Clifford Algebra

For the newcomer to CliffordAlgebra This is about mathematics which many people find so magical that they want to share it with everyone else. The problem that they have is of explaining clearly just what it is that is so special. People are also inclined to ask, "If it so good why did nobody tell me about it?" and that is a long story. My personal response when I found out about QuaternionMathematics about 15 years ago, was to ask, "What else have I not been told?" and I had to wait another 10 years or so to find the answer, CliffordAlgebra. I came back from a seminar and typed into a search engine "WilliamKingdonClifford" and entered in through a door from which I have never completely returned.

It is as though someone had a large mirror on which mathematics was written. It got broken and during the nineteenth century different mathematicians found different pieces of the mirror. They started to put the pieces together, but somehow they couldn't get them to fit and then they argued and threw some of the pieces into a dark cupboard. Only since about 1966 have different people started to find the pieces and fit them together, and we have for a hundred years been taught that the pieces are separate and unrelated. So we go on teaching the fragments, when we could teach the whole.

Hestenes, who started the "putting together" has now proposed that physics teaching should be reformed to use the whole pattern. (see HestenesOerstedMedalLecture)

For more on the history see HistoryOfVectorAnalysis and HistoryOfCliffordAlgebra

The late PerttiLounesto, author of CliffordAlgebrasAndSpinors http://users.tkk.fi/~ppuska/mirror/Lounesto/ (lots of stuff, e.g. links to dozens of researchers in applications of Clifford Algebra)

"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."
"Clifford algebra might make its curriculum debut in high schools by replacing the cross product, in undergraduate courses by replacing rotation matrices, and in graduate courses by condensing the Maxwell equations in a vacuum into a single equation in terms of Cl3."

See also GeometricAlgebra.

One very good introduction to the simplest Clifford algebra, exterior algebra, is a series of YouTube videos by David Metzler, available here: http://www.youtube.com/playlist?list=PLB8F2D70E034E9C29

For another good introduction see CliffordAlgebraaVisualIntroduction by Steven Lahar.

-- JohnFletcher

Note: Some authors talk about Geometric Algebra and some about Clifford Algebra. The usual use of this is that those who say Geometric Algebra are emphasising the geometric explanations. Hestenes (see HestenesOerstedMedalLecture) also uses the term Geometric Calculus. The algebra is the same, the difference is in emphasis on the different ways of looking at things. For more information look at this web page by Hestenes which gives a diagram of the evolution of the ideas. There is also a list of recent books. -- JohnFletcher

http://modelingnts.la.asu.edu/html/Evolution.html (broken link 20120711 but found using the WaybackMachine here: http://web.archive.org/web/20090603010039/http://modelingnts.la.asu.edu/html/evolution.html )

The use of two titles for the algebra are not new. CliffordAlgebrasAndSpinors (Lounesto) has the following p.320.

Clifford's geometric algebras were created by William K. Clifford in 1878/1882,... Clifford algebras were independently rediscovered by Lipschitz 1880/1886, who also presented their first application to geometry,...

-- JohnFletcher

Clifford Algebra Introduction

The essential core of CliffordAlgebra, and the main difference from vector algebra as taught, is that multiplying two vector objects a and b can be written as

  ab = a.b + ab

in a space of any number of dimensions. It is not assumed that

  ab = ba

and in general it will not be. Here the two parts are defined as

  a.b = (ab + ba)/2 = b.a
  ab = (ab - ba)/2 = -ba

a.b is a scalar quantity and is the equivalent of the vector dot product. If a.b = 0 for two nonzero vectors, they are orthogonal.

ab is known as the exterior product, defined in the work of Grassmann. It is related to, but not the same as, the vector cross product ab which is only defined in three dimensions. If ab = 0 for two nonzero vectors, then they are parallel.

Notice that this definition does not depend in any way on a choice of axes for the space.

(For a fuller description, see HestenesOerstedMedalLecture pages 9-14.)

-- JohnFletcher

Clifford Algebra in 2 dimensions

The power of CliffordAlgebra can be illustrated with the following, adapted from HestenesOerstedMedalLecture pages 10-11.

The algebra describes a plane containing two orthogonal unit vectors e1 and e2 which each square to +1

  e1^2 = e2^2 = 1

A unit bivector i for the plane is given by the product

  i = e1 e2 = e1e2 = -e2 e1

since

  e1.e2 = 0

The symbol i is chosen because

  i^2 = (e1 e2)(e1 e2) = -(e1 e2)(e2 e1) = -1

and thus i is a number which squares to -1. Also

  e1 = i e2
  e2 = e1 i = -i e1

Multiplication by i rotates any vector in the plane by a right angle. If a complex number z is written using this i, where a and b are scalars,

  z = a + b i

then multiplying z by e1 gives the following

  z e1 = a e1 + b e1 i = a e1 + b e2

One interpretation of this is that there is a length a along e1 followed by a length b along e1 rotated by a right angle, viz e2. Another interpretation is that ComplexNumbersArePoints once a direction is chosen for the real axis.

For me this is the beginning of the healing of the broken algebra - complex numbers and 2D vectors are rejoined. -- JohnFletcher

Clifford Algebra in 3 dimensions

A basis for the whole algebra in three dimensions can be generated from three orthogonal unit vectors e1, e2 and e3 which each square to +1.

This basis generates a unique trivector which is the pseudoscalar i for the basis. This represents an oriented unit volume.

  i = e1 e2 e3

There are three bivectors in the basis. These represent oriented unit areas.

  e1 e2 = i e3
  e2 e3 = i e1
  e3 e1 = i e2

In these equations the multiplication by i is called the dual operation. In this case the dual of a vector is a bivector and viceversa. In three dimensions the dual of an area is the vector normal to the plane.

Note that all the bivectors and the trivector square to -1.

The outer product and the vector cross product can now be related through a dual operation.

  ab = i ab

This is a key result which can be used to unify CliffordAlgebra and VectorAlgebra? in 3 dimensions. What this means is that a

b is the vector normal to the bivector a

b which defines a plane. When formulating equations in GeometricAlgebra of 3 dimensions, this is the crucial difference. Bivectors are distinct items from vectors in any object.

[edit notes] This is quite a long and important topic and will take some time. If you want to find out more without waiting, look at the HestenesOerstedMedalLecture from page 14 onwards. -- JohnFletcher

CliffordAlgebraDetails where notation used here is defined.
Clifford Algebra Examples
- CliffordAlgebraIdempotents
- CliffordAlgebraInverseDiscussion which will mature as CliffordAlgebraInverse? sometime soon.
- more examples to be added
CliffordAlgebraComputation
CliffordAlgebraApplications
CliffordAlgebraDiscussion
CliffordAlgebraResources (External Links)

Some other wiki pages which are related.

GeometricAlgebra (External links now refactored to CliffordAlgebraResources)
ComplexNumbers
ComplexNumbersArePoints
ComplexAnalysis
QuaternionMathematics
MatrixAnalysis