| I 100% agree with you in all respects — I don't come from a physics background but I hear you loud and clear. I think the 'why' is deep and psycho-historical in nature: - we're exiting the "industrial" mindset where everyone is the same making the same products, to a wider topology of knowledge and skills (more and wider horizontals, more and bigger verticals, 'average' profiles become 'scattered'). This clearly drives a need to "learn a little bit of a lot of things" even at expert level. - The walls and denial you expose here is to me but a symptom of the disease that current academia will either have to heal or die of. Seeing how Khan (and thousands of Udemy's after them, indies) changed the landscape, my money is on a major paradigm shift incoming for academia (it's already done, they just don't seem to know it yet as institutions, most of them). Lots and lots of great teachers around the world almost freely sharing incredible hands-on knowledge and insight. - Some applied domains with dramatic tension of the demand side (lots of positions to fill) don't have the luxury of elitism and massively adopt "pragmatic" approaches especially in learning. Software dev, programming and tech in general is much like that — the "one liner" installs and 1-page "getting started", all the intelligence solely put into making things intelligible and usable is, frankly, quite humbling and inspiring in that field. A very good side of the SV/Cali culture. So, examples of how to proceed next really do exist. Now when I think back of topics that I hurt my head against for months or years, that a simple 20-minute video could 'unlock'... Why, why do we not make it a staple of "teaching" to at least consider 2-3 angles to make sure everyone's got a fair chance at getting at least 1? - On the topic of hubris and laziness, this is where physics went astray, imho. Too much hubris and not enough laziness. That was back in the 1980s and it took 40 years to realize, probably 10-20 more to "fix", if ever before we build a new system (see above). That being said, > Geometric Algebra (GA) was my "lightbulb" moment where I finally understood where Dirac matrices, Pauli matrices, and the like come from and why they have the structure that they do. YES, please! Geometric algebra seems like the thing that could blow my mind too. I am very visual, to a fault maybe. Would you have a 'favorite' resource to share? (book, course, youtube, whatever?) |
The reason Physics "went wrong" is that in 3D space (only!) the mathematics of areas and vectors is coincidentally isomorphic, so it's possible to cheat and use only vectors and scalars and then everything "works". Similarly, volumes and scalars are easily confused as well, and appear to work fine.
GA has no such restrictions and the same formulas work in all dimensions, including high-dimensional or with degenerate metrics. Problems from classical geometry such as finding tangent lines to circles can be trivially extended to finding tangent hyperplanes to hyperspheres, even for very complex problems.
The formalities of GA force you to include things like the square of the unit pseudoscalar in some physics formulas that were accidentally dropped in the traditional form because in 3D this is just "1" and hence easily overlooked. This makes some formulas weirdly difficult to extend to become relativistic, when in fact the problem was just the "weak typing" of vector algebra.
Vector calculus also inherently requires a basis, which is an easy way to get bogged down in the weeds and get confused by issues with the algebra itself instead of the truly "hard" aspects of the problem.
Generally, the "lightbulb" moment for me was that Geometric Algebra has various subsets that are also closed algebras in their own right. For example, the "even" subset of a 3D GA is isomorphic to Quaternions, and the even subset of a 2D GA is basically the same thing as a Complex number. The various "named matrices" are just other subsets of 3D or 4D GAs. Physicists tend to avoid the full general case and simplify their algebras down to the special subset cases, using the historical names and greek symbols. We have to keep the symbols, you see, because otherwise you wouldn't be able to read 2000-year-old ancient greek texts, or... something.
University Physics is actually a study of the History of Physical Philosophy. The computer science equivalent would be learning about abacuses for the entire first semester, then progressing to mechanical calculators in the second semester, vacuum tubes in the second year, and so forth, only to briefly touch on transistors by the end of the third year. Postgraduate research students would be finally told about modern silicon chips and software development, but by this point they're so used to wiring up breadboards manually that it's too late to teach them how to do anything properly.
Starting with something elegant like pure functional programming in the first year is how I studied Computer Science, but I only found out about Geometric Algebra existing after I graduated Physics. It's nuts.
For background reading:
David Hestenes is one of the few physicists trying to reformulate physics in terms of GA: http://geocalc.clas.asu.edu/html/Overview.html
There's lots of papers around: http://geocalc.clas.asu.edu/html/GAinQM.html
Wikipedia is an okay starting point, but not amazing: https://en.wikipedia.org/wiki/Geometric_algebra
Enkimute's "Ganja.js" online demos are amazing, unfortunately the source was written by one of those crazy maths people who think that terseness helps readability: https://enkimute.github.io/ganja.js/examples/coffeeshop.html...
Real industrial use is few and far between, but at least a few folk have discovered that GA is ideal for robotics. Unfortunately, not everyone got the message, and most robotics software libraries are firmly vector/matrix based and have all the usual issues like numerical instability and gimbal-lock. Fun stuff.