Field Theory Fundamentals in 20 Minutes!

The most fundamental laws of nature that human beings have understood---the standard model of particle physics and Einstein's theory of gravity---are written in the language of field theory; quantum in the first case and classical in the second. The first field theory that you're likely to meet in your physics classes is electromagnetism, which describes the electric and magnetic fields through which light waves ripple all around us. But electromagnetism is actually fairly complicated; to learn field theory, it's better to start with a simpler example, and that's what I do in this video. I'll teach you the fundamentals using the Klein-Gordon theory of a real, scalar field. We'll see how to write down the Lagrangian, apply the principle of least action to obtain the Klein-Gordon field equation, how to solve it using plane waves, and how relativity is built in from the beginning.
We'll also take a peek at quantum field theory along the way.

About physics mini lessons:
In these intermediate-level physics lessons, I'll try to give you a self-contained introduction to some fascinating physics topics. If you're just getting started on your physics journey, you might not understand every single detail in every video---that's totally fine! What I'm really hoping is that you'll be inspired to go off and keep learning more on your own.

About me:

[kka]

Yes, I’m very very interested in learning all of it. Your channel is unique in that you’re able to teach very difficult topics in just 20 minutes and not only do you do it by giving us the big picture, but also by getting into the math. You’re helping us learn things in minimum time and with minimum effort. Thank you for your contribution.

[wayneyadams]

10:38 Partial derivatives seem intimidating at first, but here is a simple way to understand them. By the way, this is how I taught them to my students.

You are standing at the base of a hill. The slope changes differently depending on the direction you walk. You want to walk up the hill and retrieve your drone which just crashed on the hill northeast of you.

Let's say the hill slopes upward one foot for every five feet you walk toward the east. This is like the partial derivative, it gives you the slope in the easterly direction only, ignoring any slope to the north (or south). You begin to walk east until you are directly south of the drone, in other words, it is due north of you. Let's say you walked 20 feet to the east. How high are above your starting point? That's easy, it's simply the slope (partial derivative) multiplied by the distance you walked. Remember that the hill rises one foot for every 5 feet you walk. 1ft/5ft x 20ft = 4ft. You are now 4 feet above your starting point.

Now you turn north. At this point the hill is much steeper, sloping upward one foot for every three feet you walk toward the north. Again, this slope is like the partial derivative, it gives you the slope in the northerly direction, ignoring any slope to the east or west. This time you walk 15 feet to reach the drone. 1ft/3ft x 15ft = 5ft. You are now 9 ft up the hill, 4ft from walking east, and another 5 from walking north.

You could have walked north first then east in which case you would have to know the slopes at two points, the first to the north of your starting point then the slope to the east at the point where you turn east.

Regardless of the route you take, using the slope in one direction at a time allows you to calculate the height of each leg of your walk which you then add together to get the total height. Just like this imaginary walk up a hill, partial derivatives allow you to calculate changes in steps by calculating how the change in one variable affects the change in your target variable.

The first partial derivative on the right side tells you how the field changes with position, while ignoring the position (just like we ignored the slope northward while walking eastward). The other partial derivative tells you how the field changes in time while ignoring any changes from position. In this way the total change can be calculated.

Wayne Y. Adams
B.S. Chemistry
M.S. Physics
R&D Chemist (9yrs.)
Physics Teacher (33 yrs.)

[jacobsilver]

You've just taught the Lagrangian in the most straight forward and intuitive way that I've ever come across, and did it as a sidenote to a main topic. Thank you for your brilliant teaching. Keep up the good work.

[Byron_Vega]

The square is always so funny to me because the first time I saw it in a book I thought it was a symbol that couldn't load properly and I kept closing and opening the book until I decided to finally search for the "missing" symbol online and found out that there was never a problem. Please upload more, this was awesome!

[jpa_fasty]

This is about 2 lectures worth of work in the maths/physics degree I took. You presented it in 20 minutes, and I genuinely gained more from this than those lectures. Incredibly succinct and clear. Really good video.

[ClemoVernandez]

You're very talented at presenting and teaching. I've never seen such highly quality, clear explanations for such advanced topics. The animations and drawings add an aesthetic touch which makes it even better. You deserve so much for this kind of dedication man. Keep it up!

[speeshers]

This is, without exaggeration, one of the best videos I've ever seen. I'm a first year PhD student in physics, and this connected so many distant ideas and everything just clicked in place. Thank you so You method of explaining is extraordinary, and also really appreciate you going through some of the math and rigor, since a qualitative description can only go so far. :D

[bartpastoor]

So exited you are started posting these videos on Field Theory now, together with the handouts.
All of your videos and handouts, not shying away from math, are of an exceptionally clarity and even entertaining.
They close the gap between (semi)popular video´s and semester long university courses you sometimes too can find on YT.
But the latter, how good they may be, are very long and you easily lose the full picture of it.

I studied Theoretical physics in the eighties and then drifted to ICT.
Last couple of years taking it up again and am hoping to get some more insight into QFT, which we barely touched at the time.
After some MIT Moocs, am now into the very good book of Robinson: Symmetry and the Standard Model. It goes slow, but then I have the time.
Your videos so far and the ones hopefully to come, are of tremendous help in getting a fresh expose on the subject.

Proud to be one of your patreons.

[shubhsrivastava]

There are many science channels on YouTube which get into the detail of the topic but don't quantitatively explain the concepts. Since I like getting into the rigors of calculations, your channel helps me a lot in gaining interest in new topics. Thank you Elliot!

[fuencisclobuenasfuentes]

Superb work!!! As a physics student, I can say videos like these are extremely usefull for understanding the concept behind many tools we use. A lot of times this point of view of physical theories is lost in many textbooks in favour of mathematics and calculus (which is usefull too), so it’s extremely helpfull that people like you make this kind of videos.

Pd: english is not my native language, so sorry for any mistake I might have made in the coment :)

[SliversRebuilt]

This is everything I've ever wanted out of a physics education.

Thank you so much, man. Plus this video reminded me to buy in on your patreon now that my financial aid refunds came through (I certainly feel comfortable calling it an educational expense lol)

May many more follow in your footsteps. Think of the things we could achieve with the fruits such efforts bear across even one generation.

[cezarionescu]

This is quite impressive, even for someone who has studied the field its lovely to gain new insight on the subject matter.

[douglasstrother]

You crammed a lot of Physics into 20 minutes, Elliot!

Herbert Goldstein concludes his "Classical Mechanics" with Field Theory.

Michael Faraday's concept of the electric and magnetic fields, and his collaboration with James Maxwell were the first steps in a modern unified field theory.

[owen]

I love how so many areas of physics and mathematics come out of something so elegant yet powerful

[Cowux]

Your explanation is so GREAT. If I had these videos in my PhD years, probably I would not give up my journey in theoretical

[afshinsadeghi]

Thrilled to see this on YouTube. I always was watching the pop science clips but my search was giving zero results on formulas explanations in the YouTube. The more I searched the less I found. And I am not so in favor of reading the boring textbooks. This is well delivered. Now I can self teach a bit real field math. Thank you.

[tomgraupner]

Please, Elliot, I need "the rest" for the other spins as well. All insights in this topic is wonderful! Thank you so much for your efforts !!!

[paoloazzini]

Really appreciated this video!
This finally helps me to connect the dots of many concept i encountered during my physics degree, without having to take a full semester theoretical physics class.
I'd really appreciate a video on spinors' lagrangian too.

[fugitive]

Wow, this is amazing. I always wanted to learn about the Field Equations and now I have the basic idea where it all come from. Please continue with this series. This definitely guaranteed a sub.

[sauravkafle]

This channel is just pure gold. Although I am a high school student and did not understand most of it, I really enjoyed watching the full video. Big appreciation to you.. Wish you for amazing success..