For anyone interested in a deeper (but not dense) introduction to how and why Feynman diagrams work I’ve found a lot of pleasure going through this blog post below.
Feynman's own book 'QED: the strange theory of light and matter' is another good read, but strangely it is a bit lacking in illustrations. (And the illustrations that are there are a bit dry compared to Mattuck!)
"In particle physics, people use Feynman diagrams a lot. And these are nothing more or less than a graphical representation of summations and integrations over many variables. Also, as an ex-quantum physicist, I am a big fan of tensor diagrams [...]. You can think about it as the Einstein summation convention with no dummy indices (see Einsum in PyTorch)."
But... the Einstein summation convention is kind of fine in deep learning. The tensor manipulation in deep learning doesn't involve any of the gnarly contortions that physicists need to do like raising and lowering indices, taking covariant derivatives, or taking traces: and even that stuff moved along just fine without the need for diagramatic representations until the 1960s or so.
Drawing Theories Apart is an excellent book on Feynman Diagrams. You don't have to be actively involved in nuclear or hep to understand but the material is not for the faint of heart and a decent background in those areas goes a long way.
I don't really know how you can understand Feynman diagrams without seeing how annihilation and creation operators act on the vacuum. They are the mathematical building blocks of qft; formally they are linear operators that represent the creation or annihilation of particles.
If you apply an annihilation operator to a state containing no particles then the string gets deleted from your equation. Moreover if you swap the order of two operators you get an artifact (they're matrices after all), so a•b = b•a + c and so in qft calculations you swap your strings of operators around until you end up distilling it down to just a few non-vanishing artifacts that can be calculated. There's something called wick's theorem that the result of this process can actually be mapped onto a graph if you specify the initial and final state.
Yeah, normal ordering and contractions, if we're expanding the fields in terms of such operators when doing canonical quantization using Fock spaces. There's a lot of stuff going on there. The fact that we can draw nice pictures representing the terms of a perturbation theory doesn't mean that you can skip a regular course on QFT to understand what practitioners mean when they use them. Again, not stressing this while showing tree level Feynman diagrams as if they were straight representations of particle interactions may give wrong ideas to the unsuspecting readers.
Feynman diagrams are just pictorial representations of terms from a power series expansion of an overall 'master' equation representing a certain scattering process. A fairly simple one is electron positron annihilation, which yields two photons.
Quantum field theory is very difficult and we can't solve many problems. We can solve one in particular, free particles that do not interact. (By the way, this is like solving a simple harmonic osciallator.) Feynman diagrams use the solution to free particles as a basis - these are the lines in the diagram, like particles moving in time. Then, we add the interactions between the particles as a pertabative exapansion. This is what the parent comment refers to as the power series expansion. It is expanding in the powers of the interaction terms.
As mentioned in the video, it is not quite as simple as this however. If you do this, you will get infinity as a result of the calculation. That is not a very good perterbation theory. However the real discovery of physcists from this time was renormalization. They found out you can sum many contributions from different terms in the exapansion, corresponding to different indiviaul feynmann diagrams, and you can make these infinities cancel out. What is magic is that when this process is done you can write an effective theory for the interactions that looks exactly like the original theory you started with, and only the values of the coeffiecients are changed a little.
The magic involved here could be explained more, but in summary what it means is that the perterbation calculation you do is valid even thought it looks like it should be infinite. It also lets you make the simple analogy that the lines in the diagram are real and virtual particles rather than just the non-interacting approximation you starter with.
I left out some important details in the above explanation. So here is part 2...
Feynman discovered another very important item, the path integral formulation of physics. This was important for his derivation of Feynman diagrams and also it is a good conceptual tool.
Think of basic quantum mechanics and firing an electron through a double slit at a screen. In quantum mechanics the eletron does not have a single trajectory from the gun to the screen. Rather, it takes all the trajectories in parallel, like a wave. We can add the contribution as if the particle went over each possible trajectory and this is the same as treating the electron as if its position was given by a propogating wave. This sum over possible histories is the interpretation of Feynmans path integrals. And it is a nice way to think about quantum mechanics - multiple things are happening in parallel.
Taking the example in the video of two electrons scattering off each other, each Feynman diagram represents a possible history for the two particles, including their trajectory and any interactions between them. These interactdions are drawn as a connecting line, which is a "virtual particle" being exchanged.
More specifically, the diagram doesn't represent a single history, rather it represents all histories that have a ceratain topology, meaning here for example one photon is exchanged between the electrons. (There is an integral done to add up all the different ways this can happen.) To do the full calculation, there are many diagrams that must be included. As it is a perterbation theory, you can choose to get more accurate by include more diagrams. The expansion parameter is basically a vertex on the diagram. The more vertices you include, the more accurate you will be (assuming you include all diagrams with that number of vertices).
So basically the feynman diagram is a bookkeeping mechanism to account for all possible histories of the particles in the interaction (electon scattering here). We sum up the contribution from all these histories to find out the quantum amplitude for this scattering scenario. This is exactly analogous to adding the contribution from different paths to fine the amplitude (~probability) for our electron in the double slit experiment hitting a particular location on the screen.
To get to this intuitive result mathematically, the perturbation expansion and renormalization mentioned above are both involved.
Just to expand on this; the series is usually called a Dyson series (https://en.wikipedia.org/wiki/Dyson_series). Dyson showing the connection between perturbation theory and Feynman's formulation was very important.
In case anybody interested in graphic lambda calculus, there are 5! X 5! = 14400 possible alternatives for the beta rewrite only. So it beats me why among all the possible calculi in CS the lambda calculus is special. I made a page which gives random alternatives http://imar.ro/~mbuliga/betarand.html
An update: added more explanations about how these random variants of the beta rewrite are generated. You can see also the generated rewrites written in lambda calculus style, although they are graphical rewrites which go outside lambda calculus. I state again what I believe: that analogously with cellular automata, probably a big proportion of these random generated calculi are as rich as the original (Turing complete at least).
Agree, but I don't say that lambda calculus is not interesting, I say that there are so much more alternatives, unexplored. If we do the same with the IC of Lafont, I think that the possible (right patterns) of rewrites (which preserve the no of half-edges and nodes) are about 10^13.
A conjecture, which I don't know if it holds water, is that analoguous to CA, Wolfram style, perhaps 1/3 (or anyway a significant proportion) of the 10^13 possible graph rewrite systems, alternative to IC, are Turing universal.
Sadly, despite having read about them about a dozen times, I still don't understand how they work, or rather how to use them. Apparently, that's because of insufficient knowledge of the relevant part of physics.
Yes! I just finished it. Best vulgarization book I have ever read. Not a single equation, but still telling you exactly how the physics work, without any hand waving. I can't recommend it enough.
They're just a pictorial way of arranging calculations involving pretty technical stuff really. You're not expected to make sense of it without going through the (roughly grad school level physics) groundwork. I get that this can be frustrating, but maybe it's not a good idea sticking them in popsci just because they look intuitive.
I mean, they are, but only after you've learnt about the machinery behind them. For instance I wouldn't know how to convey the difference between fermionic lines and the bosonic ones without working through commutation and anticommutation relations, the gamma matrices, spinors, the different propagators and so on. Saying simply that fermions are straight lines with oriented arrows, bosons are wiggly lines and we'll draw them like this... doesn't help much when it comes to understanding what they mean, in my opinion.
you need to understand perturbation theory in the context of qft. that's no small feat but since it's a pretty universally studied grad core subject there's a lot of material. this book was actually suggested to me here
This guy (Dietterich Labs), does in depth (steps) derivations of cool QFT stuff. This is slightly unique as they are less handwavey than lectures (Which are often supporting written material, or at least harder to follow online) and also more informative than the calculations in some books (I think Peskin and Schroeder can be horrific unless you have someone to ask stupid questions too, if you're like me [Well read, but not that sharp on the bell curve of academic bragging rights])
Quantum Field Theory for the gifted amateur is also pretty great (And Pretty actually), Zee for the working man or something like that
I really like Quanta and I'm disappointed that they are spending resources on creating video content (and presumably reducing the written content accordingly). I don't know if it's obviously elitist of me (to wish to deny people the opportunity to educate themselves via video if that's what they like) or absurd of me to think that it could possibly be elitist to try to retain the tradition of education-via-writing-and-reading. Anyway, I like reading quanta, and I don't watch videos unless they are extremely special (3b1b) or I'm in an odd mood.
My perception (as a lapsed PhD physicist) is the community takes on a new formalism when it has to i.e. there are new results that can only be derived with the new formalism. A formalism that allows slightly better derivation of existing results does not attract attention. Twistors, Clifford algebras and other things seem to have not met this hurdle so far.
I think the person you're replying to was talking about Geometric Algebra, which is more of a project to reformulate all our current vector and tensor notation in terms of Clifford algebras and to move away from using specific representations of the algebra and instead just leverage the algebraic properties themselves.
If you look at how most physicists use the gamma matrices, they are very reluctant to treat them as algebraic objects and rely heavily on their matrix representation. A proponent of GA would say this is like using the matrix
[ 0 1
[-1 0]
everywhere in your calculations instead of just using i and remembering that i^2 = -1. Sure, it's formally equivalent but you'd still miss out on a lot of the beauty of the complex numbers.
For what it's worth, I have a soft spot in my heart from Geometric Algebra, but I think it still needs a lot of notational improvement before it'll ever see any real adoption.
If you're curious about GA as it currently stands, I'd check out Geometric Algebra for Physicists by Doran and Lasenby.
Just so you know, asking if twistors are the future in a thread about Feynman diagrams is like asking if smart watches are the future in a thread about cars. It's pretty off topic.
That said, nobody really knows about the future of twistors in physics. Twistors have some really interesting algebraic properties that may eventually get leveraged in a main-stream theory but they could also just end up being a mathematical curiosity that never has any real impact.
I think I feel pretty confident saying that twistors will never have the cultural / philosophical impact on physics that Feynman diagrams have had (for good or for ill).
https://www.quantumdiaries.org/2010/02/14/lets-draw-feynman-...