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From a mathematician's point of view, yes, you should write the Maxwell field equations, at least to see it once, that way because you're showing a very low-lev

From a mathematician's point of view, yes, you should write the Maxwell field equations, at least to see it once, that way because you're showing a very low-level symmetry that even the differential forms approach doesn't get all the way to. Differential forms is a standard approach for general relativity, e.g. MTW. I guess the people pushing this are a little pushy, but this reminds me of the whole pie fight over the Rust community. OK, so they're pushy. Nothing to do with the merits or demerits of the language (or of C for that matter). If you're a baby duck about linear algebra and geometry, there's no need to care about different formalisms. Do whatever works. But it's interesting to see how all of this stuff comes together at different levels, whether it's the geometric product, differential forms, or just linear algebra.

Certhas 4mo ago
The space time approach with E as t wedge x and B as x wedge y is purely linear algebra, not differential forms. As opposed to the weird GA form it actually makes the physically most meaningful symmetry (Lorentz transformations) explicit. That's why it's actually used in Physics. Anti symmetric space time tensors are the absolute standard. Further formulations that reveal other aspects, dualities, symmetries are much more niche and specialized subjects and not how the subject should be taught when first encountering it. https://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism https://en.wikipedia.org/wiki/Covariant_formulation_of_class...
- jsLavaGoat 4mo ago
  OK, well, MTW is a pretty standard GR textbook and it is often cited as a useful text on differential forms for math.
  - Certhas 4mo ago
    Did you mean to reply to someone else? For the record, MTW by now also shows it's age. When I was doing research in GR adjacent fields the experts were rather recommending Wald. Might have just been my bubble of course...
- Loquebantur 4mo ago
  One should teach the next generation the best way possible, and not turn them into conformists. "Standards" are things to be overcome when they've outlived their prime. Disparaging new ideas as "niche" and "specialised" when their explicit aspiration is to be better foundations is motivated reasoning.
  - Certhas 4mo ago
    The ideas of GA aren't new. As the article explains, the ideas of Clifford algebras are common place throughout research and deserve to be introduced earlier. Understanding bivectors and wedge products is much more important than, for example, Euler angles in my opinion. The geometric product on the other hand obscures much of the structure, and serves no pedagogical or fundamental purpose. Not everything that's new is better, just by virtue of being new. You might have misunderstood my point about what's niche, or I misunderstood which formulation of Maxwell the post I was replying to was referring to. Either way, it feels this discussion went off the rails rather immediately...
  - Certhas 4mo ago
    Maybe I wasn't clear: Other formulations that reveal dualities, etc... at the cost of obscuring the fundamental covariance deserve to be treated as secondary.
eigenspace 4mo ago
> From a mathematician's point of view, yes, you should write the Maxwell field equations, at least to see it once, that way because you're showing a very low-level symmetry that even the differential forms approach doesn't get all the way to. Differential forms is a standard approach for general relativity, e.g. MTW. While it's neat to write them all as one equation, I disagree that it's an enlightening perspective to learn. While it seems like writing Maxwell's equations in one equation instead of two is a step forward with even more symmetry, what is actually going on is that you are obscuring the most important part of Maxwell's equations: the gauge structure. Without this, it actually becomes much more hidden just how geometric electromagnetism is. When you write Maxwell's equations as the pair `dF = 0`, `d*F = J`, the first of those two equations is exactly what tells you that this is a gauge theory, and thus may write `F = dA` where `A` is a vector potential. This vector potential then becomes the connection which defines a covariant derivative in a fibre bundle, and one then sees that charged particles follow geodesics now in spacetime, but in an enclosing fibre bundle. This is foundationally important to modern physics, and IMO obscured by writing Maxwell's equations as `∇F = J` ____ n.b. I'm not a particularly big fan of differential forms either, I think it leaves a lot to be desired, and it's super awkward to constantly have to pull out Hodge Duals every time you want to do something that involves the metric, but I'm also unconvinced that geometric algebra is the answer here.
- jsLavaGoat 4mo ago
  What interests a mathematician isn't 100% the same as what interests the physicist. All I'm saying is there is some math there that's interesting and people should see it once for the math.
  - eigenspace 4mo ago
    I guess I'd say my point though is that the gauge structure is the mathematically interesting part of Maxwell's equations. (i.e. the fact that `F` is a closed differential form). Without it, I think it'd be of significantly less mathematical interest because it'd lose almost all of its geometric properties.
cygx 4mo ago
Note that by introducing the co-differential δ, you can write the Maxwell equations as a single expression (δ + d)F = J in the differential forms approach. However, from the perspective of Yang-Mills theory, that's rather questionable as you're stitching together the Bianchi identity and the Yang-Mills equation for no particular reason.
NooneAtAll3 4mo ago
what is MTW?
- bmacho 4mo ago
  https://en.wikipedia.org/wiki/Gravitation_(book) https://en.wikipedia.org/wiki/Gravitation_(book)

Certhas 4mo ago

The space time approach with E as t wedge x and B as x wedge y is purely linear algebra, not differential forms. As opposed to the weird GA form it actually makes the physically most meaningful symmetry (Lorentz transformations) explicit. That's why it's actually used in Physics. Anti symmetric space time tensors are the absolute standard. Further formulations that reveal other aspects, dualities, symmetries are much more niche and specialized subjects and not how the subject should be taught when first encountering it. https://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism https://en.wikipedia.org/wiki/Covariant_formulation_of_class...

jsLavaGoat 4mo ago
OK, well, MTW is a pretty standard GR textbook and it is often cited as a useful text on differential forms for math.
- Certhas 4mo ago
  Did you mean to reply to someone else? For the record, MTW by now also shows it's age. When I was doing research in GR adjacent fields the experts were rather recommending Wald. Might have just been my bubble of course...
Loquebantur 4mo ago
One should teach the next generation the best way possible, and not turn them into conformists. "Standards" are things to be overcome when they've outlived their prime. Disparaging new ideas as "niche" and "specialised" when their explicit aspiration is to be better foundations is motivated reasoning.
- Certhas 4mo ago
  The ideas of GA aren't new. As the article explains, the ideas of Clifford algebras are common place throughout research and deserve to be introduced earlier. Understanding bivectors and wedge products is much more important than, for example, Euler angles in my opinion. The geometric product on the other hand obscures much of the structure, and serves no pedagogical or fundamental purpose. Not everything that's new is better, just by virtue of being new. You might have misunderstood my point about what's niche, or I misunderstood which formulation of Maxwell the post I was replying to was referring to. Either way, it feels this discussion went off the rails rather immediately...
- Certhas 4mo ago
  Maybe I wasn't clear: Other formulations that reveal dualities, etc... at the cost of obscuring the fundamental covariance deserve to be treated as secondary.

eigenspace 4mo ago

> From a mathematician's point of view, yes, you should write the Maxwell field equations, at least to see it once, that way because you're showing a very low-level symmetry that even the differential forms approach doesn't get all the way to. Differential forms is a standard approach for general relativity, e.g. MTW. While it's neat to write them all as one equation, I disagree that it's an enlightening perspective to learn. While it seems like writing Maxwell's equations in one equation instead of two is a step forward with even more symmetry, what is actually going on is that you are obscuring the most important part of Maxwell's equations: the gauge structure. Without this, it actually becomes much more hidden just how geometric electromagnetism is. When you write Maxwell's equations as the pair `dF = 0`, `d*F = J`, the first of those two equations is exactly what tells you that this is a gauge theory, and thus may write `F = dA` where `A` is a vector potential. This vector potential then becomes the connection which defines a covariant derivative in a fibre bundle, and one then sees that charged particles follow geodesics now in spacetime, but in an enclosing fibre bundle. This is foundationally important to modern physics, and IMO obscured by writing Maxwell's equations as `∇F = J` ____ n.b. I'm not a particularly big fan of differential forms either, I think it leaves a lot to be desired, and it's super awkward to constantly have to pull out Hodge Duals every time you want to do something that involves the metric, but I'm also unconvinced that geometric algebra is the answer here.

jsLavaGoat 4mo ago
What interests a mathematician isn't 100% the same as what interests the physicist. All I'm saying is there is some math there that's interesting and people should see it once for the math.
- eigenspace 4mo ago
  I guess I'd say my point though is that the gauge structure is the mathematically interesting part of Maxwell's equations. (i.e. the fact that `F` is a closed differential form). Without it, I think it'd be of significantly less mathematical interest because it'd lose almost all of its geometric properties.

cygx 4mo ago

Note that by introducing the co-differential δ, you can write the Maxwell equations as a single expression (δ + d)F = J in the differential forms approach. However, from the perspective of Yang-Mills theory, that's rather questionable as you're stitching together the Bianchi identity and the Yang-Mills equation for no particular reason.

NooneAtAll3 4mo ago

what is MTW?

bmacho 4mo ago
https://en.wikipedia.org/wiki/Gravitation_(book) https://en.wikipedia.org/wiki/Gravitation_(book)