Understanding QFT - Episode 2: Developing the math of spin
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- čas přidán 17. 06. 2024
- This is the best explanation of spin in the internet or you get your money back
Help me pay my student debt by joining my Patreon (please...): / highlyentropicmind
Or you can help me with one time donations:
Buy me a coffee: www.buymeacoffee.com/highlyentrc
Ko-fi: ko-fi.com/highlyentropicmind
Paypal: paypal.me/HighlyEntropicMind?...
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Special thanks to my producers:
Manuel Marquez, Carlo Fazioli, Lana Montegazza, OtiumOtiosum, Valerie Hyde, T Highfill, Ryan Roberts, Cookie_T and Azure Azalia
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Credits:
Script, Host, Filming, Directing, Editing, Summoning, Production: Fernando Franco Félix
Animations and graphics: Fernando Franco Félix, Oscar Flores, Chris Diaz and Amit Mourya
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Sources:
All the scientific papers I read to make this video are listed in this article: docs.google.com/document/d/1k...
But the most important ones were:
"The electron as a vector wave" by Charles G Darwin
"On the quantum mechanics of magnetic electrons" By Wolfgang Pauli
"The experimental proof of the Directional quantization in the magnetic field" by Stern and Gerlach
"Wrong Theory-Right Experiment: The Significance of the Stern-Gerlach Experiments" by Friedel Weinert
"The Stern-Gerlach Experiment Revisited" by Horst Schmidt-Böcking and others
"Understanding Heisenberg's 'Magical' Paper of July 1925: a New Look at the Calculational Details
A lot of the research for this video also came from:
Mehra, Jagdish, and Helmut Rechenberg. The historical development of quantum theory. Vol. 1, 2 and 3, Springer Science & Business Media, 2000.
I also used Griffith's book as a guide:
Griffiths, David J., and Darrell F. Schroeter. Introduction to quantum mechanics. Cambridge university press, 2018.
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Chapters:
00:00 - Intro and recap
04:25 - Right hand rule
08:14 - Polarization of light
10:51 - Calcite is weird
14:18 - What about second polarizer?
16:19 - Polarization vector
20:51 - Electrons are polarized
23:58 - SG device as a calcite crystal
27:14 - SG device as a polarizer
28:55 - Two SG polarizers
30:46 - Epiphany (probability waves)
33:21 - Probability space
36:48 - Probability waves
38:53 - Smelling Schrödinger's cat
42:48 - Patreon stuff
44:39 - Trying to use probability space
47:13 - Probability space is dead
49:54 - LONG LIVE AMPLITUDE SPACE!
51:46 - The fuck is phase?
54:28 - The SG experiment in a single formula
56:41 - Angles of energy?
58:55 - Dot product
01:01:36 - Measurements as vectors in amplitude space
01:02:32 - Change of base
01:03:35 - Real world example
01:06:38 - Measurements are matrix multiplications
01:08:36 - Matrix multiplications are rotations
01:10:28 - Heisenberg being stupid
01:14:20 - Cancer and radiation
This is where it gets hard
01:16:21 - Spin in 3D
01:17:50 - Spheres have two degrees of freedom
01:19:41 - Inventing a new kind of number
01:24:01 - New numbers mess with amplitude
01:25:28 - New numbers mess with dot products
01:26:45 - Dirac makes the funniest joke of the century
01:27:57 - It came from the fourth dimension
01:29:16 - The return of the phase
01:31:55 - What Darwin actually discovered
01:34:23 - Measurements as rotations in 4 dimensions
01:35:23 - Determinant = 1
01:36:26 - Unitarity
01:39:45 - Group properties
01:43:16 - Information is a physical property
01:45:16 - The properties of information
01:45:16 - Intro to symmetry
01:47:21 - Recap
01:48:42 - End... Wait, Pauli matrices?
01:49:42 - End and thanks
Hello everyone, here are my thoughts on making this video
0.- First of all, thanks to all my producers who have helped me out over this time, specially Manuel Marquez, Carlo Faziolli and Lana Montegazza. Oddly enough most of their help came after I had recorded the video, so some of them are not mentioned in the video itself, but they will be thanked in every video from now on. Thank you guys, from the bottom of my heart
1.- Length of the video and length of production: This video took me twice as long as I had expected because it ended up being twice as long as I thought it would be. In hindsight I should have split the video in two, one about spin in 2D and another about spin in 3D. In the future I will ensure no video is longer than one hour
2.- Animations: Making the visuals for this video was a harrowing experience which I never want to experience again. Working with other people is hard, specially when they don't understand your vision (or don't care to understand it). Thankfully now I have the skills to make most of the animations my videos require. This increases my workload, but oddly enough it also simplifies it, and hopefully it will improve quality
3.- Research is hard man: Researching for this video was an incredibly fun but also frustrating experience, and it took a LONG time. I will make a video about that
4.- I am already writing the next one, and I can't wait for you to watch it
Finally, you can help me out with a one time donation or by joining my Patreon:
Patreon: www.patreon.com/HighlyEntropicMind
Buy me a coffee: www.buymeacoffee.com/highlyentrc
Ko-fi: ko-fi.com/highlyentropicmind
Paypal: paypal.me/HighlyEntropicMind?country.x=MX&locale.x=en_US
Okay, but did you actually know that the wave function is not the square root of probability? I just really wanted to tell you that.
@@narfwhals7843 The probability is the magnitude of the amplitude, if that's what you mean, and I do mention that in the video
@@HighlyEntropicMind My joke failed and my day is ruined. But I have the video to look forward to, so that's nice.
FINALLY
if you have a script, is there a possibility for captions?
i really enjoyed the references to other youtube videos, like hackenbush video (rest in peace), and the iron farm video!
300 million educations in one video
If you look at the chapters you can see how I structure these videos in my mind, in each section we just have to learn one thing, and each thing builds on the previous ones, or is related to them in some way, so maybe not 300 million, but around 53
❤❤
If episode 1 is comedy gold in education, episode 2 is glory. Absolutely loved the episode the whole way through, was worth the 2 hours of watching and 7 months of waiting
I am already excited for episode 3, can't wait for it, or rather, I'm ready to wait a million years for it
Seven months... ouch, I was trying to not think about that. I'll try to not take more than 3 months from now on
@@HighlyEntropicMind it's okay mate. quality over quantity
@@HighlyEntropicMindDon't sweat it. Totally worth the wait.
13:00 Ah I know this! Canadian balsam resin was used due to its clarity and the similarity of its refractive index to that of crown glass (n = 1.55). It has been regularly used in microspocy for the same reason.
Wow! Thank you! It makes so much sense
Brilliant video, can't wait for part 3!
I remember being completely lost by SU(2) and spin and all this stuff back in the days when I studied physics at university. All the maths and the Pauli matrices felt like they were just thrown at us almost entirely arbitrarily, without taking the time to really motivate their use from first principles. If I'd had more lecturers like you maybe I would've continued in physics after graduating.
The teachers failed us all, that's why I'm making these videos
Holy smokes. You just made the double slit electron measuring non interference thingy make a whole lot more sense to me. I wish there was a nice animation of the interference with waves when I was in school. That's a handy visualization.
I'm a normie loving your simplified breakdowns. Don't stop.
If this was a musical, this would be my cue to start singing "Don't stop me now"
You’re already my favourite educator on CZcams. And history tells me you’re changing your approach towards video making to match CZcams’s algorithm expectations. And you’re forgiven in advance 😂! But, if I could ask you one thing, it’d be don’t change your ways. You’re awesome just the way you are. Thank you!
Thank you, but I don't know what you mean. How do you think I changed?
100% in agreement with you!. Do not change your ways. Your style is your unique/original trademark🏆
@@HighlyEntropicMind you didn’t!! But I have followed several other educators on CZcams and it seems that, in time, they all converge to a certain “CZcams standard”. You have not changed, and I hope you never do! 😉
That was the best explanation of spin that I have ever heard. You clarified all my questions.. And the perfect treatment of why 720degrees is just once in phase space was spot on. Thanks my friend.😊
This was my goal, make this the best explanation of spin on the internet
Dude, thank you. I m looking forward to watch this tomorrow!
Finally, you uploaded it! I really enjoyed this stuff. Thanks!
finally!! this is going to be such a good watch
This video is genuinely absolutely fantastic.
What a great video. thanks a lot I was waiting for the release
You just filled in so many gaps I've been struggling with for many years. Decades, if I'm honest! Thank you so much for these videos. You are an excellent communicator. ❤
I'm gonna enjoy this video so much. I really appreciate all the effort you put into it.
Oh my god..this video feels so nostalgic considering how we've seen all your progress in the community tab. I cant believe it's finally here. Thankyou so much. Take care man!
Jesus Christ!! You made a QM video numuro uno on YT..
I watch all the qm videos on CZcams...all are unique but yours top them all!! What a riveting and revelation of QM..
Presenting such a topic with boundlessly enthousiasm, performance, acting that is not irritating.. but is entertaining...adding in just the right amount of qm along the way...
I don't think anyone can beat it easily..you understand the topic so well and even more you know how to share it and not teach it!! Which is a shock to me!! Because most nerds don't posses these both characteristics!!
So with thousands thanks please make more videos on your favourite topics...
Thank you so much for saying this, please share this video so that more people will enjoy it
This is even more wonderful than Part 1.
I never thought I would get an understanding of the mathematics underlying this. Thank you!
Great video. Commenting to support the it's presence in the algorithm.
I love this style of video, incredibly engaging ❤
Waited for this Episode so long!!
What an outstanding video! Definately one of the best videos i've seen on spin.
Thank you so much for making these videos! I really appreciate that you go so in depth and really explain the topics - not many channels are willing to do so!
Thank you
Woah, this was awesome! I absolutely loved this! So glad I clicked on this video
I'm glad you clicked on it too, now share it with your friends!
your videos are amazing, love all the slow construction you make of the concepts
Thanks. That's the only way I can learn
@@HighlyEntropicMind saame, to understand what's up with qm I actually searched for schrodinger's paper and dirac's "principles of quantum mechanics" where he builds everything, little by little, and explains the physical consequences of every single detail of the math built there, really great book
@@gabitheancient7664 I highly recommend you read this paper from Darwin, when he said that maybe space is just the integration variable for the probability wave I was fucking mindblown
royalsocietypublishing.org/doi/epdf/10.1098/rspa.1927.0179
@@HighlyEntropicMindthanks for the recommendations, will read it later
I've been looking forward to this
Thank you for your passion! This stuff is great.
You are brilliant in so many ways. The linear algebra story with Heisenberg had me in tears. Keep doing what you do!
thank you so much for saying this
No, gracias a ti por estos dos vídeos largos! Ya estaré viendo con paciencia los otros de tu canal. 👍
Highly appriciated for your work man❤
What an amazing video. Loved it
Good job Fer!!… once again, you have shown us how fun it can be to learn complex scientific concepts. You have a unique ability to understand and "translate" mathematics and physics using friendly language and analogies. You should be in the academia! Greetings🍻🥂
Hands down the best expositor of complex scientific ideas I've ever seen!
you just have to understand many simple things that together make single complex thing
FINALLY YOU UPLOADED
It's here!!
I rarely ever comment on videos but I feel like your channel is just so awesome that I have to support it in every way I can. It’s really incredible how you can convey complex information in such a fun and memorable way. I am currently in my second semester of studying physics and not too long ago I found out that I actually have ADHD which is something that I recently have started to really struggle with. Your videos are playing a huge part in keeping me motivated throughout this journey. You are an inspiration and I hope you can keep blessing us with these works of art!💫
thanks, I really needed this
The 'does the box stink or not' axis cracks me up every time.
Thank you for this awesome video!!!
you're welcome
Your videos are EXCELLENT!!!
THANK YOU
thank you for your nice comment
Just watched both episodes back to back and I must say that this was brilliant. I did my degree in Physics and while I've had a decent foundation in the principles of quantum mechanics there has been some gaps in my understanding. These videos have really helped me build a better intuition for this subject, I'm really looking forward to the rest of this series. Also, 19:53, Introduction to Electrodynamics by Griffiths is hands down my favorite Physics textbook. His writing style really feels like you're having a conversation with him and the material is pretty easy to digest.
Very very VERY great videos! I come from the math side of all of this so hearing it actually explained as physics is incredible, and your content is funny to boot! Keep up the great work!
Thank you so much, comments like these motivate me
this is so good. the explanation of moving from an asymmetrical probability space to one that's plotted on the unit circle, and how that gives rise to amplitudes that need to be squared to generate probabilities, was a real light bulb moment for me
Thank you, people like you make all that work worth it
@@HighlyEntropicMind seriously, no, thank YOU-for making the effort to write and record this, for putting it on the internet for free, and for sharing your knowledge! it's taken me a month to get through this video, just watching it in parts now and then, and it's been like reading a good book on the subject. there have been multiple light bulb moments, and i genuinely feel like i have a way better grasp of the concepts involved than i did when i started. you are a really good teacher and i can't wait for the next one!
Amazing review. Ty!
I love this style of video. It's the perfect antidote to "shut up and calculate."
I'm not going to claim to understand everything here, but I greatly appreciate having all of the historical context. Not just oh hey this is spin and this is why it's represented like this here's your equation done. But the actual full history of these are the questions they were trying to answer these are the experiments that they did to try to answer those questions these are the questions that were brought up by those experiments these are the equations that finally resulted from putting together these decades of investigation.
It really takes a topic that can seem very opaque from the outside and unfolds it into an array of colorful and accessible stories.
OMG, I love these. Will keep watching as long as you are making them. (Will probably rewatch them as well to take notes and challenge my understanding).
Your lessons remind me of Feynman
That was a really good video. I found it randomly but you interested me til the end. And now I finally understand matrices :D
Thank you so much. Please share this video with your friends
incredible accomplishment of a video! extremely clear, very engaging, detailed, the works
thank you! please share it so that more people will see it (and so that I get paid)
one of those videos im so excited to let simmer in my head for a week and then come back and watch again and repeat that process for months
Heisenberg is a great illustration of how knowledge and intelligence are separate quantities.
Excellently put
Thank you so much for this video! I'm currently halfway thru my QM class was beginning to feel a bit lost in the math. You've really helped ground my thinking.
That being said, the whole time I was watching your video I had this sense of unreality. It took me the full two hours of watching to realize that it was because my desktop wallpaper was the same as your background. I guess I was just too engrossed to notice.
You are the kind of person I made this video for. I too struggled a lot to learn these things so I decided to create the resources I wish I had had
I was waiting for a new video! Thank you, will make sure I drop you a Kofi
hell yeah watching this asap
Awesome work
When I was learning QM decade ago, we started totally from the opposite end. We were defining algebra and all those magical terms like unity, transpose, determinant. Later, much later, we started to develop QM theory and I have learned how to do the math, but I have never understood what was I doing with those spin vectors. Try to imagine associativity or unitarity just from its definition... Well, now, 5 years after I got PhD in physics, I know the answer.😅 Maybe I am just a nerd (no, I clearly have to be one), but watching videos like this makes my day. Oh crap, now I will be thinking about determinants the whole week...
1:42:19 Animations department 🤣 It is always great working with you
Finally! Lemme get a snack before I watch this
hearing you exclaim the end of the video was so disorienting. realizing 3 hours went by like that. you paced this seiries phenomenally and I'm sure to watch the remaining when you get to them.
Thank you, you have no idea
fantastic video! I just love, how you combine comedy and quantum physics, and youre spanish accent is adoroble! although.... i really do like pilot wave theory... just kidding, go on with this video style and take the time you need for them!
I was struggling with flux but this is a neat explanation. Now to understand spin...
Flux? In what context?
Great video btw :)
i really whould like to here a ful lecture series in university held by you! Youre such a good explainer!
Thanks, but I would not like to give a university lecture
This is insane. You are insane. Thank you
I don't suffer from insanity, I enjoy it
This video was polarizing 🎉
Yes!
I was waiting for this vid for so long
You are the best.
Amazing effort. Let's share, like, subscribe and comment so the algorithm notices.
id love to here, now, sincerely appreciate your video❤❤ i find it very. very! inspiring
1:11:10 Honestly I can understand how Heisenberg must have felt. I spent six months trying to come up with my own definitions and formalizations to build a system within linear algebra that could describe other geometric properties. Then I learned I was basically trying to build Clifford algebras from scratch.
Be proud, this is a problem only smart people have
I LOVE THE EXPLANATION OF i IN THIS VIDEO!!
Thank you
OMG thanks so much
Love ur videos man!,!,!,!,!
thank you
Dude.. you are such a dork. I love it! I wish I were able to be a patron but as I can not, I would love to help in other ways
thank you, share this video with your friends
Love the math for the meta verse..😮❤😂.so fine.bonus ..very much fun!
Finally out🎉 I've been waiting for this one ever since the 1st came out, I think this series is a must-watch to understand all the thought process that lies beneath all of quantum physics. Very useful, and a big part of what made me really look into the subject😊
Same i wanted this video so bad
49:41 I cracked up laughing 😂😂😂😂 great video and a great explanation of some difficult concepts.
thank you! please share!
The biggest issue in physics is that rather than naming the phenomenon measured by an experiment, we could make up a thing that explains the phenomenon without thinking of the logical consequences of thinking of that thing.
For example, in biology evolution is the process of change in heritable traits over a population. This definition of evolution is what biologists "measure", in the Fossel record or wherever void they find themselves, rather than saying "Evolution is a force that causes the change in heritable traits over a population", you see i made up a thing that explains what we measure/observe, yes both definitions are equivalent in exampling what biologist measures but one definition has logical consequence that grows/benefit the field and the other hinders the field in confusion, e.g. what kind of forces acts of the population.
So moving back to physics, rather than saying: that dark matter is matter that doesn't interact with light and is spread in a galaxy, we could say dark matter is a phenomenon in which stars in a galaxy rotate fast than expected; gravity is a field that causes objects with mass to attract each other, we could say gravity is the interaction between particles with mass; finally sticking to the video, spin is a quantum number that cause electrons to have a magnetic moment, we could say spin is the magnetic moment of an electron.
Now here is the kicker, I said making up shit hinders the field of physics in confusion BUT, a very big but, sometimes making up shit opens you to a whole world of explanation power, for example, Fardays's electromagnetic which he made up to explain how currents effect a compass at a distance. This made up idea we call fields is used to explain phenomena from relativity to particle physics.
So in conclusion, making upvery could be every powerful or harmful to the field. Thanks for reading my Ted Talk. Please like and subscribe to the Highly Entropic Mind channel he is a great storyteller.
However, In evolutionary theory, we say that mutations are responsible for the evolution organisms undergo. Suffice it to say that in some contexts, it makes sense to use empirical observation of the evolutionary changes in organisms to create a model, and in others the molecular machinery that make up said organisms is necessary to consider.
For dark matter, the latter model “invisible particles that feel gravitation” is far more useful, as the only thing you need in order to explain every resultant phenomenon such as faster rotation of galaxies, can be given by a single mass distribution function, and time evolving according to general relativity.
Now that was a commitment, great video though!
After watching this I feel like I can go back to the "Spinors for Beginners" series and actually follow it.
I FORGOT TO MENTION EIGENCHRISS! Yes, his videos are amazing, but if I knew nothing about spin I wouldn't be able to follow them
Wooow!! Que gran video
After dwelling on this video for a day, I came to an interesting realization.
If we can represent values of probablitiy arbitrarily, then consider the following representation:
1 => certainty of spin up
0 => 50% chance spin up
-1 => certainty of not spin up
Then we can see how:
-1 => certainty of spin down
0 => 50% chance spin down
1 => certainty of not spin down
Thus the probability space would match the physical state space.
With such a representation, each cardinal direction is also represented as an orthogonal state.
For example pure spin left state means the vector has value 0 in spin up/down and spin forward/back as we expect, since 0 is 50% probability in this representation.
Dang how did I not see that you posted this?!? Shucks I’ve been waiting so much longer than I needed to…
I’m sorry …
please share it so more people will see it
Hey there, wanted to say that that's an AMAZING video and an amazing continuation to the first video and thank you for making them both and making me want to study quantum physics in depth! Also have a question, ever since I learned about the sigma matrices it was always bugging me out - why does the second matrix involve (i)'s and all the other ones don't, what's so special about it and the y direction it represents in quantum state space? (if I correctly understood it)
Nice video. You ought to cover some other related topics such as Euler's Formula, Euler's Identity, Gimbal Lock, Quaternions, Fourier Series and Fourier Transforms, etc... Maybe even dive into other topics such as Lagrange and Laplace transforms, the Gamma Function, the Zeta Function and how they're all related not just mathematically but also physically.
Uff, dude, I'd liek to cover all those things, but sadly I've only got one life
@@HighlyEntropicMind Lol... Yeah, I get that, but a little bit at a time is all we can do. At least it leaves you with plenty of topics / content / cover material for potential future videos.
That core bit is so funny hahaha
Good and very nice and also cool
Just like you
I feel your frustration 😂
SU(2) is also important in Loop Quantum Gravity, after doing the 3+1 decomposition the theory is built with a SU(2) connection called the Ashtekar connection which is a linear combination of the 3D projection of the spin connection with the extrinsic curvature
Abstract algebra is so beautiful. Simply breathtaking. I cannot wat to see the beauty of SU(3) and U(1). Please.
Thank you, although first we have to talk about matter waves, about using SU(2) on protons and neutrons, about the Strong Nuclear Force and then we can finally talk about SU(3), no idea when I'm going to introduce U(1)
I have all the patience and I will wait. Thank you.@@HighlyEntropicMind
I love your shirt.
You'll love the one I'm gonna use in the next video. I plan to use a different one in each video
Pilot wave theory isn't as dubious a theory as the consensus would suggest. The nails in its coffin have since been removed, most notably, the issues surrounding localised variables of a wave. However I'm not for a minute trying to suggest that Pilot wave is in anyway a better interpretation than the others. The fact is that Quatum Theory is practically 0% complete and instead what we have is a bunch of mathematical tools we can use to accurately predict probabilities for future states and interactions of particle fields without really giving us any intuition of why.
When you discuss separating light, this is usually done with a diffraction grating rather than a prism. I suspect because the prism must absorb some light. But, when you discuss generating the state vectors from the probabilities, you do not have sufficient information to do this, as you need the relative phases of the different components, and a simple probability measurement will not do that for you (giving you just the magnitude of the component, but not the phase). The strategy to get those is called quantum state tomography and it can be done for simple systems, but the complexity grows exponentially for more complex state vectors.
You are absolutely right. That part wasn't on the script but rather I improvised it on the day of recording because I thought the video needed a more concrete example. I knew it wasn't perfect but I hope you agree with me that it does help people to understand how these vectors are real
@@HighlyEntropicMind What is particularly nice about this example is it illustrates the proper way experiments are actually done--by setting up the geometry and state preparation so that a counting experiment can measure in this case the energy of the particle. Note how this experiment is nothing like the von Neumann measurement paradigm of entangling with a classical device, nor does it have any question about when measurement occurs--it occurs when the photon is changed to an electron in the metal part of the photomultiplier tube, which is then amplified so the current can be measured to count the single photon. Collapse plays no role, because the photon is destroyed.
oh shit it's out!
Deflection is a magnetic event. A point source of force with linear displacement, electric deviation perpendicular to the line, and the magnetic loop of deflection where spin is measured.
54:33 experiment to measure the phase of an spin state: atomic clocks. Particularly something called Ramsey spectroscopy. Here, atoms are place in a given superposition of state throughout rotations of its spin and a phase is measure. Some things to google for those interested could be atomic clocks, NMR and Ramsey spectroscopy :)
34:22
Makes an entire 2h video about the mathematics of spin.
Fails to add two double digit numbers.
(To be clear, I'm joking, the video is awesome!)
Holy shit boys it dropped
Trying to figure out where the energy comes from to align the spins. Is it the kinetic energy from the partical moving. Does it slow down as it passes through the sg device?
It's not about energy, it's about momentum
Think of magnetic fields as ramps. A ramp doesn't have any energy and it doesn't apply any force, but if an object is already moving it will change its movement
This is the reason the formula for the magnetic force is F = qv x B, you need to have velocity to feel the magnetic field
Now, if an electron goes up momentum must be conserved, which means that the SG device got an equal amount of momentum down, and since it's resting on the Earth, the whole planet acquired that downwards momentum
This is the same that happens when you throw a ball, you make the entire planet have that much momentum in the opposite direction
However as you can imagine the Earth is so big the effect of this is imperceptible
I feel like quantum probability waves say something counterintuitive about the nature of time. Like maybe one day, a different understanding of time will make all this jazz make sense.
interesting, maybe
As BMO once said: Time is an illusion that helps things make sense, so that we're always living in the present tense. It seems unforgiving when a good thing ends, but you and I will always be back then
Incredible Video! Do you think you will ever do a video like this about how u(1), and localizing the this symmetry with a gauge field leads to actual physical exhanges of energy and momentum that emerge into electromagnetic interactions between charges? Im a chemistry guy and it seems like almost the entire field emerges from the existence of electric charge alone. I'd love to know more about how it fundamentally emerges.
I am totally planning on getting to U(1)
“I think the new number is going to be i” wow all the math videos i watch and enjoy and barely(read: do not) understand have led me to get one thing right lets go
* Imagine a martial arts teacher from a chinese movie saying this *
Finally, after training, you can see it too