Clifford algebra, or orthogonal Clifford algebra, is also referred to as Riemannian Clifford algebra.

Read more about Clifford Algebra:  Introduction and Basic Properties, Universal Property and Construction, Basis and Dimension, Examples: Real and Complex Clifford Algebras, Structure of Clifford Algebras, The Clifford Group Γ, Spin and Pin Groups, Spinors

Other articles related to "clifford algebra, clifford algebras, clifford, algebra":

Clifford Algebra - Applications - Computer Vision
... Recently, Clifford algebras have been applied in the problem of action recognition and classification in computer vision ... propose a Clifford embedding to generalize traditional MACH filters to video (3D spatiotemporal volume), and vector-valued data such as optical flow ... Vector-valued data is analyzed using the Clifford Fourier transform ...
Algebra Of Physical Space
... In physics, the algebra of physical space (APS) is the use of the Clifford or geometric algebra Cℓ3 of the three-dimensional Euclidean space as a model for (3+1)-dimensional space-time, representing a point in ... The Clifford algebra Cℓ3 has a faithful representation, generated by Pauli matrices, on the spin representation C2 further, Cℓ3 is isomorphic to the even subalgebra of the 3+1 Clifford algebra, Cℓ0 3,1 ... APS should not be confused with spacetime algebra (STA), which concerns the Clifford algebra Cℓ1,3(R) of the four dimensional Minkowski spacetime ...
Higher-dimensional Supergravity - Choosing Chiralities
... only of the n-dimensional Lorentz group, but also of a Lie algebra called the n-dimensional Clifford algebra ... The most commonly used basis of the complex -dimensional representation of the Clifford algebra, the representation that acts on the Dirac spinors, consists of the gamma matrices ... to as as it was first considered in the case n = 4, is not itself a member of the Clifford algebra ...
David Hestenes - Work - Geometric Algebra and Calculus
... Spacetime algebra provided the starting point for two main lines of research on its implications for quantum mechanics specifically and for mathematical physics generally ... The first line began with the fact that reformulation of the Dirac equation in terms of spacetime algebra reveals hidden geometric structure Among other ... The second line of research was dedicated to extending geometric algebra to a self-contained geometric calculus for use in theoretical physics ...
Split-complex Number - Algebraic Properties
... In abstract algebra terms, the split-complex numbers can be described as the quotient of the polynomial ring R by the ideal generated by the polynomial x2 − 1, R/(x2 − 1) ... split-complex numbers actually form a commutative and associative algebra over the reals of dimension two ... The algebra is not a division algebra or field since the null elements are not invertible ...

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