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A Clifford algebra is a unital associative algebra that contains and is generated by a vector space V over a field K, where V is equipped with a quadratic form Q : V → K.The Clifford algebra Cl(V, Q) is the "freest" unital associative algebra generated by V subject to the condition [c] = , where the product on the left is that of the algebra, and the 1 on the right is the algebra's ...
In mathematics, a geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors ...
In even dimension n, the Clifford algebra Cl n (C) is isomorphic to End(C N), which has its fundamental representation on Δ n := C N. A complex Dirac spinor is an element of Δ n. The term complex signifies that it is the element of a representation space of a complex Clifford algebra, rather than that is an element of a complex vector space.
In Clifford analysis one also considers differential operators on upper half space, the disc, or hyperbola with respect to the hyperbolic, or Poincaré metric. For upper half space one splits the Clifford algebra, Cl n into Cl n−1 + Cl n−1 e n. So for a in Cl n one may express a as b + ce n with a, b in Cl n−1.
Animation of the Bott periodicity clock using a Mod 8 clock face with second hand mnemonics taken from the I-Ching with the real Clifford algebra of signature (p,q) denoted as Cl p,q ()=Cl(p,q). As they are 2-periodic/8-periodic, they can be arranged in a circle, where they are called the Bott periodicity clock and Clifford algebra clock .
In mathematics, a Clifford bundle is an algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the algebra structure. There is a natural Clifford bundle associated to any ( pseudo ) Riemannian manifold M which is called the Clifford bundle of M .
Basil James Hiley (15 November 1935 – 25 January 2025) was a British physicist and professor emeritus of the University of London.. Long-time colleague of David Bohm, Hiley is known for his work with Bohm on implicate orders and for his work on algebraic descriptions of quantum mechanics in terms of underlying symplectic and orthogonal Clifford algebras. [1]
The real Clifford algebra and its complexification act on the complexification of A, an N-dimensional complex space. If N is even, N − 1 is odd, so the Clifford algebra has exactly two complex irreducible representations of dimension 2 N/2 − 1. So this power of 2 must divide N. It is easy to see that this implies N can only be 1, 2, 4 or 8.