Clifford algebra
Noun
Meaning
Clifford algebra (plural Clifford algebras)
(algebra, mathematical physics) A unital associative algebra which generalizes the algebra of quaternions but which is not necessarily a division algebra; it is generated by a set of
γ
i
{\displaystyle \gamma _{i}}
(with i ranging from, say, 1 to n) such that the square of each
γ
i
{\displaystyle \gamma _{i}}
is fixed to be either +1 or −1, depending on each i, and such that any product
γ
i
γ
j
{\displaystyle \gamma _{i}\gamma _{j}}
anticommutes when its factors are distinct (i.e., when
i
≠
j
{\displaystyle i\neq j}
).
Examples
Eddington believed he had identified an algebraic basis for fundamental physics, which he termed "E-numbers" (representing a certain group – a Clifford algebra ).
Component spinors Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra Cℓ(V, g) can be defined as follows.
In this way, the spinor representations are identified with certain subspaces of the Clifford algebra itself.
Geometric algebra is distinguished from Clifford algebra in general by its restriction to real numbers and its emphasis on its geometric interpretation and physical applications.
In 1999 he showed how Einstein's equations of general relativity could be formulated within a Clifford algebra that is directly linked to quaternions.
Namely, the norm operation in a certain type of algebraic system (now known as a Clifford algebra ).