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  Is Geometric Algebra/Geometric Calculus all that it's hyped up to be?

+ 2 like - 0 dislike
1207 views

There appears to be a cult following of geometric algebra/geometric calculus (GA/GC) as developed by David Hestenes. Many questions on the web regarding this but also many contradictions appear. I wanted to make this question different than others, this question hopes to bring together many theoretical physicists/mathematicians/mathematical physicists whom can contribute to the discussion of the validity of the claims and soundness of GA/GC. I want to ask the experts here on physics overflow what they know and what they think about GA/GC.

The claims are always as follows, GA/GC provides a unified formalism for theoretical physics and mathematics. It provides a framework which unifies many of the concepts in fields of mathematics such as differential geometry, algebra and many more.... Why are theoretical physicists/mathematicians not rushing into this field? If it was so universal and apparently simpler to learn (also as claimed), then why is it not as widespread? If anyone has made use of GA/GC for their research into our best theory, quantum field theory, it would be much more appealing I think. Anyone who has made use of GA/GC for quantum field theory or the standard model please share your opinions.

Now most textbooks and articles on this subject give the answer that Clifford developed this formalism concurrently with Gibbs standard formalism for vector calculus. Gibbs was a famous physicists/mathematician and worked at Princeton and so it is the reason his approach was so widespread and Cliffords approach was swept under the rug. Not until David Hestenes revived it, and now there are many descendants/followers of his whom repeat what I just said in the introduction or preface to their book or article. A prolific follower and firm believer of GA/GC is Alan MacDonald a professor of mathematics at Luther College in Decorah, IA. He wrote the following article (there are MANY more by other authors).

https://www.astro.umd.edu/~jph/GAandGC.pdf

Just read the introduction paragraph and this is generally the type of introduction which comes from most articles and books on GA/GC. It is basically a rephrasing of what Hestenes said in his pioneering work and books on GA/GC, such as his book Space-Time Algebra.

I need honest opinions from reputable mathematicians/physicists all over. It is important for us to get feedback from mathematicians and physicists outside the field of GA/GC because it seems the only people HIGHLY RECOMMENDING using GA/GC are only those who are using it exclusively. It would be good to get many opinions from mathematicians/mathematical physicists in fields related to GA/GC, such as Differential Geometry, Mathematical Physicists who make extensive use of Differential Geometry, Modern Algebraic Geometry, abstract algebra and especially those studying General Relativity and Quantum Field Theory, our two most successful and far reaching theories. 

I know it is difficult for someone to comment on other peoples work, but when something makes such big claims as Unifying much of mathematics and physics, it is important for the community to discuss this, so that opinions and ideas are shared and cult following is either justified or unjustly spreading.

I am not against GA/GC, I am curious and want to know what other professional mathematicians/physicists think of this field and their claims. Please spread this article around so that others can make their comments and we can either begin to embrace GA/GC or refute its claims.

Thank you.

Note : There are many textbooks on this subject now and here a few so you can see for yourself that it is being used by a small community people. amazon.com/Geometric-Algebra-Physicists-Chris-Doran-ebook/dp/… , springer.com/gp/book/9780817682828 , link.springer.com/book/10.1007/978-1-84628-997-2 –

asked May 4, 2020 in Theoretical Physics by Kay Bei (10 points) [ revision history ]
edited May 4, 2020 by Kay Bei

I am no expert, but seems to me impossible to unify physics and maths; So many subjects and in an infinite universe only more "new" theories will pop up in the future.

2 Answers

+ 2 like - 0 dislike

Serious mathematicians and theoretical physicists know and use Clifford algebras and spinors extensively, because they are an important link between algebra, geometry, and topology.  They just don't fly the "geometric algebra" flag over this work, because - as you note - that's the banner of David Hestenes and his followers.   

A good example is the proof of the Atiyah-Singer index theorem.   It's a general theorem relating solutions of partial differential equations to topology, but the proof boils down to considering a special case, namely the Dirac equation, which is formulated in terms of Clifford algebras and spinors.  But there are many other examples!

You can learn about this material here:

  •  H. Blaine Lawson and Marie-Louise Michelsohn, Spin Geometry, Princeton University Press, 1989.

but I'll warn you right now that this is not an easy book.  It would help to start with the material on Clifford algebras and spinors here:

  •  Yvonne Choquet-Bruhat, Cecile DeWitt-Morette, and Margaret Dillard-Bleick, Analysis, Manifolds, and Physics (2 volumes), North-Holland, 1982 and 1989.

You can read my super-sketchy introduction to the Atiyah-Singer index theorem here.

answered Feb 17 by John Baez (395 points) [ no revision ]

@JohnBaez , @PatrickPowers  thank you John and Patrick, this clarifies some things but now I have more questions, :) . Sorry for all the questions, if you have time i would greatly appreciate your advice on any, none or all  of them :) I am hoping this thread could be useful for anyone wanting to study geometric algebra in relation to theoretical physics:

So your saying mathematicians and physicist often use geometric algebra without knowing it, correct? 

You are saying Geometric Algebra is a term flagged by Hestenes to describe all of these things? Would you say Geometric Algebra is a theory which has fundamental principles which can give rise to all these tools that physicists and mathematicians use such as Spinors, Clifford Algebras... Or is it merely a name given to a set of independent tools?

I still cant tell if your trying to say Geometric Algebra by Hestenes is a new theory or just that he collected all these tools/objects/structures and put them into on nice package? I also cant tell if you think his theory is logically consistent? Are there any errors in his theory?

Have you heard of the book by Emil Artin titled Geometric Algebra, is this different? 

So Hestenes work is not new. Is he just trying to put it all into one nice package?

 Would it then not be useful for a mathematican or physicist to study geometric algebra as a field in itself, considering they will be using these tools anyway?

I actually find the book of Bruhat at too high of a level for me at the moment. Can you recommend a book at a lower level than this one even. Do you recommend learning from a geometric viewpoint or algebraic viewpoint there are books which emphasize one or the other?

"So your saying mathematicians and physicist often use geometric algebra without knowing it, correct?"

No, I'm saying they prefer to call it the theory of Clifford algebras, because using the term "geometric algebra" tends to flag you as a fan of Hestenes, who has a particular attitude toward Clifford algebras that most mathematicians and physicists don't like.

"I still can't tell if you're trying to say Geometric Algebra by Hestenes is a new theory or just that he collected all these tools/objects/structures and put them into on nice package? I also can't tell if you think his theory is logically consistent? Are there any errors in his theory?"

I haven't studied Hestene's work carefully, because what I've read did not excite me: it seemed like a somewhat clumsy way of talking about things that I already knew.   I should read it carefully sometime.  I imagine it must have some new elements, but I could see instantly that it also has lot which is not new.   I've never heard of any logical inconsistencies or errors in his theory, though of course we all make mistakes here and there.

"Have you heard of the book by Emil Artin titled Geometric Algebra, is this different?"

Judging from the description on Amazon, and what I know about Emil Artin (a famous mathematician), this book is quite different: some other ways of applying algebra to geometry.

"Would it then not be useful for a mathematican or physicist to study geometric algebra as a field in itself, considering they will be using these tools anyway?"

There are too many useful things to study them all.   You have to just make up your own mind what you want to study.

I actually find the book of Bruhat at too high of a level for me at the moment. Can you recommend a book at a lower level than this one even?  Do you recommend learning from a geometric viewpoint or algebraic viewpoint there are books which emphasize one or the other?

To really understand math you have to fuse the geometric and algebraic viewpoints and see them as flip sides of the same coin.  How you get to this point is a matter of individual taste.

If you want to learn about Clifford algebras and spinors and Choquet-Bruhat, DeWitt-Morette and Dillard-Bleick's book is too hard, maybe this will be easier:

  •  Pertti Lounesto, Clifford Algebras and Spinors, Cambridge University Press, 2001.

Or read Hestenes!  Just remember that most mathematicians have a somewhat different way of talking about the same ideas - and try to learn that other way sometime.

+ 1 like - 0 dislike

"Why are theoretical physicists/mathematicians not rushing into this field? If it was so universal and apparently simpler to learn (also as claimed), then why is it not as widespread?"

Theoretical physicists already have mathematical techniques that work.  GA may be simpler to learn, but that is no advantage to them.  They have already learned the trad techniques.  The easy thing for them to do is to keep using them.  There is no cost to this.  Learning GA would be a cost. 

According to Thomas Kuhn, nothing new will come into a field until there is something the new technique can do that the old one can't.  GA doesn't qualify.   It can give you no answers that the old ways don't.  Simpler, easier answers in some cases, but not to the degree that would attract great enthusiam.

OK, suppose a student learned GA first.  He/she would still need to learn all the trad techniques in order to have access to the vast body of literature.  MORE would have to be learned: all the trad techniques plus GA.  Would this approach be an advantage?  As far as I know such a student does not yet exist so we can't say.

What I can share is my own personal experience.  My unusual interest is what physics would be like if we had more spatial dimensions.   Much of the trad math does not work at all there and can't be adapted.  In particular,  applications of the cross product work only in 3D.   I had to use geometric algebra instead.

I haven't gotten that far, but already I can tell you that I found this quite valuable in having given me insight into trad 3D magnetism.   Describing magnetism in terms of planes of rotation is natural.  That's what magnetism is:  a force that causes curves and rotations.    Describing magnetism in terms of vectors produced by a cross product is unnatural.  Sure, it works.  But having experienced both, I can tell you that the trad approach gives the wrong intuition.  Going further, quantum spin is notoriously unintuitive in 3D.  It only makes sense as a rotation or cycle in even-dimensional space, and GA handles that easily.  Maxwell's equations make more sense in GA.  The signs come out in a sensible way, which they don't in the trad method.  I'm not sure why, but suspect it is because GA doesn't have the arbitrary right hand rule.  

Now a thing about any field of study is this.  Things are taught a certain way.  The field then promotes students who easily learn in this way, while rejecting students unable to learn in this way.  Perhaps those students could learn in some other way, but we shall never know.   Those accepted students eventually become teachers, and teach in the way that they found easy to learn.  So the system quite naturally perpetuates itself.  If some new way of learning comes along, well, teachers are very busy.  Do they want to become beginners once again?  Usually not.  Even if they would like to do this, the time is not there.  So the system naturally resists change.  It will continue to do so until, as Kuhn explained, something comes along that is so much better than the old system that the considerable effort to change is seen as worthwhile.   

Now a bit about my personal experiences.  I learned in math grad school (rather to my surprise) that I was no good at math.  I had done well in lower math but that was by unconsciously using homebrew geometrical methods.  That can be very effective on an SAT test but not in higher math.  I couldn't do it.  All I could do was geometry.  That's how I think.  So GA is perfect for me.  I find it easy to imagine subspaces and so forth.  My officemate couldn't do that at all.  GA might have no attaction to him, nor to great majority of mathematicians suspect I.  Perhaps they would find it impenetrably abstract.  Like I noted before disciplines tend to select a certain type of person, and this perpetuates itself.

There is hope for GA though.  Books that teach physics in GA are starting to come into existence.  There is nothing to stop people like me from learning in this way. 

answered Feb 16 by PatrickPowers (10 points) [ no revision ]

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