A geometric algebra is the Clifford algebra of a vector space over the field of real numbers endowed with a quadratic form.

Given a finite dimensional real quadratic space with quadratic form, the geometric algebra for this quadratic space is the Clifford algebra Cℓ ( V, Q ).

Moreover, the algebra generated by the three matrices is isomorphic to the 3-dimensional Euclidean real Clifford Algebra.

In detail, if V is a finite-dimensional complex vector space with nondegenerate bilinear form g, the Clifford algebra is the algebra generated by V along with the anticommutation relation.

The Clifford algebra Cℓ < sub > n </ sub >( C ) is algebraically isomorphic to the algebra of complex matrices, if is even ; or the algebra of two copies of the matrices, if is odd.

The Lie algebra is embedded as a Lie subalgebra in equipped with the Clifford algebra commutator as Lie bracket.

Irreducible representations over the reals in the case when V is a real vector space are much more intricate, and the reader is referred to the Clifford algebra article for more details.

The most typical type of spinor, the Dirac spinor, is an element of the fundamental representation of the complexified Clifford algebra, into which the spin group Spin ( p, q ) may be embedded.

Some simple examples of spinors in low dimensions arise from considering the even-graded subalgebras of the Clifford algebra.

The Clifford algebra Cℓ < sub > 2, 0 </ sub >( R ) is built up from a basis of one unit scalar, 1, two orthogonal unit vectors, σ < sub > 1 </ sub > and σ < sub > 2 </ sub >, and one unit pseudoscalar.

As a real algebra, Cℓ < sup > 0 </ sup >< sub > 2, 0 </ sub >( R ) is isomorphic to field of complex numbers C. As a result, it admits a conjugation operation ( analogous to complex conjugation ), sometimes called the reverse of a Clifford element, defined by

The Clifford algebra Cℓ < sub > 3, 0 </ sub >( R ) is built up from a basis of one unit scalar, 1, three orthogonal unit vectors, σ < sub > 1 </ sub >, σ < sub > 2 </ sub > and σ < sub > 3 </ sub >, the three unit bivectors σ < sub > 1 </ sub > σ < sub > 2 </ sub >, σ < sub > 2 </ sub > σ < sub > 3 </ sub >, σ < sub > 3 </ sub > σ < sub > 1 </ sub > and the pseudoscalar i

equivalence of these constructions are a consequence of the uniqueness of the spinor representation of the complex Clifford algebra.

Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra can be defined as follows.

Then the assignment extends uniquely to an algebra homomorphism by sending the monomial in the Clifford algebra to the product of matrices and extending linearly.

In this construction the representation of the Clifford algebra Cℓ ( V, g ), the Lie algebra so ( V, g ), and the Spin group Spin ( V, g ), all depend on the choice of the orthonormal basis and the choice of the gamma matrices.

These are subspaces of the Clifford algebra of the form Cℓ ( V, g ) ω, admitting the evident action of Cℓ ( V, g ) by left-multiplication: c: xω → cxω.

The action of an even Clifford element on vectors, regarded as 1-graded elements of Cℓ < sub > 2, 0 </ sub >, is determined by mapping a general vector to the vector

Cℓ < sup > odd </ sup > ω on restriction to the action of the even Clifford algebra.

The Clifford algebra Cℓ ( V, Q ) is the " freest " algebra generated by V subject to the condition < ref > Mathematicians who work with real Clifford algebras and prefer positive definite quadratic forms ( especially those working in index theory ) sometimes use a different choice of sign in the fundamental Clifford identity.

In fact, if then the Clifford algebra Cℓ ( V, Q ) is just the exterior algebra Λ ( V ).

It is then straightforward to show that Cℓ ( V, Q ) contains V and satisfies the above universal property, so that Cℓ is unique up to a unique isomorphism ; thus one speaks of " the " Clifford algebra Cℓ ( V, Q ).

The universal characterization of the Clifford algebra shows that the construction of Cℓ ( V, Q ) is functorial in nature.

In mathematics, the split-complex numbers are members of the Clifford algebra Cℓ < sub > 1, 0 </ sub >( R ) = Cℓ < sup > 0 </ sup >< sub > 1, 1 </ sub >( R ) ( the superscript 0 indicating the even subalgebra ).

Below it is shown that distinct Clifford algebras may be algebra isomorphic, as is the case of Cℓ < sub > 2, 0 </ sub >( R ) and Cℓ < sub > 1, 1 </ sub >( R ) which are both isomorphic to the ring of two-by-two matrices over the real numbers.

We will denote the Clifford algebra on C < sup > n </ sup > with the standard quadratic form by Cℓ < sub > n </ sub >( C ).

In physics, the algebra of physical space ( APS ) is the use of the Clifford or geometric algebra Cℓ < sub > 3 </ sub > of the three-dimensional Euclidean space as a model for ( 3 + 1 )- dimensional space-time, representing a point in space-time via a paravector ( 3-dimensional vector plus a 1-dimensional scalar ).

The second approach is to construct a vector space using a distinguished subspace of V, and then specify the action of the Clifford algebra externally to that vector space.

Specifically, a Clifford algebra is a unital associative algebra which contains and is generated by a vector space V equipped with a quadratic form Q.

The definition of a Clifford algebra endows it with more structure than a " bare " K-algebra: specifically it has a designated or privileged subspace that is isomorphic to V. Such a subspace cannot in general be uniquely determined given only a K-algebra isomorphic to the Clifford algebra.

A Clifford algebra as described above always exists and can be constructed as follows: start with the most general algebra that contains V, namely the tensor algebra T ( V ), and then enforce the fundamental identity by taking a suitable quotient.

Clifford multiplication together with the privileged subspace is strictly richer than the exterior product since it makes use of the extra information provided by Q.

Further, the isometry groups of Q and − Q are the same ( O ( p, q ) ≈ O ( q, p )), but the associated Clifford algebras ( and hence Pin groups ) are different.

In general a Clifford algebra C is a central simple algebra over some field extension L of the field K over which the quadratic form Q defining C is defined.

Sandman said: `` The announcement that Sen. Clifford Case Aj, has decided to spend all his available time campaigning for Mr. Mitchell is a dead giveaway.

This expression is related to the development of Bessel functions in terms of the Bessel – Clifford function.

He used this rather disparaging term in his 1830 novel Paul Clifford: He is certainly a man who bathes and ‘ lives cleanly ’, ( two especial charges preferred against him by Messrs. the Great Unwashed ).

Bulwer-Lytton's name lives on in the annual Bulwer-Lytton Fiction Contest, in which contestants think-up terrible openings for imaginary novels, inspired by the first line of his novel Paul Clifford: It was a dark and stormy night ; the rain fell in torrents — except at occasional intervals, when it was checked by a violent gust of wind which swept up the streets ( for it is in London that our scene lies ), rattling along the housetops, and fiercely agitating the scanty flame of the lamps that struggled against the darkness.

Clifford Cocks, an English mathematician working for the UK intelligence agency GCHQ, described an equivalent system in an internal document in 1973, but given the relatively expensive computers needed to implement it at the time, it was mostly considered a curiosity and, as far as is publicly known, was never deployed.

He was apparently of above average height: according to Clifford Brewer he was but his remains have been lost since at least the French Revolution, and his exact height is unknown.

The contemporary historians Clifford Brown and Carol Wilson believe it is likely that he died of natural causes.

It therefore has a unique irreducible representation ( also called simple Clifford module ), commonly denoted by Δ, whose dimension is 2 < sup > k </ sup >.

where γ < sup >*</ sup > is the conjugate of γ, and the product is Clifford multiplication.