Clifford Algebra Details

CliffordAlgebraDetails is a page with some of the detailed stuff from CliffordAlgebra. This needs more attention which I am giving it bit by bit. -- JohnFletcher

Some Information

There is not one CliffordAlgebra, there are many, dependent on the choice of the dimension of the space for the algebra.
If the dimension is n, then there will be n independent basis vectors in the algebra.
Each algebra will have a total number of terms 2^n.
The algebra is generated by the products of distinct basis vectors.
- The product of no basis vectors is the scalar 1. This is called grade 0.
- The product of one basis vector is the vector itself. This is called grade 1.
- The product of two orthogonal distinct basis vectors is a bivector. This is called grade 2.
- The number of distinct products at each grade is determined by the binomial theorem.
- There is one object which is the product of all the distinct basis vectors. This is called the pseudoscalar and has grade n.
Nonzero objects in the algebra can be made up by adding terms which are scalar multiples of any of the terms, drawn from any of the grades.
- Note that this violates the principle in vector analysis that a scalar cannot be added to a vector.
- In practice two common sorts of object are vectors (all grade 1) and objects which combine a scalar and a bivector.

Notation for the basis vectors

For some applications there is no need to specify particular basis vectors.
In other applications there is need to have a set of n specified basis vectors.
- These are usually orthogonal unit vectors.
- Note that although in ordinary vector algebra all unit vectors square to +1, in CliffordAlgebra some or all may square to -1.
- It is part of the distinction made between CliffordAlgebra and geometric algebra that there is a physical significance to objects which square to -1.
- The number of positive square and negative square basis vectors adds up to the dimension of the algebra.
There are a number of different notations in use for these vectors. This is partly history and partly the preferences of different authors.
- Some authors use the letters i,j and k for the first three basis vectors.
- In another convention, the basis vectors with positive square are labelled as e1 e2 etc and the ones with negative square as f1 f2 etc.
- Various special notations are used for the pseudovector, such as i or I. Care must be taken. In some dimensions it squares to +1 and in some to -1.
- Multiplication of a vector by the pseudovector is called forming the dual. The grade of the object formed is n-1.
  - example in 3 dimensions, the dual of a vector (grade 1) is a bivector (grade 2) and vice-versa.

Geometrical Interpretation

It is sometimes reasonable to regard basis vectors that square to -1 as indicating negative curvature in the dimensions in question. See also WickRotation (e.g. http://en.wikipedia.org/wiki/Wick_rotation and discussion on blog of string theorist Luboš Motl (pronounce: "Loo-bosch Maw-tull") at http://motls.blogspot.com/2005/02/wick-rotation.html, and a different blog with discussion of Motl at http://www.math.columbia.edu/~woit/blog/archives/000160.html which includes the amusing comment, "Do you think it's statistically significant that the two most prominent string theorists with weblogs are both incredibly arrogant and incapable of admitting that anyone who disagrees with them might have a point?" -- DougMerritt

This part needs revision as there are some errors in it. -- JohnFletcher

1D Clifford Algebra. You get reals (1) and 1D unit vectors (i). 1*1=1, i*i=1, i*1=i. Not very interesting.
2D Clifford Algebra. You get reals (1), two orthogonal unit vectors (i and j), and imaginaries or pseudovectors (i). 1*1=1, 1*i=i, i*i=-1, i*i=1, i*j=i etc.
3D Clifford Algebra. You get reals, three orthogonal unit vectors, the i, j and k of quaternions, and the unit volume element, whose symbol I can't remember.
4D Clifford Algebra, where the fourth dimension is time and doesn't quite behave like all the others. This is completely out of control, as far as I can remember.

Can someone who remembers all this better than I do, fill in the details? There was something terribly clever about expressing all Maxwell's equations in four symbols, or something. -- PeterHartley

The problem is that for each "dimensional" entry in the table above, there are several choices.

Cl(2) is actually isomorphic to the quaternions, not Cl(3), though Cl(3) has several sub-algebras isomorphic to the quaternions. Of the 1-d Clifford algebras, one is described above, and the other is isomorphic to the complex numbers.

Some general comments not yet revised, was originally on CliffordAlgebra.

The central definition in the algebra is multiplication. For three dimensions, this unifies the dot product and cross product of Gibbs vector theory into one operation.
Terms involving two basis vectors are also called bivectors.
The term which is the product of all the basis vectors forms a pseudoscalar.
In simple systems, the basis vectors will be orthogonal, but this need not be so.
For orthogonal systems, each basis vector will have an old style dot product of zero with all of the others.
This is where it gets interesting The dot product of any basis vector with itself can be either +1 or -1. This means that there are several different algebras at each value of n, depending on the choices made.
So, in the table above, at n = 4 we can have
Clifford (4) - all positive, four dimensional algebra.
Clifford (3,1) - 3 positive, one negative. This is one used in physics a lot.
Clifford (2,2) - 2 positive, 2 negative. This is called a balanced algebra.
Clifford (1,3) - 1 positive, 3 negative, also used in physics.
Clifford (0,4) - 4 negative.