Wacker Art Clifford Algebra Wappen der Familie Wacker
Edersee - Fähre Scheid-Rehbach
Bild: "Edersee - Fähre Schein-Rehbach"

Prolog

"... geometry, you know, is the gate of science, and the gate is so low and small that one can only enter it as a little child."

William Kingdon Clifford


In mathematics the name Clifford Algebra is used but Clifford himself named his algebra Geometric Algebra. For an Overview on the subject visit my Geometric Algebra page. On this page it is assumed that the reader is familiar with the basic concepts of geometric algebra.

Edersee
Bild: "Edersee"

Vectors involved in the Polarisation Identity

Clifford Algebra

The Inner Product

We start with the geometric product of the two vectors u and v:

Geometric Product

Exchanging the vectors u and v gives the relation:

Geometric Product Commutated

By adding the two equations we get the scalar product expressed with the help of the geometric product:

Scalar Product

Or in a more general form:

Scalar Product general Form.

Building the square of a vector with the geometic product we get:

Quadratic Form

Which is the scalar product of the vector. That means we get the square length of the vector.

The relation of geometric products of the ortogonormal basic vectors, can be written with the help of geometic products as follows:

Clifford Product

Or in a more general form with the metric tensor when the base vectors are not orthogonal.

Metric Tensor

If we square the sum of two vectors (using the geometric product) we get:

Sum of two vectors squared   (1)

Reordering the elements gives:

Clifford Product   (2)

If we square the difference of two vectors we get:

Difference of two vectors squared   (3)

Reordering the elements gives:

Clifford Product   (4)

Summing up equation (2) and (4) gives equation (5):

Clifford Product   (5)

We now have the following relations for the scalar product of two vectors:

Clifford Product   (2)

Clifford Product   (4)

Clifford Product   (5)

The Outer Product

The outer product can be defined in dependence from the geometric product as follows:

Outer Product

Using the following product relation:

Complex Length   (6)

Gives the following relation for the outer product:

Outer Product   (7)

Using the following product relation:

Complex Length   (8)

Gives a further relation for the outer product:

Outer Product   (9)

The diverens between the given two relations for the outer product results in the folowing expression:

Outer Product   (10)

We now have the following relations for the outer product of two vectors:

Outer Product

Outer Product

Outer Product

Geometric Product

When we sum up the results for the inner (5) and the outer product (10) we come back to the geometric product:

Geometric Product

Geometric Product

Geometric Product

Geometric Product

Geometric Algebra and Lie-Algebra (Commutator Symbol)

Commutator Symbol

The commutator symbol is defined as [a,b] = ab-ba.

If we apply the commutator symbol on vectors a, b with the geometric product as operation, we get the following expression for the outer product:

2ab = [a, b] = ab-ba;

2ab = [a, b] = -[b, a] = -(ba-ab) = -2ba;

2aa = [a,a] = aa - aa = 0;

λ[a, b] = [λa, b] = λ(ab - ba) = [a, λb];

[a+b,c] = [a,c] + [b,c] = (a+b)c - c(a+b) = ac + bc - ca - cb = (ac - ca) + (bc - cb);

[a, [b, c]] = a(bc - cb) - (bc - cb)a = abc - acb - bca + cba;

[[a, b], c] = (ab - ba)c - c(ab - ba) = abc - bac - cab + cba;

Jacobi Identity:

[a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0;

[a, bc - cb] + [b, ca - ac] + [c, ab - ba] = 0;

(abc - acb) - (bca - cba) + (bca - bac) - (cab - acb) + (cab - cba) - (abc - bac) = 0;

abc - abc + acb - acb + bca - bca + cba - cba + bac - bac + cab - cab = 0;

Anticommutator Symbol

2ab = {a, b} = ab + ba;

The Geometric Product with Commutator and Anticommutator Symbol

2ab = 2ab + 2ab = {a, b} + [a, b] = ab + ba + ab - ba;

Motivation for a Lie-Algebra

With a matrix A we can define a element a = (IA) and a second element b = (IB) from a matrix B, both close to the identity element I for small ε.

The product will be:

ab = (IA)(IB) = I + ε(A+B) +ε2AB;

If we ingnore the AB term because ε2 is very small we get:

ab = I + ε(A+B);

With a-1 = (1+εA)-1 = I - εA + ε2A2 - ε3A3... ;

For non commutating elements a,b the expression aba-1b-1 is not equal I. We can express this term as follows:

aba-1b-1 = (IA)(1+εB)(I - εA + ε2A2)(I - εB + ε2B2);

Suppressing terms with εn and n>2 will give:

aba-1b-1 = I + ε2(AB - BA).

Definiton of a Lie Group can be found on the Abstract Algebra page:

Edersee - Weißer Stein
Bild: "Edersee - Weißer Stein"

Vector Bivector Products I

The following example of a vector and a bivector product can be generalised to arbitary vectors.

Vector-Bivector-Relations

Vector-Bivector-Relations

The result of the product is a vector part and a trivector part.

Vector-Bivector-Relations

The result of the product is a vector part and a trivector part.

Vector-Bivector-Relations

For the example we get the following result that can be generalised:

Vector-Bivector-Relations

The difference separates the vector part and the sum separates the trivector part.

Vector Bivector Difference Formular

Vector Bivector Sum Formular

We can know define the inner and the outer product for a vector with a bivector.

Vector Bivector Inner and Outer Product

The scalare product of a vector with a bivector is anticommutative and the outer product is commutative.

Commutative

Now we can write down the general form of the product of a vector with a bivector:

Vector Bivector Product

Edersee
Bild: "Edersee"

Vector Bivector Products II

Double Products

Double Product

 

Double Product

Trible Products

Trible Product

Trible Product

Trible Product

Trible Product

Trible Product

Trible Product

The Vector Bivector Products

Vector Bivector Product

Vector Bivector Product  (1)

Vector Bivector Product

Vector Bivector Product  (2)

Vector Bivector Product

Vector Bivector Product

If we build the difference between (1) and (2) we come to the following expression:

Vector Bivector Product

Bivector

With the letter B for the bivector (bc) it is possible to rewrite this equation in the following form.

Vector Bivector Product

On the right side is a Vector quantity. That means this operation is grade lowering.

The grade lowering property of the expression is used to define the scalare product of a vector with a bivector.

Vector Bivector Inner Product

This definition will result in the following relation:

Vector Bivector Product

Vector Bivector Outer Product

Vector Bivector Geometric Product

Vector - Bivector Relations

We start with the following relation between two vectors:

Vector Blade Relation 1

From this relation we get:

Vector Blade Relation 1

Renaming the vector b to b1. And then multiplying from the right with the vector b2 gives:

Vector Blade Relation 2

Decomposing the third term in the same manner as before, results in the following expression:

Vector Blade Relation 3

Vector Blade Relation 4

Vector Blade Relation 4

From this equations we come to the following general formular for the inner product of a vector with a blade:

Vector Blade Relation 4

The bi with the inverted circumflex is omitted from the product.

Fürstental
Bild: "Edersee bei Niedrigwasser - Fürstental"

The Bivector Square

Preparation

Basic relations used in the deduction of the formular of the bivector square.

Expressing the Wedge Product by the Scalare Product and the Geometric Product

Pythagorean Relation

Insert in (1)

Pythagorean Relation

Insert in (1)

Pythagorean Relation

Pythagorean Relation

Insert in (7)

Pythagorean Relation

Insert in (8)

Deduction of the basic relations of the Bivector Square

Bivector Square

Scheid Edersee
Bild: "Edersee"

The next page is about Geometric Algebra and Matrices.

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27. August 2017 Version 1.0
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