Something New

Saturday, October 15, 2005
Time for a geekfest. Nongeeks are warned to look elsewhere. Now.

Every undergraduate in science or engineering has learned to add the concept of vectors to his repertoir, expanding mathematics beyond the application of numbers only. Vectors are liberating; they form a tremendously powerful approach to physics, constituting the backbone of almost all science and engineering today. A vector is defined as a little arrow, i.e., a directed line segment. The addition of vectors is defined in a natural and physically intuitive way by moving one vector to the top of the other. Then draw the vector that starts at the bottom of the first and ends at the top to the second. That's the sum of the original two.

Multiplication of vectors, by contrast, is problematic. There are two ways of "multiplying" vectors, neither of them philosophically satisfactory. The dot product, v ⋅ w, and the cross product, v × w, are both problematic. The dot product takes you out of the realm of vectors into the realm of scalars and the cross product only exists in dimension 3. Worse, it takes what is an essentially 2-dimensional situation, a relationship between two vectors which are sitting in the same plane, and pops you up to an apparently extraneous third dimension.

The multiplication of vectors is so unsatisfactory, in fact, that in most treatments of vectors in higher mathematics it is ignored entirely, vectors being thought of merely as objects which it is possible to add (and stretch), but not as a general rule to multiply.

There is a new way to look at multiplication between vectors however, a new foundation of vector relations which allows the laws of physics to be rewritten much more succinctly and intuitively. The name is "Geometric Algebra", and detailed references can be found here, here, here, and here. The new formalism appears to be very powerful, allowing such disparate topics as Maxwell's equations, the Pauli spin matrices familiar from quantum mechanics, and relativity theory, for three examples, to be reformulated much more clearly than ever before. Maxwell's equations can now be reduced to a single line as shown below, for example.
F = J
That single line contains all of Maxwell's equations in one fell swoop!

Like all revolutionary acts of brilliance, the fundamental ideas of geometric algebra are quite simple yet very powerful.

The first idea is to allow a few more objects than simply scalars and vectors into our theory. We now also include directed areas (parallelograms), called bivectors. "Directed" in this case means that we have chosen either a clockwise or a counterclockwise direction to traverse the parallelogram, and associated it to the area. If we start with the vectors v and w, then we define the bivector
v ∧ w
to be the parallelogram formed by moving w along v, with the orientation taken to be that given by first moving along v and then along w. This new product, called the wedge product, is an area normal to the cross product, having area equal to the length of the cross product. It is defined in every dimension and doesn't remove us from the 2-dimensional situation we started with. It is in fact the replacement for the cross product in the new formalism. The wedge product is antisymmetric, i.e.,
v ∧ w = - w ∧ v,
(because the bivector on the left is oriented clockwise while the one on the right is oriented counterclockwise), whereas the dot product is symmetric,
v ⋅ w = w ⋅ v.
This first new idea, bivectors, can be likewise generalized to trivectors, a trivector being an oriented parallelepiped, and similar oriented objects in higher dimensions.

The second great insight of the new scheme is that the multiplication of vectors should contain both a symmetric and an antisymmetric component at the same time. Therefore a new product, the geometric product of two vectors is defined by the simple formula
vw = v ⋅ w + v ∧ w.
This definition still suffers from the drawback that a pair of 1-dimesional objects (vectors) has given rise through "multiplication" to a 0-dimensional object (a scalar) and a 2-dimensional object (a bivector). In a sense, this seems to balance out and to be more satisfying than either the dot product or the cross product alone. But the real solution is to change the scope of operations. Instead of considering vectors as our fundamental objects, we now consider multivectors to be the fundamental objects of mathematical physics, where multivectors are simply abstract sums of the form
a + bu + c(v ∧ w),
with a, b, and c being ordinary scalars, u an ordinary vector, and v ∧ w a bivector. As with the definition of the complex numbers, the disparate pieces of the multivectors are added only with like pieces, allowing us to stay in the arena of multivectors when adding, and the geometric product defined above allows us to remain in the same arena when multiplying.

In dimension 3, the proper objects are now considered to be the set of all multivectors of the form
a + bu + c(v ∧ w) + d(s ∧ t ∧ r),
with a, b, c, d real numbers. Similarly, in dimension 4 we throw in a quadvector term on the end.

Experience has painfully taught us that the best long-term progress in our theories comes from improvements in our fundamental notations themselves, in what mathematicians call the "machinery" which we use to conceptualize reality. Geometric Algebra seems to hold the potential to be an earth-shattering paradigm shift in physics, engineering, and mathematics. Keep an eye on it.

Update: Graphics for wedge and dot products fixed as per Seneca's suggestion, using the code found on this useful page.

14 comments:

chuck said...

Yeah,

But geometric algebra requires a non-degenerate inner product so as to make use of the resulting identity between a vector space and its dual. Consequently, it is not quite so general as the exterior algebra, nor, IMHO, are Clifford algebras as easy to use on a computer. However, there is a lot of fun in translating traditional formulas from mechanics to the geometric algebra version and trying to understand the result. It's a great way to pass time. I will also admit that complex numbers and quaternions look quite spiffy dressed up in geometric algebraical clothing.

RogerA said...

God--this is like the math courses I had at the academy--WHY are you doing this to me. Get help!!!

chuck said...

rogera,

Didn't Gen. Bradley relax by doing integrals? Math does have its uses ;)

terrye said...

I doubt if I am alone when I say I don't have the least idea what you people are talking about.

MeaninglessHotAir said...

Roger and Terrye, You were warned!

SneakyFeet said...

But you stopped just as it was getting interesting.

Syl said...

Well, hallelujah! I think I actually understand the concept! Not that I'm able to play around with it. Nor can I turn around and explain it to someone else!

(Maybe that means I don't understand at all...LOL)

Well, I got an A in differential calculus while simultaneously using both differential and integral calculus in my physics course.

But, you must understand, and you can probably see by my terminology, that that was over ::gasp:: forty years ago!

And I have dabbled very little since.

But, damn, you're right on when you say:

"Experience has painfully taught us that the best long-term progress in our theories comes from improvements in our fundamental notations themselves, in what mathematicians call the "machinery" which we use to conceptualize reality."

Therein lies the beauty, the utility, and all those AHA moments.

terrye said...

meaningless:

Yeah but I just had to look.

It is kind of like when you are watching the horror movie and the silly girl hears the sound behind the door and you are saying "Don't open the door!!"

but she does.

I just had to look.

Seneca the Younger said...

MHA: ⋅ and ×, thus v ⋅ w and v × w.

See Elizabeth Castro's handy table.

Seneca the Younger said...

God--this is like the math courses I had at the academy--WHY are you doing this to me. Get help!!!

Some people like rap music.

I doubt if I am alone when I say I don't have the least idea what you people are talking about.

"In mathematics you don't understand things. You just get used to them." — John von Neumann

chuck said...

Hmmm...

I think a simpler way to describe the algebra is to start with an orthogonal set of unit vectors. The multiplication rules are then:

u_i × u_i = 1,

and

u_i × u_j = − u_j × u_i , i ≠ j.

Another way to think of it is in terms of linear maps and wedge products. The product

u ∧ F,

where F is of degree i, can be regarded as a linear map from terms of degree i into the terms of degree i + 1 with kernel precisely those terms in F containing u. Remove u from those terms and you have a mapping of the kernel into the terms of degree i - 1. The geometric algebra product puts these maps together, modulo some sign changes, to get a bijection F → ker ⊕ coker. It is not hard to see why a second multiplication reverses this map upto a scalar factor.

SneakyFeet said...

Chuck, I think

u_i × u_i = 0.

no?

chuck said...

sneakyfeet,

Chuck, I think

u_i × u_i = 0.

no?


I am using the × for the geometric product, not the cross product. The wedge product (∧) is the part that corresponds to the usual three dimensional cross product. One small correction would to note that in some cases (special relativity) one could have u_i × u_i = -1 instead of u_i × u_i = 1.

offworld said...

Intriguing. Thank you, MHA. I should find time to play with this.