Spinors, Clifford, and Cayley Algebras

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We remark that all properties of the inner product are fulfilled. The matrix associated to the quadratic form q H n R is. We obtain that the quadratic form q H n R is positive definite and the Clifford algebra C H n R associated to the tensor algebra T H n R is isomorphic to Cl 2 , 0 K which is isomorphic to a split quaternion algebra. Theorem 3.

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We obtain that the quadratic form q H n R is positive definite and the Clifford algebra C H n R associated to the tensor algebra T H n R is isomorphic with Cl 2 , 0 K which is isomorphic to a split quaternion algebra. Example 3. The quadratic form q H n R is negative definite, therefore the Clifford algebra C H n R associated to the tensor algebra T H n R is isomorphic to Cl 0 , 2 K which is isomorphic to the quaternion division algebra H 1 , 1. We also have given an algorithm which allows us to find a division quaternion algebra starting from a split quaternion algebra and vice versa.

Estadual de Campinas, Sao Paulo; Clifford Algebras , 9 1 Clifford Algebras , 20 2 Springer, Berlin; Smith TL: Decomposition of generalized Clifford algebras. Clifford Algebras , 23 1 Academic Press, New York; Wiley-Interscience, New York; Horadam AF: A generalized Fibonacci sequence. Clifford Algebras , 23 3 Download references. I want thank the anonymous referees for their remarkable comments, suggestions and ideas which helped me to improve this paper.

Since V comes equipped with a quadratic form, there is a set of privileged bases for V : the orthogonal ones. An orthogonal basis is one such that. The fundamental Clifford identity implies that for an orthogonal basis. This makes manipulation of orthogonal basis vectors quite simple. Given a product of distinct orthogonal basis vectors of V , one can put them into standard order while including an overall sign determined by the number of pairwise swaps needed to do so i.

The most important Clifford algebras are those over real and complex vector spaces equipped with nondegenerate quadratic forms. For a complete classification of these algebras see classification of Clifford algebras. The geometric interpretation of real Clifford algebras is known as geometric algebra. Every nondegenerate quadratic form on a finite-dimensional real vector space is equivalent to the standard diagonal form:. The pair of integers p , q is called the signature of the quadratic form. The real vector space with this quadratic form is often denoted R p , q.

One can also study Clifford algebras on complex vector spaces. Every nondegenerate quadratic form on a complex vector space is equivalent to the standard diagonal form. Let the vector space V be real three dimensional space R 3 , and the quadratic form Q be derived from the usual Euclidean metric. Then, for v , w in R 3 we have the quadratic form, or scalar product,.

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Now introduce the Clifford product of vectors v and w given by. This formulation uses the negative sign so the correspondence with quaternions is easily shown. Denote a set of orthogonal unit vectors of R 3 as e 1 , e 2 , and e 3 , then the Clifford product yields the relations. The basis elements can be identified with the quaternion basis elements i , j , k as.

In this section, dual quaternions are constructed as the even Clifford algebra of real four dimensional space with a degenerate quadratic form. Let the vector space V be real four dimensional space R 4 , and let the quadratic form Q be a degenerate form derived from the Euclidean metric on R 3.

For v , w in R 4 introduce the degenerate bilinear form. This degenerate scalar product projects distance measurements in R 4 onto the R 3 hyperplane. The Clifford product of vectors v and w is given by.

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Denote a set of orthogonal unit vectors of R 4 as e 1 , e 2 , e 3 and e 4 , then the Clifford product yields the relations. One can show that this map is independent of the choice of orthogonal basis and so gives a natural isomorphism. If the characteristic of K is 0, one can also establish the isomorphism by antisymmetrizing. This map can be shown to be a linear isomorphism, and it is natural. The associated graded algebra. In the following, assume that the characteristic is not 2.

Clifford Algebra

Clifford algebras are Z 2 - graded algebras also known as superalgebras. This Z 2 -grading plays an important role in the analysis and application of Clifford algebras. Elements that are pure in this Z 2 -grading are simply said to be even or odd. The degree of a Clifford number usually refers to the degree in the N -grading.

In particular over the reals this implies that. Recall that the tensor algebra T V comes with an antiautomorphism that reverses the order in all products:. We call this operation Clifford conjugation denoted. Of the two antiautomorphisms, the transpose is the more fundamental. Note that all of these operations are involutions. In fact, all three operations depend only on the degree modulo 4. That is, if x is pure with degree k then. A basis independent definition of one such extension is.

Clifford Algebra Techniques in Linear Algebra

One can show that. One can check that this reduces to the original bilinear form when restricted to V. That is,. In this section we assume that the vector space V is finite dimensional and that the bilinear form of Q is non-singular. A central simple algebra over K is a matrix algebra over a finite dimensional division algebra with center K. For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions.


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The structure of Clifford algebras can be worked out explicitly using the following result. Goodreads helps you keep track of books you want to read. Want to Read saving…. Want to Read Currently Reading Read. Other editions. Enlarge cover. Error rating book. Refresh and try again.

Spinors, Clifford and Cayley Algebras - Robert Hermann - Google книги

Open Preview See a Problem? Details if other :. Thanks for telling us about the problem. Return to Book Page. Clifford Algebras and Spinors by Pertti Lounesto ,. Cassels Editor.


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  • Hitchin Editor. This second edition of a popular and unique introduction to Clifford algebras and spinors has three new chapters.

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    The beginning chapters cover the basics: vectors, complex numbers and quaternions are introduced with an eye on Clifford algebras. The next chapters, which will also interest physicists, include treatments of the quantum mechanics of the electron, electromagnet This second edition of a popular and unique introduction to Clifford algebras and spinors has three new chapters. The next chapters, which will also interest physicists, include treatments of the quantum mechanics of the electron, electromagnetism and special relativity. A new classification of spinors is introduced, based on bilinear covariants of physical observables.

    This reveals a new class of spinors, residing among the Weyl, Majorana and Dirac spinors. Scalar products of spinors are categorized by involutory anti-automorphisms of Clifford algebras.

    This leads to the chessboard of automorphism groups of scalar products of spinors.