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I do love Geometric Albegra and hope it gets further adoption. In my field (physics), the project of moving to geometric algebra currently looks hopeless due to

I do love Geometric Albegra and hope it gets further adoption. In my field (physics), the project of moving to geometric algebra currently looks hopeless due to the inertia of other formalisms and lack of interest from established physicists. The real shame is that the best opportunity to introduce physicists to geometric algebra was a hundred years ago when we were first discovering spinors. First the Pauli algebra and then the Dirac algebra. If physicsts at the time had been made aware of the geometric algebra approach to these things, I think it could have been an appropriate motivation to switch to geometric algebra. Instead, people came up with matrix representations of the Pauli and Dirac algebras and the field is stuck using those and saying "meh. Good enough."

qubex 8y ago
Not “meh, good enough” but rather “eh, computationally equivalent, why bother reformulating something that looks like a black box but is actually a gimmick we can rely upon to give equivalent results and that is understood by everybody in the field?”
- klodolph 8y ago
  Two ways of approaching a problem can be computationally equivalent and yet one can be a clear winner. For example, take a look at Maxwell’s equations—as originally formulated there were 20 different equations, with many equations duplicated for x, y, and z. Heaviside, Gibbs, and Hertz reformulated these using vectors and reduced the number of equations to four. Computationally equivalent, but nobody uses the 20-equation version of Maxwell’s equations any more. Now consider relativity, which was originally motivated by Maxwell’s equations. We can describe flat spacetime as a 4D space with the Minkowski metric, which we can extend into a geometric algebra Cl_1,3(R). We can then describe field and current as multivectors, and we end up with the equation (singular!) ∇ F = µ_0 c J This has the additional property that it captures how observations of electromagnetism change under Lorentz transforms. If you think of a Lorentz transform as just a change of basis in spacetime algebra, and if you think of electricity and magnetism as together being the basis for electromagnetism, then it’s obvious that (for example) a moving observer would see a magnetic field generated by a stationary charge. This is obvious because the basis change from a stationary to moving observer will directly correspond to a basis change from an electric field to a magnetic field. This is simplifying a bit but I find it easier to remember and reason about the geometric algebra version of many of these formulas.
deleted 8y ago
[deleted]
petermcneeley 8y ago
Do you think nature requires such an exotic representation?
- eigenspace 8y ago
  I would argue that there is nothing exotic about geometric algebra and that its a much more natural representation of nature than our current hodge-podge of traditional vectors, tensors, complex numbers, differential forms and spinors.
keldaris 8y ago
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...
impendia 8y ago
Can I ask you (and keldaris) what precisely you mean by geometric algebra, and how you would propose to teach it? I'm a math professor. To me "geometric algebra" refers to any use of geometry in mathematics to illuminate algebraic principles. There are very very many of these -- this article illustrates one good example. I try to incorporate intuition from geometry into my teaching whenever possible. It seems that the two of you are referring to some particular unifying framework, and/or set of definitions and notation. Could I ask you to elaborate on what you mean? Thanks.
- cartep 8y ago
  I'm not any of the two you asked, but given that @eigenspace is a physicist, what you are looking for (notations used, etc) is possibly here: http://geocalc.clas.asu.edu/html/Evolution.html#References http://geocalc.clas.asu.edu/html/Evolution.html#References under "Recent Books". Notably "Clifford Algebra to Geometric Calculus" and "New Foundations for Classical Mechanics" by Hestenes. PD: if you ever attempt to read any of those two, I'd advise to go to the latest (corrected) reprint.
- nanofortnight 8y ago
  https://en.wikipedia.org/wiki/Geometric_algebra https://en.wikipedia.org/wiki/Geometric_algebra Less ambiguously called "Clifford algebras", though Clifford himself prefered the term geometric algebra.
- deleted 8y ago
  [deleted]
- dahart 8y ago
  Geometric algebra has a definition that is more specific than any use of geometry to teach Elementary Algebra on real numbers. https://en.m.wikipedia.org/wiki/Geometric_algebra https://en.m.wikipedia.org/wiki/Geometric_algebra https://en.m.wikipedia.org/wiki/Clifford_algebra https://en.m.wikipedia.org/wiki/Clifford_algebra For teaching geometric algebra, there are curricula and textbooks available... http://geometry.mrao.cam.ac.uk/home/introduction-to-ga/ http://geometry.mrao.cam.ac.uk/home/introduction-to-ga/ https://arxiv.org/abs/1205.5935v1 https://arxiv.org/abs/1205.5935v1 http://www.faculty.luther.edu/~macdonal/GA&GC.pdf http://www.faculty.luther.edu/~macdonal/GA&GC.pdf http://assets.cambridge.org/052148/0221/sample/0521480221WS.pdf http://assets.cambridge.org/052148/0221/sample/0521480221WS.... http://geometricalgebra.org/downloads/ga4cs_toc.pdf http://geometricalgebra.org/downloads/ga4cs_toc.pdf https://www.amazon.com/dp/0792355148/ https://www.amazon.com/dp/0792355148/

qubex 8y ago

Not “meh, good enough” but rather “eh, computationally equivalent, why bother reformulating something that looks like a black box but is actually a gimmick we can rely upon to give equivalent results and that is understood by everybody in the field?”

klodolph 8y ago
Two ways of approaching a problem can be computationally equivalent and yet one can be a clear winner. For example, take a look at Maxwell’s equations—as originally formulated there were 20 different equations, with many equations duplicated for x, y, and z. Heaviside, Gibbs, and Hertz reformulated these using vectors and reduced the number of equations to four. Computationally equivalent, but nobody uses the 20-equation version of Maxwell’s equations any more. Now consider relativity, which was originally motivated by Maxwell’s equations. We can describe flat spacetime as a 4D space with the Minkowski metric, which we can extend into a geometric algebra Cl_1,3(R). We can then describe field and current as multivectors, and we end up with the equation (singular!) ∇ F = µ_0 c J This has the additional property that it captures how observations of electromagnetism change under Lorentz transforms. If you think of a Lorentz transform as just a change of basis in spacetime algebra, and if you think of electricity and magnetism as together being the basis for electromagnetism, then it’s obvious that (for example) a moving observer would see a magnetic field generated by a stationary charge. This is obvious because the basis change from a stationary to moving observer will directly correspond to a basis change from an electric field to a magnetic field. This is simplifying a bit but I find it easier to remember and reason about the geometric algebra version of many of these formulas.

deleted 8y ago

[deleted]

petermcneeley 8y ago

Do you think nature requires such an exotic representation?

eigenspace 8y ago
I would argue that there is nothing exotic about geometric algebra and that its a much more natural representation of nature than our current hodge-podge of traditional vectors, tensors, complex numbers, differential forms and spinors.

keldaris 8y ago

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...

impendia 8y ago

Can I ask you (and keldaris) what precisely you mean by geometric algebra, and how you would propose to teach it? I'm a math professor. To me "geometric algebra" refers to any use of geometry in mathematics to illuminate algebraic principles. There are very very many of these -- this article illustrates one good example. I try to incorporate intuition from geometry into my teaching whenever possible. It seems that the two of you are referring to some particular unifying framework, and/or set of definitions and notation. Could I ask you to elaborate on what you mean? Thanks.

cartep 8y ago
I'm not any of the two you asked, but given that @eigenspace is a physicist, what you are looking for (notations used, etc) is possibly here: http://geocalc.clas.asu.edu/html/Evolution.html#References http://geocalc.clas.asu.edu/html/Evolution.html#References under "Recent Books". Notably "Clifford Algebra to Geometric Calculus" and "New Foundations for Classical Mechanics" by Hestenes. PD: if you ever attempt to read any of those two, I'd advise to go to the latest (corrected) reprint.
nanofortnight 8y ago
https://en.wikipedia.org/wiki/Geometric_algebra https://en.wikipedia.org/wiki/Geometric_algebra Less ambiguously called "Clifford algebras", though Clifford himself prefered the term geometric algebra.
deleted 8y ago
[deleted]
dahart 8y ago
Geometric algebra has a definition that is more specific than any use of geometry to teach Elementary Algebra on real numbers. https://en.m.wikipedia.org/wiki/Geometric_algebra https://en.m.wikipedia.org/wiki/Geometric_algebra https://en.m.wikipedia.org/wiki/Clifford_algebra https://en.m.wikipedia.org/wiki/Clifford_algebra For teaching geometric algebra, there are curricula and textbooks available... http://geometry.mrao.cam.ac.uk/home/introduction-to-ga/ http://geometry.mrao.cam.ac.uk/home/introduction-to-ga/ https://arxiv.org/abs/1205.5935v1 https://arxiv.org/abs/1205.5935v1 http://www.faculty.luther.edu/~macdonal/GA&GC.pdf http://www.faculty.luther.edu/~macdonal/GA&GC.pdf http://assets.cambridge.org/052148/0221/sample/0521480221WS.pdf http://assets.cambridge.org/052148/0221/sample/0521480221WS.... http://geometricalgebra.org/downloads/ga4cs_toc.pdf http://geometricalgebra.org/downloads/ga4cs_toc.pdf https://www.amazon.com/dp/0792355148/ https://www.amazon.com/dp/0792355148/