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 twicedifferentiable 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, twicedifferentiable) 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 handwavy) Description of Gauge Invariance

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