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Clifford algebras and spin representations I
Clifford Algebras and Spinors
Pertti Lounesto My research focuses on algebras emerging from problems in geometry and physics, called Clifford algebras. In physics, the concept of Clifford algebra, as such or in a disguise, is a necessity in the description of electron spin, because spinors cannot be constructed by tensorial methods, in terms of exterior powers of the vector space. In geometry, information about orientation of subspaces can be encoded in simple multivectors, which can be added and multiplied. Physicists are familiar with this tool in the special case of one-dimensional subspaces, which they manipulate by vectors not by projection operators, which lose information about orientations. I am also interested in misconceptions of research mathematicians , while they enter unexplored domains.
Please direct questions, comments or concerns to feedback inspirehep. Information References 48 Citations 1 Files Plots. Clifford Algebras 21 Update these references. Is quantum space-time infinite dimensional? The extended relativity theory in Born-Clifford phase spaces with a lower and upper length scales and Clifford group geometric unification - Castro, C.
A historical review of spinors is given together with a construction of spinor spaces as minimal left ideals of Clifford algebras. Spinor spaces of euclidean spaces over reals have a natural linear structure over reals, complex numbers or quaternions. Clifford algebras have involutions which induce bilinear forms or scalar products on spinor spaces. The automorphism groups of these scalar products of spinors are determined and also classified. Unable to display preview.
The geometry of 3-dimensional space is traditionally described with the help of the scalar product and the cross product.
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In mathematics , a Clifford algebra is an algebra generated by a vector space with a quadratic form , and is a unital associative algebra. As K -algebras , they generalize the real numbers , complex numbers , quaternions and several other hypercomplex number systems. Clifford algebras have important applications in a variety of fields including geometry , theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford. The most familiar Clifford algebras, the orthogonal Clifford algebras , are also referred to as pseudo- Riemannian Clifford algebras , as distinct from symplectic Clifford algebras. The idea of being the "freest" or "most general" algebra subject to this identity can be formally expressed through the notion of a universal property , as done below.