Is Geometric Algebra inconsistent/circular?

[malawi_glenn]

I am trying to learn Geometric Algebra from the textbook by Doran and Lasenby.

They claim in chapter 4 that the geometric product $ab$ between two vectors $a$ and $b$ is defined according to the axioms
i) associativity: $(ab)c = a(bc) = abc$
ii) distributive over addition: $a(b+c) = ab+ac$
iii) The square of any vector is a real scalar

Then they claim that the inner and outer product are defined as
$a \cdot b = \frac{1}{2} (ab+ba)$
$a \wedge b = \frac{1}{2} (ab-ba)$
so that
$a b = a \cdot b + a \wedge b$

My problem is that if you are given two vectors, say
$a = 1e_1 + 3 e_2 - 2e_3$
$b = 5e_1 -2 e_2 + 1e_3$

How to actually compute $ab$?

I mean, you then have to specify how either $a \cdot b$ and $a \wedge b$ works, or how (in detail) $a b$ are to be performed.

This is in my view, circular.

From another point of view, say that you start backwards by defining the inner and outer product, and then define the geometric product as $a b = a \cdot b + a \wedge b$

Then how to show that the geometric product is associative? The usual definition of the outer product is associative, but the usual definition of the inner (dot) product is NOT associative. So how to show that the geometric product is associative if you take the inner and outer product as starting point for the geometric product?

Thank you very much in advance for any kind of feedback

 

[micromass]

I am trying to learn Geometric Algebra from the textbook by Doran and Lasenby.

They claim in chapter 4 that the geometric product $ab$ between two vectors $a$ and $b$ is defined according to the axioms
i) associativity: $(ab)c = a(bc) = abc$
ii) distributive over addition: $a(b+c) = ab+ac$
iii) The square of any vector is a real scalar

This definitely does not define a unique product. So if that is the definition your book gives, then that is incorrect. It is certainly possible to define the geometric product axiomatically, but I think the easiest way is to do as you noticed and define the geometric product "backwards". But for information, the full list of axioms takes a bilinear form $Q$ (usually taken to be the inner product) and defined the geometric product
1) Associative
2) Distributive
3) $a^2 = Q(a,a)$
4) The geometric product on scalars agrees with the usual product

Then how to show that the geometric product is associative? The usual definition of the outer product is associative, but the usual definition of the inner (dot) product is NOT associative. So how to show that the geometric product is associative if you take the inner and outer product as starting point for the geometric product?

Thank you very much in advance for any kind of feedback

The problem here is that taking the geometric product of two vectors does not give you a vector anymore! It gives you a combination of a scalar and a 2-form. So the definition $xy = x\cdot y + x\wedge y$ is not good enough anymore since it only holds true for scalars. You'll need a definition that holds true for all $n$-forms if you wish to show associativity.

 

[malawi_glenn]

Thank you for the reply!

This is the book I am using http://www.cambridge.org/se/academi...hysics/geometric-algebra-physicists?format=PB
My impression was that the authors are considered (?) to be world leading experts on this matter, that is why I choose to study this book

As you point out, $ab$ is no longer a vector but instead a scalar plus a bivector. This means that in order to compute $(ab)c$ one needs to define the geometric product of scalars-vectors and bivectors-vectors (and so forth).

Then my question is, how do you define such geometric products? My impression is that if you can define the geometric product between some "basic elements", then you can recursively work out the geometric product between any "multivector"

 

[micromass]

It's a shame if the book doesn't tell you how to compute the geometric product for multivectors! In any case, you can see a fully rigorous approach here http://arxiv.org/pdf/1205.5935v1.pdf

 

[micromass]

An excellent book that gives a lot of intuition is the book by Dorst et al: https://www.amazon.com/dp/0123749425/?tag=pfamazon01-20
You'll find both the axiomatic approach in that book, as the general definition of the geometric product in terms of multivectors. But the book is not theorem/proof style.

 

[malawi_glenn]

Thank you, I don't know if the book tells me this or not, I am just starting at chapter 4. Perhaps it becomes more clear and complete in the latter chapters. I just felt that I needed some feedback.

The book by Dorst was also something I considered. I had a brief examination on google books and my impression was that it was too detailed (left contraction, right contraction etc) but perhaps this is the way to do it rigorously?

 

[micromass]

Thank you, I don't know if the book tells me this or not, I am just starting at chapter 4. Perhaps it becomes more clear and complete in the latter chapters. I just felt that I needed some feedback.

The book by Dorst was also something I considered. I had a brief examination on google books and my impression was that it was too detailed (left contraction, right contraction etc) but perhaps this is the way to do it rigorously?

Well, it is exactly this left/right contraction that you use to define the geometric product in general!

 

[malawi_glenn]

I guess I am more into physics-applications (I have PhD degree in theoretical particle physics), that was also a reason for why I didn't want to spend 70$ on a book inclined towards computer graphics. Perhaps I should just "bite the bullet" and use the book by Dorst to straighten out my fundamental issues and then move back to a more "physics application approach" text after that? What do you suggest?

 

[micromass]

I guess I am more into physics-applications (I have PhD degree in theoretical particle physics), that was also a reason for why I didn't want to spend 70$ on a book inclined towards computer graphics. Perhaps I should just "bite the bullet" and use the book by Dorst to straighten out my fundamental issues and then move back to a more "physics application approach" text after that? What do you suggest?

I guess it depends on how much you want to take on faith. I don't think this left/right contraction stuff is actually very important in physics, so it might not be very useful for you to learn it. I think you should try to read the Arxiv link I provided above to see how things are done rigorously. I think that should be enough. You should only get Dorst if you want an intuition behind the myriad of different operations in geometric algebra. But I don't think it's necessary to actually get the book if you're into physics. Although it is a really good one.

 

[malawi_glenn]

Thanks, I will definitively follow your advice

What is your impression on the books by Hestenes?

 

[micromass]

I'm not a fan. There's definitely a lot of information in them. But they don't give enough intuition to be useful to me.

 

[malawi_glenn]

Ok, thanks!

My goal of learning this is just to have some fun :)

 

[micromass]

Another good resource are the two books by MacDonald:
https://www.amazon.com/dp/1453854932/?tag=pfamazon01-20
https://www.amazon.com/dp/1480132454/?tag=pfamazon01-20

But maybe you find these too easy for your level.

 

[malawi_glenn]

as long as things are done in logical and pedagogical consistent manners, I don't care if the level is "too easy" :)

 

[malawi_glenn]

In the ArXiV paper you sent me, it says:

"I begin with a formal product of vectors uv that obeys the usual rules for multiplication; for example,
it’s associative and distributive over addition."

now this directly confuses me, what IS the "usual" rules for multiplication? I often think that multiplication is commutative, but the geometric product is not commutative in general

 

[mnb96]

I think in that context the "usual" rules for multiplication are exactly those that the author means by "usual", i.e. associativity and distributivity over addition. Commutativity holds for multiplications between real scalars but it is certainly not an "usual" property of matrix multiplication, for instance.

I would not focus too much attention on the choice of word "usual" and I think that you can mentally replace the string "the usual rules" with "the following rules".

When the author said "the usual rules for multiplication", he probably had in mind the rules of multiplication in an arbitrary ring (not necessarily in the ring of reals), where multiplication is not necessarily commutative.

 

[fresh_42]

One could also object that "usual" implies a $1$ and an inverse. This shows how personal the term is. If you're used to mappings, rings and algebras then usual becomes something more general. Personally, I even regard associativity as a special condition and would be satisfied with the distributivity.

 

[malawi_glenn]

ok so usual in the ring kind of sense?

 

[fresh_42]

ok so usual in the ring kind of sense?

More in the sense of an algebra (= a vector space with multiplication) since you still have vectors and a scalar field like ℝ or ℂ that allows to stretch the vectors.

 

[micromass]

In the ArXiV paper you sent me, it says:

"I begin with a formal product of vectors uv that obeys the usual rules for multiplication; for example,
it’s associative and distributive over addition."

now this directly confuses me, what IS the "usual" rules for multiplication? I often think that multiplication is commutative, but the geometric product is not commutative in general

That is the introduction. The introduction is for motivation only. You should start reading at section 2 page 10.

 

[malawi_glenn]

Hello again

Chapter 2 was a gem :)

 

[fresh_42]

I had a short glimpse on the paper. Thanks for that tip. Why don't we teach this approach at school? This would close the huge gap between school and nearly all natural science a little. And I doubt that it would cause more trouble than the concepts to solve linear equation systems that are actually used these days.

 

[micromass]

I had a short glimpse on the paper. Thanks for that tip. Why don't we teach this approach at school? This would close the huge gap between school and nearly all natural science a little. And I doubt that it would cause more trouble than the concepts to solve linear equation systems that are actually used these days.

I know. It wouldn't take much trouble at all to teach geometric algebra. Sadly, this isn't done at all nowadays, even though I'm 100% convinced that the geometric algebra approach is superior than what we do now. It's the power of tradition of course. There are many other things in mathematics that I feel should be basic knowledge for undergrads, but that is somehow something that is only known to a select few.

 

[malawi_glenn]

I know. It wouldn't take much trouble at all to teach geometric algebra. Sadly, this isn't done at all nowadays, even though I'm 100% convinced that the geometric algebra approach is superior than what we do now. It's the power of tradition of course. There are many other things in mathematics that I feel should be basic knowledge for undergrads, but that is somehow something that is only known to a select few.

Can't we start a thread where this is discussed? "Non-mainstream but useful math to know" and provide suggestions for learning material, books etc?

 

[fresh_42]

There are many other things in mathematics that I feel should be basic knowledge for undergrads,...

This always reminds me on an occasion when a six-year old proudly presented me how she could count backwards. As she stopped at zero, I asked her why? Five minutes later she knew negative numbers and hadn't the least problem with it.

 

[fresh_42]

Can't we start a thread where this is discussed? "Non-mainstream but useful math to know" and provide suggestions for learning material, books etc?

Good idea. I couldn't find a punchline so I didn't. (Sorry, for telling this in your thread.)

 

[malawi_glenn]

Good idea. I couldn't find a punchline so I didn't. (Sorry, for telling this in your thread.)

Ah I just meant that it would be a nice thing to do, so that it becomes easier for other people finding out about the stuff we will post regarding non-mainstream math :)

 

[micromass]

Can't we start a thread where this is discussed? "Non-mainstream but useful math to know" and provide suggestions for learning material, books etc?

I'm actually planning to do an insight series about it too!

 

[malawi_glenn]

I'm actually planning to do an insight series about it too!

awesome! :D

 

[chiro]

Hey malawi_glenn.

Do you understand the idea of a division algebra and its construction?

The original stuff by Grassmann focused on this idea of making it possible to have an algebra on vectors that satisfied multiplication and divisibility but the consequences of making this consistent were such that you got all these other constraints (like non-commutativity).

Making it general is the difficulty that these algebras have and linear algebra forms a lot of the basis of generalizing it so that it scales as dimensions increase.

Making it rigorous is difficult no doubt and there have been baby steps over roughly 150 years or so but the idea is actually straight forward algebraically.

 

[mnb96]

I'm actually planning to do an insight series about it too!

Wonderful idea! I am really looking forward to see it published soon.

 

[robphy]

Although I've postponed learning geometric algebra (GA) [putting my efforts into differential forms instead],
I have found myself having to learn to it in order to understand/decode some calculations.

In writing an Insight,
it might be useful to include translations between statements and calculations in GA
and statements and calculations in (say) more-standard tensor-calculus and/or vector-calculus,
accompanied by some diagrams.
To me, the notation found in various sources in quite dense, rather abstract for a beginner, and not quite standardized,
forcing the reader to juggle the various objects and operations.

My $0.03.

 

[mnb96]

I guess the original idea for the insight was not to focus only on the details of GA, but more generally on "Non-mainstream but useful math to know". As micromass said:

There are many other things in mathematics that I feel should be basic knowledge for undergrads, but that is somehow something that is only known to a select few.