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In my personal experience physicists are usually rather pragmatic about these things. Not many will support a "project of moving to geometric algebra" for its o

In my personal experience physicists are usually rather pragmatic about these things. Not many will support a "project of moving to geometric algebra" for its own sake, or the sake of perceived mathematical elegance - I know I wouldn't, even though I spent some time studying it just for fun. On the other hand, find a problem where geometric algebra gets you to a useful insight in a quicker or more natural way and few will object to treating it in that language. For most of the usecases associated with the examples you mention - Pauli and Dirac algebras - I don't think it matters very much. For instance, I haven't seen a decent QM course that doesn't devote at least a lecture or two to discussing the origin of Pauli matrices and the associated abstract algebra and group theory - as they should - but I don't think the course would be any more concise or useful (in terms of the physical content) if it then proceeded to treat QM in the language of geometric algebra. There's nothing wrong with the matrix representations (especially in a more computational context), though obviously one should know where the representation comes from and how to derive it.

eigenspace 8y ago
> On the other hand, find a problem where geometric algebra gets you to a useful insight in a quicker or more natural way and few will object to treating it in that language. I would argue that E&M and anything involving rotations (or Lorenzt boosts!) are two great examples where GA is miles ahead of anything other than differential forms but still has modest advantages of differential forms as well. The main advantage I see in geometric algebra is that its a unifying framework. The same langauge used to understand E&M in GA is also useful in quantum mechanics, general relativity, complex analysis, linear algebra, etc. I'd argue that physicists don't mind this mish-mash of notations because they've already put in the work to learn them all separtely but I think there are real advantages (especially pedagogical) to unifying them into one framework.
- keldaris 8y ago
  I think that last point really needs to start with a thorough reform of basic mathematical education (starting from undergrad calculus) to have any chance of real success. The present mish-mash of notations, however disjointed, lends itself very well to teaching people who don't know much beyond basic real analysis and differential equations. You can't just throw them in a GA-based E&M class without completely changing the entire curriculum first. Your first point got me curious, though. I found E&M in differential forms quite satisfying from an intellectual point of view (although of little practical benefit to the problems I was interested in compared to the classical approach, which I was plenty familiar with already). What advantages do you see for GA vs. differential forms in this context? This might make for a few interesting seminars with our math folks...
  - wolfgke 8y ago
    > You can't just throw them in a GA-based E&M class without completely changing the entire curriculum first. Preparing for the additional requirements of the lectures of the coming semester is whar the semester break is for.
    - keldaris 8y ago
      Sure, as long as you don't expect anyone to complete the course or are willing to play catch up for most of the semester. In other words, I'd love to plan courses that way, but no university administrator would ever let me (instead I restrict myself to graduate seminars and beyond).
    - jacobolus 8y ago
      Nah, you really want to start students learning the GA formalism in their linear algebra and vector calculus classes, if not in high school. Especially important in high school is to dive into not only the “complex numbers” but also the “split complex numbers”.
  - eigenspace 8y ago
    I should be clear here that my longing for a physics community that embraces GA should not be confused with any level of optimism that it'll actually happen. I agree its a monstrously difficult task and our current formalisms are more or less good enough. A man can dream though! One advantage I see in the GA treatment of E&M is in radiation. If one takes the Faraday bivector for a stationary charge (ie. the Coulumb field) and then just 'rotates' it into a moving frame using the GA rotors you can find the standard radiation fields of E&M quite easily. This is cool because it's not only a neat proceedure mathematically, but its computationally quite efficient, say if you need to find the fields produced by N moving charged particles. I'm sure there are more examples of things like this but it's been a while since I last looked at E&M in GA so none are coming to mind right now.
    - keldaris 8y ago
      That's a neat example, thanks!
- bsder 8y ago
  How does Geometric Algebra link into the 4-vector representation of E&M? https://mitpress.mit.edu/books/collective-electrodynamics https://mitpress.mit.edu/books/collective-electrodynamics

eigenspace 8y ago

> On the other hand, find a problem where geometric algebra gets you to a useful insight in a quicker or more natural way and few will object to treating it in that language. I would argue that E&M and anything involving rotations (or Lorenzt boosts!) are two great examples where GA is miles ahead of anything other than differential forms but still has modest advantages of differential forms as well. The main advantage I see in geometric algebra is that its a unifying framework. The same langauge used to understand E&M in GA is also useful in quantum mechanics, general relativity, complex analysis, linear algebra, etc. I'd argue that physicists don't mind this mish-mash of notations because they've already put in the work to learn them all separtely but I think there are real advantages (especially pedagogical) to unifying them into one framework.

keldaris 8y ago
I think that last point really needs to start with a thorough reform of basic mathematical education (starting from undergrad calculus) to have any chance of real success. The present mish-mash of notations, however disjointed, lends itself very well to teaching people who don't know much beyond basic real analysis and differential equations. You can't just throw them in a GA-based E&M class without completely changing the entire curriculum first. Your first point got me curious, though. I found E&M in differential forms quite satisfying from an intellectual point of view (although of little practical benefit to the problems I was interested in compared to the classical approach, which I was plenty familiar with already). What advantages do you see for GA vs. differential forms in this context? This might make for a few interesting seminars with our math folks...
- wolfgke 8y ago
  > You can't just throw them in a GA-based E&M class without completely changing the entire curriculum first. Preparing for the additional requirements of the lectures of the coming semester is whar the semester break is for.
  - keldaris 8y ago
    Sure, as long as you don't expect anyone to complete the course or are willing to play catch up for most of the semester. In other words, I'd love to plan courses that way, but no university administrator would ever let me (instead I restrict myself to graduate seminars and beyond).
  - jacobolus 8y ago
    Nah, you really want to start students learning the GA formalism in their linear algebra and vector calculus classes, if not in high school. Especially important in high school is to dive into not only the “complex numbers” but also the “split complex numbers”.
- eigenspace 8y ago
  I should be clear here that my longing for a physics community that embraces GA should not be confused with any level of optimism that it'll actually happen. I agree its a monstrously difficult task and our current formalisms are more or less good enough. A man can dream though! One advantage I see in the GA treatment of E&M is in radiation. If one takes the Faraday bivector for a stationary charge (ie. the Coulumb field) and then just 'rotates' it into a moving frame using the GA rotors you can find the standard radiation fields of E&M quite easily. This is cool because it's not only a neat proceedure mathematically, but its computationally quite efficient, say if you need to find the fields produced by N moving charged particles. I'm sure there are more examples of things like this but it's been a while since I last looked at E&M in GA so none are coming to mind right now.
  - keldaris 8y ago
    That's a neat example, thanks!
bsder 8y ago
How does Geometric Algebra link into the 4-vector representation of E&M? https://mitpress.mit.edu/books/collective-electrodynamics https://mitpress.mit.edu/books/collective-electrodynamics