A Clifford algebra is an associative algebra associated with an vector space field quadratic form
(1)
for every multiplicative identity .
More precisely, the Clifford algebra is the quotient tensor algebra ideal field characteristic
of
(2)
where symmetric bilinear form associated
with
For Euclidean space and orthonormal
basis
for vector basis of the
Clifford algebra. Its dimension is therefore
Clifford algebras are generally not commutative . When exterior
algebra of
Clifford algebras are used to define spinors .
In the Wolfram Language , CliffordAlgebra NonCommutativeExpand canonical form .
See also Algebra ,
Dirac Matrices ,
Exterior Algebra ,
Geometric
Algebra ,
Hypercomplex Number ,
Quadratic
Form ,
Quaternion ,
Spinor ,
Spinor Field ,
Tensor
Algebra ,
Vector Space
Portions of this entry contributed by Todd
Rowland
Explore with Wolfram|Alpha
References Abłamowicz, R. "Hecke Algebra, SVD, and Other Computational Examples with CLIFFORD." 14 Oct 1999. https://arxiv.org/abs/math/9910069 . Ablamowicz,
R.; Lounesto, P.; and Parra, J. M. Clifford
Algebras with Numeric and Symbolic Computations. Huang, J.-S. "The Clifford Algebra." §6.2 in Lectures
on Representation Theory. Iyanaga,
S. and Kawada, Y. (Eds.). "Clifford Algebras." §64 in Encyclopedic
Dictionary of Mathematics. Lawson,
H. B., Jr. and Michelsohn, M.-L. Spin
Geometry. Lounesto,
P. Clifford
Algebras and Spinors, 2nd ed. Lounesto, P. "Counterexamples to Theorems Published and Proved
in Recent Literature on Clifford Algebras, Spinors, Spin Groups, and the Exterior
Algebra." http://www.helsinki.fi/~lounesto/counterexamples.htm . Penrose,
R. §11.5 in Road
to Reality: A Complete Guide to the Laws of the Universe. Referenced on Wolfram|Alpha Clifford Algebra
Cite this as:
Weisstein, Eric W. , with contributions by Todd Rowland . "Clifford Algebra." From MathWorld https://mathworld.wolfram.com/CliffordAlgebra.html
Subject classifications
-dimensional
vector space 
over a field 
and a quadratic form 
on 
. In the sign convention used here, it is generated by 
subject to the multiplication rule
,
where 
is the multiplicative identity.
of the tensor algebra 
by the two-sided ideal 
generated by 
for 
. When the field characteristic
of 
is not 2, the defining rule also gives
![B(v,w)=[Q(v+w)-Q(v)-Q(w)]/2](/images/equations/CliffordAlgebra/Inline15.svg)
is the symmetric bilinear form associated
with 
.
with 
an Euclidean space and 
, an orthonormal
basis 
satisfies
.
The products 
with 
,
together with the empty product 
, form a vector basis of the
Clifford algebra. Its dimension is therefore 
.
is identically zero, the Clifford algebra is the exterior
algebra of 
.
, 
, or 0. NonCommutativeExpand
puts expressions in these algebras into canonical form.
