On The Use of Clifford Algebras

[Xamien]

... or, as an alternative title: I Like Making Things Difficult For Myself.

I will preface this post with the fact that I'm majoring in physics and math, so as to help explain my motivations.

Specifically, I'm interested in the use of Clifford algebras to do many of the calculations common throughout the course of a college career in physics. I've seen suggestions for its use (specifically called geometric algebra or calculus, by one rather passionate individual) and I've seen arguments against. I would like to test the hypothesis myself to see if there really is any benefit to using the Clifford algebras as a linguistic underpinning to my studies, mostly because I am drawn to the alleged computational efficiency benefits.

I've procured some introductions to the subject through the use of Google, but I want more without having to spend $200 at Princeton's Uni press webpage to get the benefit of a hardcopy. Can anyone point me in the right direction? Also, are there any specific subjects that would be invaluable to this endeavor? My level of knowledge is four semesters of Calculus, the basic physics courses for physics majors, with some self-teaching in linear algebra and ordinary differential equations.

To answer any possible questions, I am aware that I am making things difficult for myself taking on more work, but I'm doing it because I enjoy the topics and I have a little brain bandwidth left to spare.

 

[granpa]

have you seen this:
http://geocalc.clas.asu.edu/

 

[MrNerd]

I have a book that has some attempt to teach something about a lot of math necessary in physics. It's called The Road to Reality, by Roger Penrose. It's not exactly a textbook, but it only costs about $20-$30 or so, and you get a lot of information. The entire first third of the book is dedicated to explaining the math in the book, although it's not the greatest thing in the world. Still, it's inexpensive and you get a lot of extra information that you may want to learn about now or later.

And yes, I distinctly remember coming across Clifford Algebra in it.

 

[Sankaku]

What does "making things difficult" have anything to do with it? If it is interesting, study it.

have you seen this:
http://geocalc.clas.asu.edu/

Hestenes' own site is a good one. There are also some things here:

http://www.mrao.cam.ac.uk/~clifford/

in particular:
http://www.mrao.cam.ac.uk/~clifford/publications/abstracts/imag_numbs.html

If you do look at one of Hestenes' books, "New Foundations..." is more accessible than "Clifford Algebra to Geometric Calculus."

 

[Xamien]

have you seen this:
http://geocalc.clas.asu.edu/

Yes, I have. I'm still exploring it.

I have a book that has some attempt to teach something about a lot of math necessary in physics. It's called The Road to Reality, by Roger Penrose. It's not exactly a textbook, but it only costs about $20-$30 or so, and you get a lot of information. The entire first third of the book is dedicated to explaining the math in the book, although it's not the greatest thing in the world. Still, it's inexpensive and you get a lot of extra information that you may want to learn about now or later.

And yes, I distinctly remember coming across Clifford Algebra in it.

Thank you! I'll look it up.

What does "making things difficult" have anything to do with it? If it is interesting, study it.

Hestenes' own site is a good one. There are also some things here:

http://www.mrao.cam.ac.uk/~clifford/

in particular:
http://www.mrao.cam.ac.uk/~clifford/publications/abstracts/imag_numbs.html

If you do look at one of Hestenes' books, "New Foundations..." is more accessible than "Clifford Algebra to Geometric Calculus."

I've lurked very quietly on a lot of forums and paid attention to the discussions of others in different mediums, and as a result have gotten used to seeing a pooh-pooh response in many cases. I appreciate your suggestions and will look through them eagerly.

 

[twofish-quant]

I have a book that has some attempt to teach something about a lot of math necessary in physics. It's called The Road to Reality, by Roger Penrose. It's not exactly a textbook, but it only costs about $20-$30 or so, and you get a lot of information.

You have to be careful about Penrose. The man is brilliant, but he has some ideas are considered seriously crankish by most people.

 

[MrNerd]

You have to be careful about Penrose. The man is brilliant, but he has some ideas are considered seriously crankish by most people.

That's mostly twister theory, right? He did tell the reader quite a few times when he believed that he was diverging from the most commonly viewed things(like string theory), in a rather polite way. He also writes a couple times that he isn't completely covering everything on the subject, like in the alternatives to string theory area. It was mostly one or two alternatives there, I think.

Wasn't Galileo considered crankish by the people in power(church) at the time? I assume you mean crankish as thinking outside the mainstream(like a crackpot). However, they can probably help you to think outside the box. Without a crackpot, who would ask "Is the world orbited by the sun, or does the world orbit the sun?" A crackpot is good(in my opinion) if they don't say things are true because it has to be, but instead use facts and/or math.

Without the "What if?", science never changes(aside from things like, hey everyone, this is related to that thing)

However, some crackpots truly are cracked pots.

 

[Sankaku]

You have to be careful about Penrose. The man is brilliant, but he has some ideas are considered seriously crankish by most people.

Maybe for physics, I doubt so in Mathematics.

I suspect the application of Clifford algebra to Physics is not a "crackpot" idea. Why bother bringing his other (less conventional) opinions into a thread about Clifford algebra?

 

[Xamien]

So, I wanted to be clear on something for which I've gotten one opinion already that I'm understanding correctly. For working with vectors in a plane and determining the bivector from them, I understand it is sufficient to use PQ = P(dot)Q + P(wedge)Q where P(dot)Q = (PQ + QP)/2 and P(wedge)Q = (PQ - QP)/2.

For P(dot)Q, is the calculation correct to use the otherwise normal dot operation and for P(wedge)Q is roughly the same as P X Q, such that the resulting bivector would be written simply as PQ = (dot product) + (cross product)?

Sorry I'm not using the notation code. I haven't gotten used to using it, yet.