Ciro Santilli @cirosantilli 34

Some courses at least allow you to see material for free, e.g.: www.coursera.org/learn/quantum-optics-single-photon/lecture/UYjLu/1-1-canonical-quantization. Lots of video focus as usual for MOOCs.

Some are paywalled: www.coursera.org/learn/theory-of-angular-momentum?specialization=quantum-mechanics-for-engineers

It is extremely hard to find the course materials without enrolling, even if enrolling for free! By trying to make money, they make their website shit.

The comment section does have a lot of activity: www.coursera.org/learn/statistical-mechanics/discussions/weeks/2! Nice. And works like a proper issue tracker. But it is also very hidden.

November 2023 topics:

A charismatic, perfect-English-accent (Received Pronunciation) physicist from University of Cambridge, specializing in quantum field theory.

He has done several "vulgarization" lectures, some of which could be better called undergrad appetizers rather, a notable example being Video "Quantum Fields: The Real Building Blocks of the Universe by David Tong (2017)" for the prestigious Royal Institution, but remains a hardcore researcher: scholar.google.com/citations?hl=en&user=felFiY4AAAAJ&view_op=list_works&sortby=pubdate. Lots of open access publications BTW, so kudos.

The amount of lecture notes on his website looks really impressive: www.damtp.cam.ac.uk/user/tong/teaching.html, he looks like a good educator.

David has also shown some interest in applications of high energy mathematical ideas to condensed matter, e.g. links between the renormalization group and phase transition phenomena. TODO there was a YouTube video about that, find it and link here.

Ciro Santilli wonders if his family is of East Asian, origin and if he can still speak any east asian languages. "Tong" is of course a transcription of several major Chinese surnames and from looks he could be mixed blood, but as mentioned at www.ancestry.co.uk/name-origin?surname=tong it can also be an English "metonymic occupational name for a maker or user of tongs". After staring at his picture for a while Ciro is going with the maker of tongs theory initially.

Harvard University + MIT combo.

As of 2022:Fuck that.

Also, they have an ICP.

November 2023 course search:

www.youtube.com/watch?v=WB8r7CU7clk&list=PLUl4u3cNGP60TvpbO5toEWC8y8w51dtvm by Iain Stewart. Basically starts by explaining how quantum field theory is so generic that it is hard to get any numerical results out of it :-)

But in particular, we want to describe those subtheories in a way that we can reach arbitrary precision of the full theory if desired.

The variables of the Lagrangian, e.g. the angles of a double pendulum. From that example it is clear that these variables don't need to be simple things like cartesian coordinates or polar coordinates (although these tend to be the overwhelming majority of simple case encountered): any way to describe the system is perfectly valid.

In quantum field theory, those variables are actually fields.

Thought experiment that illustrates the path integral formulation of quantum field theory.

Mentioned for example in quantum field theory in a nutshell by Anthony Zee (2010) page 8.

Ciro Santilli's favorite music genre.

Ciro's 2020 perfect Friday evening: jazz fusion + study quantum field theory on an Amazon Kindle. Ahhhhhh.

When we particles particles, the action is obtained by integrating the Lagrangian over time:

In the case of field however, we can expand the Lagrangian out further, to also integrate over the space coordinates and their derivatives.

Since we are now working with something that gets integrated over space to obtain the total action, much like density would be integrated over space to obtain a total mass, the name "Lagrangian density" is fitting.

E.g. for a 2-dimensional field $f(x,y,t)$:

Of course, if we were to write it like that all the time we would go mad, so we can just write a much more condensed vectorized version using the gradient with $x\in {\mathbb{R}}^2$:

And in the context of special relativity, people condense that even further by adding $t$ to the spacetime Four-vector as well, so you don't even need to write that separate pesky $t$.

The main point of talking about the Lagrangian density instead of a Lagrangian for fields is likely that it treats space and time in a more uniform way, which is a basic requirement of special relativity: we have to be able to mix them up somehow to do Lorentz transformations. Notably, this is a key ingredient in a/the formulation of quantum field theory.

Originally it was likely created to study constrained mechanical systems where you want to use some "custom convenient" variables to parametrize things instead of global x, y, z. Classical examples that you must have in mind include:

When doing lagrangian mechanics, we just lump together all generalized coordinates into a single vector $\overrightarrow{q}(t):\mathbb{R}\to {\mathbb{R}}^n$ that maps time to the full state:where each component can be anything, either the x/y/z coordinates relative to the ground of different particles, or angles, or nay other crazy thing we want.

The Lagrangian is a function ${\mathbb{R}}^n\times {\mathbb{R}}^n\times \mathbb{R}\to \mathbb{R}$ that maps:to a real number.

Then, the stationary action principle says that the actual path taken obeys the Euler-Lagrange equation:This produces a system of partial differential equations with:

The mixture of so many derivatives is a bit mind mending, so we can clarify them a bit further. At:the $q_i$ is just identifying which argument of the Lagrangian we are differentiating by: the i-th according to the order of our definition of the Lagrangian. It is not the actual function, just a mnemonic.

Then at:

However, people later noticed that the Lagrangian had some nice properties related to Lie group continuous symmetries.

Basically it seems that the easiest way to come up with new quantum field theory models is to first find the Lagrangian, and then derive the equations of motion from them.

For every continuous symmetry in the system (modelled by a Lie group), there is a corresponding conservation law: local symmetries of the Lagrangian imply conserved currents.
Genius: Richard Feynman and Modern Physics by James Gleick (1994) chapter "The Best Path" mentions that Richard Feynman didn't like the Lagrangian mechanics approach when he started university at MIT, because he felt it was too magical. The reason is that the Lagrangian approach basically starts from the principle that "nature minimizes the action across time globally". This implies that things that will happen in the future are also taken into consideration when deciding what has to happen before them! Much like the lifeguard in the lifegard problem making global decisions about the future. However, chapter "Least Action in Quantum Mechanics" comments that Feynman later notice that this was indeed necessary while developping Wheeler-Feynman absorber theory into quantum electrodynamics, because they felt that it would make more sense to consider things that way while playing with ideas such as positrons are electrons travelling back in time. This is in contrast with Hamiltonian mechanics, where the idea of time moving foward is more directly present, e.g. as in the Schrödinger equation.

Furthermore, given the symmetry, we can calculate the derived conservation law, and vice versa.

And partly due to the above observations, it was noticed that the easiest way to describe the fundamental laws of particle physics and make calculations with them is to first formulate their Lagrangian somehow: why do symmetries such as SU(3), SU(2) and U(1) matter in particle physics?s.

TODO advantages:

Bibliography:

One of the things Ciro Santilli most deeply despises.

Real luxury is to understand quantum field theory and number theory.

Clothing/jewelry/car luxury is at worst a way to show off. And at best a replacement for nature/the countryside. People living in big cities have lost nature, and to some, looking at luxury goods (or watching television) serves as a (unsatisfactory) replacement.

Ciro Santilli would like to fully understand the statements and motivations of each the problems!

Easy to understand the motivation:

Hard to understand the motivation!

Used a lot in quantum mechanics, where the equations are really hard to solve. There's even a dedicated wiki page for it: en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics). Notably, Feynman diagrams are a way to represent perturbation calculations in quantum field theory.

Let's gather some of the best results we come across here:

This is the only way to truly understand and appreciate the subject.

Understanding the experiments gets intimately entangled with basically learning the history of physics, which is extremely beneficial as also highlighted by Ron Maimon, related: there is value in tutorials written by early pioneers of the field.

"How we know" is a basically more fundamental point than "what we know" in the natural sciences.

In the Surely You're Joking, Mr. Feynman chapter O Americano, Outra Vez! Richard Feynman describes his experience teaching in Brazil in the early 1950s, and how everything was memorized, without any explanation of the experiments or that the theory has some relationship to the real world!

Although things have improved considerably since in Brazil, Ciro still feels that some areas of physics are still taught without enough experiments described upfront. Notably, ironically, quantum field theory, which is where Feynman himself worked.

Feynman gave huge importance to understanding and explaining experiments, as can also be seen on Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979).

This is one of the first examples in most quantum field theory.

It usually does not involve any forces, just the interpretation of what the quantum field is.

www.youtube.com/watch?v=zv94slY6WqY&list=PLSpklniGdSfSsk7BSZjONcfhRGKNa2uou&index=2 Quantization Of A Free Real Scalar Field by Dietterich Labs (2019)

Formulated as a quantum field theory.

Theory that describes electrons and photons really well, and as Feynman puts it "accounts very precisely for all physical phenomena we have ever observed, except for gravity and nuclear physics" ("including the laughter of the crowd" ;-)).

Learning it is one of Ciro Santilli's main intellectual fetishes.

While Ciro acknowledges that QED is intrinsically challenging due to the wide range or requirements (quantum mechanics, special relativity and electromagnetism), Ciro feels that there is a glaring gap in this moneyless market for a learning material that follows the Middle Way as mentioned at: the missing link between basic and advanced. Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979) is one of the best attempts so far, but it falls a bit too close to the superficial side of things, if only Feynman hadn't assumed that the audience doesn't know any mathematics...

The funny thing is that when Ciro Santilli's mother retired, learning it (or as she put it: "how photons and electrons interact") was also one of her retirement plans. She is a pharmacist by training, and doesn't know much mathematics, and her English was somewhat limited. Oh, she also wanted to learn how photosynthesis works (possibly not fully understood by science as that time, 2020). Ambitious old lady!!!

Experiments: quantum electrodynamics experiments.

Combines special relativity with more classical quantum mechanics, but further generalizing the Dirac equation, which also does that: Dirac equation vs quantum electrodynamics. The name "relativistic" likely doesn't need to appear on the title of QED because Maxwell's equations require special relativity, so just having "electro-" in the title is enough.

Before QED, the most advanced theory was that of the Dirac equation, which was already relativistic but TODO what was missing there exactly?

As summarized at: youtube.com/watch?v=_AZdvtf6hPU?t=305 Quantum Field Theory lecture at the African Summer Theory Institute 1 of 4 by Anthony Zee (2004):

That video also mentions the interesting idea that:Therefore, for small timescales, energy can vary a lot. But mass is equivalent to energy. Therefore, for small time scale, particles can appear and disappear wildly.

QED is the first quantum field theory fully developed. That framework was later extended to also include the weak interaction and strong interaction. As a result, it is perhaps easier to just Google for "Quantum Field Theory" if you want to learn QED, since QFT is more general and has more resources available generally.

Like in more general quantum field theory, there is on field for each particle type. In quantum field theory, there are only two fields to worry about:

Looks very impressive! Last update marked 2011 as of 2020.

Goes up to "A.15 quantum field theory in a Nanoshell", Ciro have to review it to see if there's anything worthwhile in that section.

Personal page says he retired as of 2020: www.eng.fsu.edu/~dommelen/ But hopefully he has more time for these notes!

And he appears to have his own lightweight markup language that transpiles to LaTeX called l2h: www.eng.fsu.edu/~dommelen/l2h/

Second quantization also appears to be useful not only for relativistic quantum mechanics, but also for condensed matter physics. The reason is that the basis idea is to use the number occupation basis. This basis is:

Bibliography:

TODO. Can't find it easily. Anyone?

This is closely linked to the Pauli exclusion principle.

What does a particle even mean, right? Especially in quantum field theory, where two electrons are just vibrations of a single electron field.

Another issue is that if we consider magnetism, things only make sense if we add special relativity, since Maxwell's equations require special relativity, so a non approximate solution for this will necessarily require full quantum electrodynamics.

As mentioned at lecture 1 youtube.com/watch?video=H3AFzbrqH68&t=555, relativistic quantum mechanical theories like the Dirac equation and Klein-Gordon equation make no sense for a "single particle": they must imply that particles can pop in out of existence.

Bibliography: