Clifford Algebra and its contributions to physics

[Phylosopher]

Hello,

I see a lot of people enthusiastic about Clifford algebra and its future role for physics, yet I also see a lot of people frustrated opinions about it. Contents in the internet seems really really small compared with other mathematical topics, not to mention less books about it.

All this makes me less confident about learning this topic.

Well, this is what I feel honestly! Now I can ask some questions.

1 Does Clifford algebra brings any good to physics or is it just reformulation?

2 If it did/does, then in what areas of physics does Clifford algebra work as master tool above all others. (I will appreciate it if you provided me with resources, especially scientific papers)
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Background: I am an undergraduate physics student "interested in mathematical physics" considering taking my senior project on Clifford algebra. My supervisor is really enthusiastic about, but I feel that it might not be a good project! Since I am not yet convinced with it, because of all the controversial opinions in the internet.

 

[jedishrfu]

The wikipedia article on Clifford Algebras has some Physics examples:

https://en.wikipedia.org/wiki/Clifford_algebra#Applications

 

[Phylosopher]

The wikipedia article on Clifford Algebras has some Physics examples:

https://en.wikipedia.org/wiki/Clifford_algebra#Applications

I am not an expert, but from what I read. A lot of these applications can be achieved with other mathematical tools that are more common and not just clifford algebra (some people claims). Which makes people think about it as an accessory.

"That what I understood from people discussions on the internet"

 

[romsofia]

This page talks about an application:
http://physics.oregonstate.edu/coursewikis/GDF/book/gdf/spinors

 

[jedishrfu]

Often times in math its necessary to have different tools available to approach a problem.

Sometimes you will hear of a mathematician seeing a connection to another body of mathematics and recasting the problem in that new form and then go on to proof it.

Clifford algebras is one such tool.

 

[FactChecker]

One of the interesting Clifford Algebras is Geometric Algebra (GA). GA consolidates a lot of mathematical constructs involving different dimensions into one construct and the mathematics of that construct. The result is that it includes many specially derived algebras that were invented for specific applications. It helps greatly to consolidate all the mathematical manipulations into one logical, methodical system. Then it turns out that so much of physics is just the mathematical manipulations and the basic physics concepts are much simpler in the form of GA. For instance, Maxwell's equations become one simple GA equation. That one GA equation might represent many mathematical manipulations (like a simple matrix equation represents a lot of multiply/adds), but those manipulations are routine and systematic.

That being said, there is a learning curve to master GA and the payoffs are not immediate. Also, a lot of the traditional specialized algebras developed for specific applications are not always easy to translate into GA.

 

[Phylosopher]

Thank you for your answers, they were helpful. I will suggest for people interested in Clifford Algebra to look at this too:
https://www.reddit.com/r/math/comments/5bd2ei/lets_talk_about_geometric_algebra_not_algebraic/

One more question! What is the best book to study Clifford Algebra from a Mathematical point of view(Most of the books out there are for applications). There is "Clifford Algebra to Geometric Calculus" By Hestenes but it seems that it is a hard one to read.
There is also Macdonald's books "Linear and Geometric Algebra" and "Vector and Geometric Algebra", but I don't know how good they are.

 

[FactChecker]

I was not impressed with the Hestenes book. He seemed to be talking in a different language. The Macdonald books seem good. (That's Vector and Geometric Calculus) I got them but have not read very far into them. They do not seem to go very far into the applications. Macdonald put a good summary article on the internet at http://faculty.luther.edu/~macdonal/GA&GC.pdf )

 

[hunt_mat]

One of the interesting Clifford Algebras is Geometric Algebra (GA). GA consolidates a lot of mathematical constructs involving different dimensions into one construct and the mathematics of that construct. The result is that it includes many specially derived algebras that were invented for specific applications. It helps greatly to consolidate all the mathematical manipulations into one logical, methodical system. Then it turns out that so much of physics is just the mathematical manipulations and the basic physics concepts are much simpler in the form of GA. For instance, Maxwell's equations become one simple GA equation. That one GA equation might represent many mathematical manipulations (like a simple matrix equation represents a lot of multiply/adds), but those manipulations are routine and systematic.

That being said, there is a learning curve to master GA and the payoffs are not immediate. Also, a lot of the traditional specialized algebras developed for specific applications are not always easy to translate into GA.

I spend three years working on applications of geometric algebra to rigid body dynamics in which is is extremely powerful and far superior to anything out there at the moment. The book by Doran and Lasenby is becoming a classica but my feeling that it is not well written in some places as it should be and a much better job can be done.

 

[hunt_mat]

Thank you for your answers, they were helpful. I will suggest for people interested in Clifford Algebra to look at this too:
https://www.reddit.com/r/math/comments/5bd2ei/lets_talk_about_geometric_algebra_not_algebraic/

One more question! What is the best book to study Clifford Algebra from a Mathematical point of view(Most of the books out there are for applications). There is "Clifford Algebra to Geometric Calculus" By Hestenes but it seems that it is a hard one to read.
There is also Macdonald's books "Linear and Geometric Algebra" and "Vector and Geometric Algebra", but I don't know how good they are.

They're actually quite readable. They're a little quick for my liking but the exercises are relatively straight forward and can be done rather easily.

 

[Phylosopher]

They're actually quite readable. They're a little quick for my liking but the exercises are relatively straight forward and can be done rather easily.

Does they (Macdonald's books) use the same notation as Hestenes books? The notation in which Geometric Algebra is written in, differ from book to book I believe.

 

[hunt_mat]

Does they (Macdonald's books) use the same notation as Hestenes books? The notation in which Geometric Algebra is written in, differ from book to book I believe.

I have found that notations differ from book to book. It is still a very readable text though.