demagogue on 16/2/2024 at 16:43

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Numberphile has come out with not one but two new videos on the weird apparent phenomenon of -1/12 lurking inside infinite series.

But I really like the last part of the 2nd video... There's a reason why -1/12 pops up in string theory and QFT, and the way he put it, the functions that use it can tame our models that have these infinite contributions & symmetry breaking with smooth weighting functions that preserve the symmetries throughout the series and get the results we actually see in experiments, as if it's nature's way of protecting us from everything blowing up to infinity & breaking the underlying symmetry generating the thing.

I like it because it points to mysteries beyond our understanding, but you get hints about which direction to look; and there's still work to be done to find more key pieces and put them together. Or maybe it's a red herring, but it sure seems like something's in there.

I watched the 2nd video 1st, but now I've watched the 1st video, and it sets up analytic continuation and also ends on a similar punchline.
Anyway, the two videos complement each other well, and here they are in order:

[video=youtube;FmLIGN8ZGdw]https://www.youtube.com/watch?v=FmLIGN8ZGdw[/video]

[video=youtube;beakj767uG4]https://www.youtube.com/watch?v=beakj767uG4[/video]

demagogue on 17/2/2024 at 15:07

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The recommended viewing order is the order they were released & the order I posted them, so the top then the bottom video. Sorry, I shouldn't have even put that into question. It's just that I started drafting my post about #2 before I even knew there was a video #1 posted just before it, and then I tweaked the wording when I found out, but the order they were released seems best.

Cipheron on 18/2/2024 at 11:49

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Keep in mind there are -1/12 videos on that an other channels going a way back, so this is just the latest installment.

demagogue on 1/3/2024 at 19:00

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While Dune is in the news & I saw this video being posted by my favorite math YouTube guru, this is how I felt especially this last week with this QM course I'm going through. XD

[video=youtube;08EyioJ2oSg]https://www.youtube.com/watch?v=08EyioJ2oSg[/video]

Just one part of this last week's problem took me 6 pages of algebra to muddle through. (The rest of it took another 4 pages.)
For reference it was just one part of checking Heisenberg's Uncertainty for a traveling free particle (a wave packet) in 1D.
All I had to do was take the wave function at one slice of time, Ψ(x,0), and get the time dependent version of it Ψ(x,t) and then square it, then I could get everything else I needed to check the uncertainty.

You'd think that'd be simple. The initial function I started with was simple enough.
But holy hell did that get into the weeds!
At one point one exponential had 12 terms inside of it, and that was only one out of 4 terms in the whole equation.
What's crazy is at the end, suddenly everything starts cancelling out, and applied to <x> and <p>, position and momentum, after all of that, you end up with <x>= &#295;lt/m and <p>= &#295;l, lol.

It's not even that it's hard. Each individual step is more or less standard calculus & algebra.
Sometimes there's a very specific trick you have to remember.
But there's just so damn much of it.

demagogue on 2/3/2024 at 19:50

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A free particle is just in open space with nothing else around.
Or if you wanna think about it like Feynman, it's nothing but the holes.

It's actually the most complicated of all the first term problems.
It's a lot easier when some potential blocks the particle somewhere somehow.