The Guy Made Most Physics Theories Redundant.

His discoveries made famous physicists' theories redundant... but also a lot easier to solve!

Hermann Weyl contributed a lot to physics and math, including showing how Maxwell's Electromagnetism could be perfectly combined with Einstein's Relativity. However outside the physics and math circles, he isn't exactly famous.

In this video we start by looking at scalar and vector fields - regions of space that can be described by a numerical value and a vector value at each point in space respectively. We also look at the gradient and curl operators, written with a downward pointing triangle known as a the "nabla" or "del" operator.

A fun mathematical fact: The curl of the gradient of any scalar field is always zero (as long as the scalar field is continuous and twice-differentiable - but all our scalar fields are defo those things lol).

This is fun for physicists too - because this leads to something known as Gauge Invariance.

In the theory of electromagnetism, magnetic fields are used to describe how forces act on other magnets placed in the field. They are vector fields, and are usually labelled with the letter B. Now it turns out that B fields can sometimes be more simply described by another type of vector field known as the "vector potential", written with the letter A. If we take the curl of A, we get the B field.

But this must mean that for any given B field, there are multiple possible (and allowed) A fields. Because if we find one A field that works, such that its curl DOES give the B field we are studying, then we can also find multiple other A fields simply by adding on the gradient of any (continuous, twice-differentiable) scalar field.

Because this way, the curl of our new field (A' or "A prime"), will be the curl of A + the curl of the added bit - which we said is zero. So we get the same physical magnetic field B from our new A' field, as we did with our old A field.

And since the scalar field could be ABSOLUTELY ANYTHING we want, we can make infinitely many allowed A fields if we know just one.

This idea is gauge invariance - and it's a redundancy in the math of physics theories such as electromagnetism, relativity, and even quantum physics. It allows us to do very exciting things such as solve mathematically different problems simply by "switching to another gauge", (such as by finding a new A' to work with).

An intuitive explanation of gauge invariance comes from looking at electric fields, themselves written as the gradient of a "scalar potential". These scalar potentials are also called "electric potential" fields, and the difference between the values at two points within one such field gives us potential difference. This is the same as the voltage we study when we learn about circuits!

Now two different potential fields can be used to describe the same electric field. We can generate two different fields simply by shifting the value of one field at every point by the same amount. This way, the potential difference between two points still stays the same. We have, in essence, seen two different gauges that give us the same physical results (i.e. the same potential difference).

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Chapters:
0:00 - Hermann Weyl: Making Physics Redundant
1:12 - Scalar and Vector Fields, Gradient and Curl Operators
3:37 - A Fun Mathematical Coincidence
4:05 - The Vector Potential in Electromagnetism
5:24 - Gauge Invariance - the Redundancy!
7:16 - An Intuitive (but slightly hand-wavy) Description of Gauge Invariance

Videos in Cards:

[ParthGChannel]

Hi friends, thank you so much for watching! As always, please let me know what topics you'd like me to cover in the future, and hit that subscribe button for more fun physics content. I really appreciate your support :)

[EricKolotyluk]

Either I am getting smarter, or you are getting better at explaining things... This is one of my favorite channels.

[akhilmarar]

Parth.. you're awesome Brother. As a physics teacher, I show your videos to my students. You explanations are visual, clear and simplified. Kudos to you as well as gratitude from me and my students. 😊

[TerryBollinger]

Parth, thank you for a great and easily understood explanation!

Your addition of 3 to all of the numbers reminded me of a gauge invariance analogy I've used a couple of times: When you are paddling a boat at the surface of a lake it doesn't matter whether the lake is 10 meters deep or 1000 meters deep — that is, your boat paddling is “gauge invariant” for the depth of the lake. As long as your wave activity remains confined to near the surface the depth of the lake doesn't matter.

Yet that depth is real in terms of the energy it contains, e.g., if the lake is drained to a lower level. Such a deep lake storing lots of energy corresponds to a high electric field and energizes an entire region of space in which smaller-scale electromagnetic activity is occurring.

[robertschlesinger]

Excellent video, Parth, as always. Very interesting, informative and worthwhile video.

[quitchiboo]

The Moment you started to do your little plug for subscribing to the channel, I realized how valuable you had made those 4 min for me already. I clicked all the things, you deserve it! The way you explained curl and gradient instantly made it click for me, thanks man!

[catman]

THANKS!! Very interesting, and nice to see you again on YouTube
🖖😊

[swamihuman]

- As a fellow teacher, I appreciate your clear/concise/insightful presentation.
- Keep up the great content...

[jeremylevitt]

This was very well structured and equally well explained. Thanks for taking the time to make this!

Although, the youtube algo's a little scary, LOL. Just started reading his Theory of Groups and QM a couple weeks back.

[FrancisFjordCupola]

Really love video's like these. Thank you for highlighting an important part of physics that's often glossed over.

[physicsadhyaapak]

Although, I know about Gauge Invariance before, still the way he describes get you thr new insights of this same topic.

[bryangohmppac]

Wait, so this video is about Hermann Weyl's contribution or just plain ordinary explanation of E and B field and its corresponding scalar and vector potential

[eqwerewrqwerqre]

Very well made video, good content and explanation. My E&M professor kinda skipped over all that gauge stuff and left us scratching our heads at why we were allowed to do that so this is the first actual explanation of why I've seen.

A note though: this is the first video of yours I've seen and i was strongly avoiding it due to the "clickbaity" nature of the title. Whenever i see things like "... changed physics forever" or "physics will never be the same" or something like that it turns me off cause there's just so many people out here trying to trick people who have no formal education into feeling like they received some profound knowledge that really doesn't mean anything and that the creator will just package up and ship out to them over and over again at least once a week.

You're not that, but the title could've fooled me lmao. Had the title been something more addressed to the material like "Herman Weyl and Gauge Invariance" i would've clicked on it at light speed. You could even keep the flashy title and the information like, "The guy who made physics theories redundant - Herman Weyl and gauge invariance".

To end, i understand. The algorithm is a fickle thing, and i really do appreciate you making this very good video, and i do understand I'm probably in the infinitesimal minority here. And i love you, and i hope you're doing well.

[shwifty]

Oh this is so interesting. When I studied classical field theory gauge invariance took such a central spot, didnt know it was discovered only decades after SRT

[stephenpalmer]

you really help ed with my understanding of the Vector Potential. thanks.

[picksalot]

Excellent, lucid explanation. I'll be watching more of your videos. Thanks 👍

[TNYDCK]

To summarise "the gradient of the curl of a scalar field" ...

Something like a temperature field would be a scalar field and because the temperature does not have a direction it does not rotate. So, rotation is always zero therefore the slope is always zero.

[teacupxcupx]

during the graduation i only found books with lenghty derivations without such a deep physical understanding.now i understand the meaning of these terms.thank you sir.

[mienzillaz]

Maybe if people were using his name instead calling him "a guy" would help with recognition.

[alanx]

extended electrodynamics has the arbitrary gauge transformations undone and now the equations have terms of and predict scalar waves, scalar long. waves, and curl free gradient driven current densities, they have experimentally verified it.