Into Clifford algebra (part 1): Let's say we're all familiar enough with vectors in 2d or 3d. In vector calculus there's the well known dot product,

a·b = a₁b₁ + a₂b₂ + a₃b₃

and in 3d, there's also the lesser known cross product a×b. It goes back to Graßmann's exterior algebra that comes with a wedge product a∧b which, in 3d, happens to be the cross product. But there are problems with these products...

Geometric algebra, as William Kingdon Clifford originally named his result, combines those ideas into a single geometric product that algebraically behaves much better than the two vector products. Again, not quite as an historian might put it...


some history

While Carl Friedrich Gauss did not bother to publish about him discovering quarternions in 1819, they were finally brought to light by William Rowan Hamilton in 1843. Shortly after, 1844, Herrmann Günther Graßmann defined his wedge product.

When James Clerk Maxwell captured electromagnetism in 1865, he did so using quarternions in a mess of twenty equations. The race for the best formalism to do physics was on...

Clifford's Geometric algebra entered the scene unnoticed in 1878, while Josiah Willard Gibbs published about vectors as late as 1880. The latter leading the pack ever since Oliver Heaviside reformulated Maxwell's equations using vectors 1884. Bringing them down to four, impressively outperforming quarternions!

Wolfgang Pauli and Paul Adrien Maurice Dirac described the electron spin in 1927/28 using matrices. But those spinors are unwieldy to describe with vectors, where should they live and how to generalize?

It wasn't until the 1980's when David Hestenes cast many physics problems in Clifford algebras, getting them to the attention they deserve. Using Clifford algebra electromagnetism can intuitively be put down in a single equation!

Well, P.R.Girard refers in his 1984 essay "The quaternion group and modern physics" to a modern description using a quarternionic potential function of only one variable in a single differential equation, citing back to Ludwik Silberstein... To my regrets my historical knowledge ends here, so let's get back to the math:


what are those problems with vector calculus?

For one, the dot product  does not yield a vector, but an object of different type - a number. And there is no useful way to add a number to a vector (independent of the chosen basis to represent it).

Same happens with the cross product, you get something called a bivector, which just happens to look like a vector in 3d. In the other cases you also get an object of different type.

An n-dimensional vector lives in n-space and consists of n numbers (or components), each on equal footing, together giving the coordinate data of a point in n-space. Here's how a 3d vector is usually seen:

/1\
| 2 | (a vector in R³)
\3/

Clifford elements (let's call them cliffs) come with much more information: For dimension n you get 2^n numbers. That's quite some room, certainly enough to accomodate vectors, isn't it? Indeed, they're thriving in there. But what's the other data for?

Before i show you a cliff i'd like to have a better notation because Clifford algebras, lacking type problems such as we've just seen, are much more fun in algebraic notation. Experts know to cautiously interpret the n numbers of a vector as multiples of n basis vectors, written like so:

1e₁ + 2e₂ + 3e₃ (a vector in R³)

Here it comes. To ease the eye, let's do a 2-dimensional cliff first. Here's a simple one called diagonal element because all components are equal to 1:

1 + e₁ + e₂ + e₁₂ (diagonal element of Cl(2,0))

Cl(2,0) is the span of four basis elements and the first component is called scalar. Note that i didn't write an extra unit symbol to place besides the scalar, as pseudo basis element or somesuch.

Scalars are cliffs of grade 0, vectors are grade 1, bivectors of grade 2, in 3d one encounters trivectors, and so on... So e₁ and e₂ really are just vectors. And the last component can always be called counit, but here e₁₂ is also a bivector.

You can get a bivector by multiplying two vectors. In general, any bivector can be written as linear combination of the 2-graded generators. In Cl(2,0) e₁₂ is the counit so the bivector subspace is 1-dimensional.

Whatever the dimension, you can calculate the extent of a bivector in it's subspace just as you can compute the length of a vector in it's space. But with a bivector one is supposed to associate an area.


finding e₁₂ with highscool algebra

The following train of thought i daydreamed while watching part 1 of Eckhard Hitzer's lecture linked below. Not sure what to make of it, i hope you find it inspiring.

Algebraists like to start with some rules and elements to construct new elements. Let's begin with an orthonormal basis for R² given by unit vectors e₁ and e₂. On these, the scalar product (or dot product) works like this:

(0) e₁·e₂ = e₂·e₁ = 0  (orthonormal)
(1) e₁·e₁ = e₁² = 1 (unitary)
(2) same for e₂

What can we do to construct new elements? We're allowed to add vectors, so one simple thing is to try and compute (e₁+e₂)². Distributive laws hold, we can make this look like an exercise in highscool algebra:

(e₁+e₂)·(e₁+e₂) = e₁² + e₂² + e₁·e₂ + e₂·e₁ = 2

That's almost the definition of Clifford's associative geometric product! Because we know e₁² = e₂² = 1, the term set in italic has to be zero! That is, the following must hold:

(a) e₁·e₂ + e₂·e₁ = 0
which can also be written as:
(a') e₁·e₂ = –e₂·e₁

That's called anticommutative. It means, when swapping the order in a multiplication, we have to also flip the sign. That area can be negative! The geometric product of vectors a·b really is Graßmann's wedge product a∧b when a and b are orthogonal. But as (1) illustrates, it equals the dot product a·b for parallel vectors, and is commutative in that case.

Rule (a) is an alternative to rule (0) in the sense that it won't ruin what vectors can do. Okay, we have constructed a new value e₁·e₂ (i called e₁₂ before), and can now compute the remaining products:

e₁·e₁₂ = e₂
e₂·e₁₂ = –e₁
and so on...

Look, there's a subspace of quarter rotations (90°), making e₁₂ look like the imaginary complex number i:

       1·e₁₂ = e₁₂
    e₁₂·e₁₂ = –1
    –1·e₁₂ = –e₁₂
 –e₁₂·e₁₂ = 1

That oriented area you get by multiplying two vectors a·b specifies how much the result will rotate a towards b. See the picture below for an illustration of the types of generators in Cl(3,0). Of these it has 1 scalar, 3 vectors, 3 bivectors, and 1 trivector. Now look at Pascal's triangle below...

There's a much more delightful stuff to tell, but this post is getting too long already. Just put your questions in the comment section or follow the


fun references

Unfortunately there is background noise in Eckhard Hitzer's otherwise fascinating lecture. It's steady, and in a quiet environment you might even forget it's there. Go and try, lots of information in here!
Tutorial 1 on Clifford's Geometric Algebra

Some of his work seems to be available on his homepage, but for some reason i couldn't access it. So here's an external link list instead:
http://gaupdate.wordpress.com/2012/03/06/e-hitzer-online-preprints-2010-2012-video-lectures-and-mathematical-reviews/

+John Baez has highly interesting stuff to show at his lookouts around "The Octonions". If you find the beginning of the following page confusing, don't give up and try just a bit further down. Or the next page.
http://www.math.ucr.edu/home/baez/octonions/node6.html

slehar's blog post (2014) has much material to offer, all explained in very basic terms. The author is also keen to interpret Clifford algebras to benefit the study of consciousness... The mathematics is certainly inspiring!
"Clifford Algebra: A Visual Introduction"
http://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/

Wikipedia offers a nice heap of introductions to Cl(3,0) here:
http://en.wikipedia.org/wiki/Geometric_algebra

There's a Clifford algebra master page for grown-ups:
http://en.wikipedia.org/wiki/Clifford_algebra

And they have a "Comparison of vector algebra and geometric algebra":
http://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra

Maxwell's original formulation appeared in his paper "A Dynamical Theory of the Electromagnetic Field":
http://en.wikipedia.org/wiki/A_Dynamical_Theory_of_the_Electromagnetic_Field

The n-vector illustration i cut up is by User:Maschen, referenced here:
http://en.wikipedia.org/wiki/Exterior_algebra

The picture of William Kingdon Clifford i cut from here:
http://en.wikipedia.org/wiki/William_Kingdon_Clifford

Pascal's triangle by User:Drini, part of which you can see below, found here:
http://en.wikipedia.org/wiki/Pascal_triangle

Part 2: Multiplication in geometric algebra
https://plus.google.com/115434895453136495635/posts/NKqs5BwsCqq

Part 3: _Cayley graphs and a strange clock_
https://plus.google.com/115434895453136495635/posts/bkHQAZLzpt8

Before, i wrote about "Rotations and Spinors"
https://plus.google.com/115434895453136495635/posts/UoB2YNEvHK8

#scienceeveryday : #geometric and #clifford #algebra   #part1