You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
Tim Hosgood (Apr 19 2020 at 16:07):I have a copy of Paugum's "Towards the Mathematics of Quantum Field Theory" (from a book sale a long time ago), and the content has always seemed super interesting to me. I've never properly studied it though, since whenever I've mentioned it to any senior mathematicians, they have unanimously told me that it is very hand-wavy, and verging on wrong in some places (in fact, this is probably the _nicest_ review I've heard of it). Has anybody got any more concrete things they could say about this book? e.g. "these chapters are good, and if you read these then just be careful of these specific points"
Tim Hosgood (Apr 19 2020 at 16:08):(I'm not really looking for other book recommendations for learning QFT (I think I would probably turn to Zee), but am interested in this book specifically because it lumps together a lot of concepts that I've not seem lumped together before)
John van de Wetering (May 20 2020 at 18:14):Don't have an answer to Tim, but I am myself looking for a good introduction book to QFT. Something that isn't too hand wavy, but also isn't so much focused on the mathematics that it looses sight of the physics. Any suggestions?
John Baez (May 20 2020 at 18:16):Quantum field theory takes a long time to learn, and the best way to learn it depends hugely on ones individual nature, like how much physics one knows already and how much mathematical rigor one needs.
John Baez (May 20 2020 at 18:17):Since none of the most important QFTs have been proved consistent (except in a limited sense), getting to actual QFT as physicists do it requires at some point a plunge into non-rigor.
John Baez (May 20 2020 at 18:18):For mathematical people I recommend this:
John Baez (May 20 2020 at 18:18):For physicists this is a good introduction:
John Baez (May 20 2020 at 18:20):About this: "since whenever I've mentioned it to any senior mathematicians, they have unanimously told me that it is very hand-wavy, and verging on wrong in some place..."
Personally I would not listen too much about mathematicians when it comes to books on QFT. Do you ask physicists advice for what are good math books?
Math and physics are quite different subjects. The books actual physicists use are completely nonrigorous and hand-wavy, but they teach you the stuff you need to know to actually understand QFT!
John Baez (May 20 2020 at 18:21):The project of learning physics is very different from the project of making it mathematically rigorous, and it should probably come first, because making physics rigorous is extremely hard, and almost impossible to follow if you don't first understand the physics.
sarahzrf (May 20 2020 at 19:04):man, physics drives me fucking batty
sarahzrf (May 20 2020 at 19:04):im not sure ive ever heard anybody acknowledge that physicists mean something different by a "variable" than mathematicians do
sarahzrf (May 20 2020 at 19:04):kinda had to realize that myself
John Baez (May 20 2020 at 19:49):I think experts know that physicists use "quantities" like energy and decide later which quantities are functions of which, and change their minds quite a lot.
John Baez (May 20 2020 at 19:49):Because this is what the physical world is like: you don't know what's a function of what.
sarahzrf (May 20 2020 at 19:53):there u go: things "being a function of" other things 🤬
sarahzrf (May 20 2020 at 19:53):it's enough to make u go nuts
sarahzrf (May 20 2020 at 19:53):is 5 a function of 3???
John Baez (May 20 2020 at 19:55):No, but energy can be a function of temperature and pressure.
sarahzrf (May 20 2020 at 19:56):well, i know what that means now, but it's a pretty different concept from how you operate in math :thinking:
sarahzrf (May 20 2020 at 19:56):that's really my main thrust
Eduardo Ochs (May 20 2020 at 20:02):John Baez said:
I think experts know that physicists use "quantities" like energy and decide later which quantities are functions of which, and change their minds quite a lot.
I've had lots of problems with this too - and practically every time that I tried to ask physicists they told me to stop trying to think in terms of mathematical models for a while and think on the "real world"...
Eduardo Ochs (May 20 2020 at 20:04):sarahzrf said:
im not sure ive ever heard anybody acknowledge that physicists mean something different by a "variable" than mathematicians do
Do you know this book?
https://en.wikipedia.org/wiki/Structure_and_Interpretation_of_Classical_Mechanics
Daniel Geisler (May 20 2020 at 20:15):Just dropping a ad for mathematical physics - Einstein's contraction and Noether's Theorem are two examples of mathematic's dominance of physics.
sarahzrf (May 20 2020 at 20:16):Eduardo Ochs said:
sarahzrf said:
im not sure ive ever heard anybody acknowledge that physicists mean something different by a "variable" than mathematicians do
Do you know this book?
https://en.wikipedia.org/wiki/Structure_and_Interpretation_of_Classical_Mechanics
i started it once but i did not get very far before getting distracted
sarahzrf (May 20 2020 at 20:16):honestly just assume that's my response to any such questions in the future :upside_down:
John Baez (May 20 2020 at 20:36):Eduardo Ochs said:
John Baez said:
I think experts know that physicists use "quantities" like energy and decide later which quantities are functions of which, and change their minds quite a lot.
I've had lots of problems with this too - and practically every time that I tried to ask physicists they told me to stop trying to think in terms of mathematical models for a while and think on the "real world"...
I always studied both math and physics on an equal footing in school - I could never quite decide what I liked best, though it became clear I was better at math. So I learned both ways of thinking. It's easy for people who mainly do math to see all the equations and mathematical concepts used in physics and underestimate how different the ways of thinking are... until they try to actually learn physics, or talk to physicists!
John Baez (May 20 2020 at 20:38):The funniest example I recall was: a mathematician came up to me, and said "You know math, and you know physics, so maybe you can answer this. I've been wondering for a long time. What's the definition of an electron?"
John Baez (May 20 2020 at 20:39):I laughed and told them an electron was not something we get to define - it's a thing out there, and we can make different models of it, and there are a lot of great mathematical models of electrons, and in each one "electron" is defined in a different way.
John Baez (May 20 2020 at 20:42):So you gotta learn about the Bohr atom, and Schrodinger's equation, and the Dirac equation, and quantum electrodynamics, and the Standard Model... each of which have their own things to say about electrons.
Arthur Parzygnat (May 20 2020 at 20:42):sarahzrf said:
well, i know what that means now, but it's a pretty different concept from how you operate in math :thinking:
It's not entirely different from how math is done. For example, it's possible the mathematical language for QTF is not available yet (another possibility is that it is, and we have yet to put the proper words together). When we do math, it's not always rigorous and precise at first, but it gets to that stage through some (often serious) effort. QFT is still currently in this stage---the structure of the theory is still not fully understood and is going under revision (and I'm talking about physicists)! Even Noether's theorem is thought of differently today than in (modern!) textbooks.
Arthur Parzygnat (May 20 2020 at 20:46):John van de Wetering said:
Don't have an answer to Tim, but I am myself looking for a good introduction book to QFT. Something that isn't too hand wavy, but also isn't so much focused on the mathematics that it looses sight of the physics. Any suggestions?
I actually think Folland's book is pretty decent for an introduction for mathematicians that actually discusses the physics. Have you looked at it?
John Baez (May 20 2020 at 20:53):Hmm, I haven't seen Folland's book Quantum Field Theory: a Tourist Guide for Mathematicians.
John Baez (May 20 2020 at 20:54):I like his book Harmonic Analysis in Phase Space, which is a very mathematical treatment of some nice ideas coming from quantum mechanics.
Arthur Parzygnat (May 20 2020 at 20:59):I have not read a Folland book I did not thoroughly enjoy :) I highly recommend all of them.
Okay, well technically I never finished any of his books (nothing wrong with Folland---I'm just unfortunately not good at reading books), but I have looked at many chapters and learned a lot from what I did read.
John Baez (May 20 2020 at 21:04):The book by Folland that I find most frustrating is the one that I teach from every year - his book on real analysis. The proofs are too terse for the students to follow them. But we use that book because it covers all the topics we want... and it does have a lot of nice material very nicely organized.
Arthur Parzygnat (May 20 2020 at 21:07):Oh @John van de Wetering , Wald's book "QFT in curved spacetime..." may also interest you as a sort of introduction to QFT with a C*-algebra emphasis (though it's focus is more about black holes, so it might not be the kind of thing you're looking for).
John Baez (May 20 2020 at 21:20):By the way, my complete list of recommendations for books on QFT is here. I think the Ticciati book may be the best for mathematicians who don't want to take a deep dive into the delightful mire of actual physics.
John van de Wetering (May 20 2020 at 21:51):John Baez said:
Hmm, I haven't seen Folland's book Quantum Field Theory: a Tourist Guide for Mathematicians.
Looking at the review, this seems to be the book for me! Thanks!
Gershom (May 20 2020 at 22:30):I have heard very good things about Renormalization and Effective Field Theory by Costello, and a friend of mine was (before the pandemic hit) running a reading group on it.
Tim Hosgood (May 20 2020 at 23:37):@John Baez i get that mathematicians shouldn’t be the judge of physics books, but this really is (from what i can tell) actually a maths book, and not a physics one. my interest in it isn’t even for the physics, but instead some of the infinity-categorical formalisms of stuff
(sorry, can’t figure out how to directly reply/quote on mobile)
Nikolaj Kuntner (May 21 2020 at 01:03):I see 13 hours of lectures by Costello are also available here since January (https://www.youtube.com/playlist?list=PL7aXC0jU4Qk6PABeoSBjWAJ5sxfEhI9DQ).
While the people who work in physics departments and the people who work in mathematics departments ask different questions and write books in different styles, I'm unsure if there's different things ("theoretical physics" and "mathematics") beyond this sociological split. To quote Vladimir Arnold and his extreme position in one direction, "Mathematics is the part of physics where experiments are cheap." (https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html).
I don't agree with Arnold's take in that essay, but still: Maybe a mathematician is just a theoretical physicist cut off from the experimental physicist, free to roam in ever deeper corners of what was once somewhat related to geometry and time.
Todd Trimble (May 21 2020 at 04:05):John Baez said:
The funniest example I recall was: a mathematician came up to me, and said "You know math, and you know physics, so maybe you can answer this. I've been wondering for a long time. What's the definition of an electron?"
I feel a secret sympathy for that poor benighted mathematician. Long ago we had begun a conversation on representations of the Poincare group, and in that thread I had wanted to know what was the representation corresponding to the electron, and I wonder whether the mathematician of the story was trying to ask the same thing.
Arthur Parzygnat (May 21 2020 at 05:10):Nikolaj Kuntner said:
I see 13 hours of lectures by Costello are also available here since January (https://www.youtube.com/playlist?list=PL7aXC0jU4Qk6PABeoSBjWAJ5sxfEhI9DQ).
While the people who work in physics departments and the people who work in mathematics departments ask different questions and write books in different styles, I'm unsure if there's different things ("theoretical physics" and "mathematics") beyond this sociological split. To quote Vladimir Arnold and his extreme position in one direction, "Mathematics is the part of physics where experiments are cheap." (https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html).
I don't agree with Arnold's take in that essay, but still: Maybe a mathematician is just a theoretical physicist cut off from the experimental physicist, free to roam in ever deeper corners of what was once somewhat related to geometry and time.
This is great. I had no idea Costello had videos on this!
Arthur Parzygnat (May 21 2020 at 07:41):John Baez said:
The funniest example I recall was: a mathematician came up to me, and said "You know math, and you know physics, so maybe you can answer this. I've been wondering for a long time. What's the definition of an electron?"
Omg, I have a similar story, which I just remembered now. I was at MIT one summer for a research project as an undergraduate (I think between my second and third year). I met a big shot in the high energy side of the physics department (I can't remember who it was though, probably for the better), and I was telling him I'm interested in both the mathematical and physical aspects of QFT, though I hadn't yet taken a course in QFT (I had just finished quantum). And he was talking to me about some ideas in QFT and asked me if I knew what a gluon was. I replied no. He scoffed at me and asked, "Well, do you know what an electron is???" At the time I just said yes, because I felt a little embarrassed, but a part of me wished my response was, "Do you know what an electron is?"
John Baez (May 21 2020 at 16:49):Todd Trimble said:
John Baez said:
The funniest example I recall was: a mathematician came up to me, and said "You know math, and you know physics, so maybe you can answer this. I've been wondering for a long time. What's the definition of an electron?"
I feel a secret sympathy for that poor benighted mathematician. Long ago we had begun a conversation on representations of the Poincare group, and in that thread I had wanted to know what was the representation corresponding to the electron, and I wonder whether the mathematician of the story was trying to ask the same thing.
I'm sure they wanted some answer like that - "the electron transforms in the blah-di-blah representation of the Poincare group SU(3) x SU(2) x U(1)".
John Baez (May 21 2020 at 16:53):But that's just a portion of the description of what the electron is like in the Standard Model, and there are lots of other very useful models of electrons; for example a physical chemist would never want to use the Standard Model.
John Baez (May 21 2020 at 16:54):If I'd been in a different mood I might have taken the time to explain this general stuff, and then give them some particular specification of an electron, maybe in QED.
John Baez (May 21 2020 at 16:54):But this was a guy who just walked up to me in a department tea.
John Baez (May 21 2020 at 16:55):Tim Hosgood said:
my interest in it isn’t even for the physics, but instead some of the infinity-categorical formalisms of stuff.
Oh, okay. :cry:
John Baez (May 21 2020 at 16:59):Arthur Parzygnat said:
Omg, I have a similar story, which I just remembered now. I was at MIT one summer for a research project as an undergraduate (I think between my second and third year). I met a big shot in the high energy side of the physics department (I can't remember who it was though, probably for the better), and I was telling him I'm interested in both the mathematical and physical aspects of QFT, though I hadn't yet taken a course in QFT (I had just finished quantum). And he was talking to me about some ideas in QFT and asked me if I knew what a gluon was. I replied no. He scoffed at me and asked, "Well, do you know what an electron is???" At the time I just said yes, because I felt a little embarrassed, but a part of me wished my response was, "Do you know what an electron is?"
Funny.
Of course for a physicist "do you know what an electron is???" is a very different question than "do you know the definition of an electron?"
It's like you might ask someone recovering from a coma "do you know who your wife is???" - but never "do you know the definition of your wife?"
John Baez (May 21 2020 at 17:01):In physics an electron is something you're supposed to be familiar with in many ways.
Amar Hadzihasanovic (May 21 2020 at 18:46):I did a bunch of theoretical physics courses including a few on QFT as a student of mathematics, and I was never particularly bothered by the physics-standard of rigour. But I struggled quite a lot with the conceptual side of QFT perhaps for the opposite reason, that I could not really “see” where it was coming from.
As in, both the intro-to-quantum mechanics and the intro-to-relativity courses started from these really striking experimental or thought-experimental problems -- the ultraviolet catastrophe, the photoelectric effect, Michelson-Morley etc -- whereas QFT, at least the way I was taught and found in most books, would start from a much more formal place:
“how do we reconcile CFTs with quantum mechanics?” “how do we reconcile special relativity with quantum mechanics?” etc
And I could never quite understand what that was for... experiments would enter the picture a bit as an afterthought, as 'tests of the theory'.
So I guess even if I could get a grip on the formal aspects and calculations, I never felt like I knew what was going on. (Perhaps studying some condensed-matter physics could have helped?)
So, now I'm wondering, could anyone recommend an account of QFT which is “for theoreticians”, but develops the subject 'in response to experiments' and not 'in response to theoretical inconsistencies'? :grinning:
Amar Hadzihasanovic (May 21 2020 at 18:54):(Oops, CFT -> classical field theories, not conformal field theories)
John Baez (May 21 2020 at 19:11):The basic idea of quantum field theory showed up right at the start of quantum mechanics: we need to treat individual vibrational modes of the electromagnetic field as quantum harmonic oscillators or we'll get hit with the ultraviolet catastrophe, i.e. there won't be a well-defined state of thermal equilibrium at any temperature greater than absolute zero.
John Baez (May 21 2020 at 19:12):Since the electromagnetic field is a bunch of uncoupled harmonic oscillators (before we introduced charged matter) it's very easy to quantize.
John Baez (May 21 2020 at 19:12):I don't know the early history of how it was done, but it could have been done in the Bohr-Sommerfeld approach before Schrodinger and Heisenberg came along.
John Baez (May 21 2020 at 19:13):Probably most courses on QFT are too eager to blow right past this stuff to give you any intuition for what the quantized electromagnetic field is like. It's probably easier to read about it in books on "quantum optics", which is a hot topic right now.
John Baez (May 21 2020 at 19:17):So for example it's good to think about how how it makes no sense to measure either the electric or magnetic at a single point: you need to integrate against a "test function", like a compactly supported smooth function on space... and the uncertainty principle prevents one from measuring both the electric and magnetic field exactly, integrated against the same test function... and hand how measuring either of them in the vacuum state gives a Gaussian random variable.
John Baez (May 21 2020 at 19:17):And then I guess it's also good to think about what these observables are like in single-photon states, and in coherent states, and so on.
John Baez (May 21 2020 at 19:18):All this uses math that's a combination of Maxwell's equations, Fourier analysis and the quantum harmonic oscillator.
John Baez (May 21 2020 at 19:20):Anyway, I'm not answering Amar's question about an experimentally motivated intro to QFT yet. All this stuff about free fields must have shown up very early, because it's just combining a famous linear classical field theory (the vacuum Maxwell equations) with the basic idea of the quantum harmonic oscillator.
sarahzrf (May 21 2020 at 19:20):i remember once i saw a meme that was like
sarahzrf (May 21 2020 at 19:21):student: "im sick of studying the quantum harmonic oscillator all day when can we learn qft"
teacher: :sad:
John Baez (May 21 2020 at 19:21):Heh. "I'm sick of learning how to add - I want to learn how to subtract!"
Amar Hadzihasanovic (May 21 2020 at 19:24):John Baez said:
Probably most courses on QFT are too eager to blow right past this stuff to give you any intuition for what the quantized electromagnetic field is like. It's probably easier to read about it in books on "quantum optics", which is a hot topic right now.
Yes! I did a really cool, really idiosyncratic course on quantum optics, taught by Lorenzo Maccone -- maybe you know him. That helped a lot with that part.
John Baez (May 21 2020 at 19:24):All the hard part shows up when you look at interacting (i.e. nonlinear) field theories. And here we get the history of QED, where experiment and theory went hand in hand. Here it pays to spend time thinking about the problems people had with negative-frequency solutions of the Klein-Gordon equation, and how through a comedy of misunderstandings this led Dirac first to the Dirac equation and then his prediction of the existence of antimatter. The usual textbook accounts are both historically inaccurate (for example positrons had been seen before Dirac gathered the courage to predict their existence, but people couldn't believe it so they said their photographic plates had been flipped over by accident!) and weak on the physics and math.
John Baez (May 21 2020 at 19:25):Dirac's "hole" theory seems a bit naive now, and we now know that the Dirac equation didn't help at all with the problem of negative-frequency solutions: you just need to pick the correct complex structure on your Hilbert space of solutions to get the energies to come out positive.
John Baez (May 21 2020 at 19:25):Physics books often describe this in terms of a somewhat mysterious "switching annihilation and creation operators".
John Baez (May 21 2020 at 19:26):So all of this is worth a lot of thought, and you have to pick it up here and there.
John Baez (May 21 2020 at 19:26):But the real fun starts when you let your quantized electron-positron field interact with your quantized photon field!
sarahzrf (May 21 2020 at 19:26):someday i want u to teach me some physics 🥺
John Baez (May 21 2020 at 19:27):I'm doing it. :upside_down:
sarahzrf (May 21 2020 at 19:27):i can figure out descent on my own probably with enough time and nlab pages
John Baez (May 21 2020 at 19:27):Yeah, well I'm learning descent right now so I've been more eager to explain that!
sarahzrf (May 21 2020 at 19:28):John Baez said:
I'm doing it. :upside_down:
i mean kinda but this is all stuff that's way above my level so i'm not Actually absorbing it :mischievous:
John Baez (May 21 2020 at 19:28):And I'll get back to that today - I saw your comments here yesterday.
John Baez (May 21 2020 at 19:28):I think with physics you have to hear the music a few times before you start trying to play it.
sarahzrf (May 21 2020 at 19:29):i do have some experience already with physics, i took a classical mechanics course a couple years ago
sarahzrf (May 21 2020 at 19:29):it forced me to finally get semi competent with sitting down and grinding out derivatives of symbolic expressions >.>
sarahzrf (May 21 2020 at 19:30):and like one of those intro level "modern physics" classes where they have you solve the time independent schrödinger equation but they dont actually explain jack shit
John Baez (May 21 2020 at 19:30):Anyway, I was gonna recommend that Amar read these:
Emilio Segre, From X-Rays to Quarks: Modern Physicists and Their Discoveries, W. H. Freeman, San Francisco, 1980.
Abraham Pais, Inward Bound: of Matter and Forces in the Physical World, Clarendon Press, New York, 1986. (More technical.)
John Baez (May 21 2020 at 19:31):These are good historical accounts of the development of quantum field theory in response to experiments in particle physics - written by good physicists!
John Baez (May 21 2020 at 19:33):A more focused account with an incredibly sexist but probably accurate title:
John Baez (May 21 2020 at 19:34):I find all this historical stuff really helpful for understanding physics... and when I was kid it made me want to be a physicist!
John Baez (May 21 2020 at 19:36):sarahzrf said:
i do have some experience already with physics, i took a classical mechanics course a couple years ago
it forced me to finally get semi competent with sitting down and grinding out derivatives of symbolic expressions >.>
and like one of those intro level "modern physics" classes where they have you solve the time independent schrödinger equation but they dont actually explain jack shit
It's good to get competent at grinding out computations, so that course wasn't a total waste. But you'd probably have a lot of fun trying to actually understand quantum mechanics, because it's absolutely mind-blowing.
John Baez (May 21 2020 at 19:37):And understanding it requires a bit of computation but mainly lots of abstract math and also thinking about the physics of what the formalisms actually mean.
John Baez (May 21 2020 at 19:40):All the vague chat about interpretations of quantum mechanics makes me want to vomit.
sarahzrf (May 21 2020 at 19:41):i know a lil
sarahzrf (May 21 2020 at 19:43):im familiar w/ the idea that an observable being a hermitian operator means that u have an orthonormal basis of states with real "definite values" of the observable, those being the eigenvalues of those basis vectors
sarahzrf (May 21 2020 at 19:44):& when u observe u collapse to a single one of those basis vectors w/ probability proportional to the squared modulus of the coefficient
sarahzrf (May 21 2020 at 19:45):i dont rly know the stuff about, like, commutators n shit
sarahzrf (May 21 2020 at 19:45):i know that systems combine by ⊗ rather than ⊕ and hence entanglement
John Baez (May 21 2020 at 19:50):Commutators show up when you think about Lie groups of symmetries of a quantum system: every Lie group has a Lie algebra, and the commutators are the Lie bracket. I happen to be turning these slides of mine into a paper right now:
http://math.ucr.edu/home/baez/noether/
and it's all about how "observables as things you measure" are related to "observables as Lie algebra elements that generate symmetries".
John Baez (May 21 2020 at 19:51):Maybe I should talk about the paper I'm writing a bit here... just to keep from getting lonely with it.
sarahzrf (May 21 2020 at 19:52):i know a lil tiny bit about lie algebras
sarahzrf (May 21 2020 at 19:52):but not that much
John Baez (May 21 2020 at 19:53):This paper is more about "what does it all mean?" than any hard-core math.
Fabrizio Genovese (May 21 2020 at 20:27):sarahzrf said:
i do have some experience already with physics, i took a classical mechanics course a couple years ago
I took classical mechanics in my second year. The first sentence the professor said was "An affine space is a set A wth a structure of vector space on it, such that its additive group acts on the underlyng set freely and transitively." That was the dopest course intro I had seen so far, and it turned out that the entire course was basically differential geometry, done in a heavily formal way. At that time I thought physics was the dopest thing ever. Unfortunately I took another course a couple of years later where we just used lagrangians to spit out equations of motion for pendulums and stuff. Having seen how beautiful the subject could have been if thaught in the right way, that second course was one of the most depressive academic experiences I've ever had
Fabrizio Genovese (May 21 2020 at 20:30):sarahzrf said:
i know a lil
If you struggle with the "lack" of formalism in physics, and you aren't supergood with books, I cannot advise enough to watch Frederic Schuller's lectures on General Relativity and Quantum Mechanics on Youtube. He's by far the _best_ teacher I've ever seen. Plain increadible. And everything he does is completely formal, cristal clear and beautifully presented.
Fabrizio Genovese (May 21 2020 at 20:32):There's this course called "The geometrical anatomy of theoretical physics" where in a semester he starts with propositional logic, then does:
Arthur Parzygnat (May 21 2020 at 20:56):John Baez said:
A more focused account with an incredibly sexist but probably accurate title:
- Silvan Schweber, QED and the Men Who Made It, Princeton U. Press, Princeton, 1994.
@Amar Hadzihasanovic, I second John's recommendation for "QED and the men who made it." I honestly recommend going through that book with a bunch of friends and also reading the main papers that they mention by Dirac, Pauli, Weisskopf, Schwinger, etc. Turn it into a learning seminar if possible. It's hard to do it on one's own. I also like Weinberg's "The discovery of subatomic particles," which almost exclusively talks about the experiments and is relatively easy to read (and short!).
Oscar Cunningham (May 21 2020 at 21:01):One thing I remember from the QFT course I took was that we spent a lot of time learning classical things! In particular: the Klein-Gordon equation, Green's functions, spinors, the Dirac equation, relativistic electromagnetism, gauge invariance, the Higgs mechanism. These could all be taught in a course called 'Classical Field Theory'. Then once you knew them, QFT would be much easier.
John Baez (May 21 2020 at 21:04):Yes, a lot of what you learn in a QFT course is classical. Don't forget Noether's theorem - that's perfectly classical, but a lot of students only get hit with it in quantum field theory.
I think all these things tend to make QFT seem more bulky and intimidating than it really is. (It's already quite bulky and intimidating on its own.)
John Baez (May 21 2020 at 21:05):All this talk is making me remember how much I like QFT. I did my PhD thesis on it.
Amar Hadzihasanovic (May 21 2020 at 21:21):John, Arthur, thanks for the recommendations! Reading historical accounts would have been a good idea. At the time I was too busy passing exams to do that.
I remember reading Max Born's autobiography at some point, and finding it striking how different and at the same time more plausible and vivid the account of the “birth of quantum mechanics” was, compared to the story told in most textbooks...
John Baez (May 21 2020 at 21:27):Don't say it "would have been a good idea". Make me happy: say "it's a good idea". :upside_down:
It's lots of fun and helpful at any age.
Amar Hadzihasanovic (May 21 2020 at 21:32):Ah, of course, it is a good idea! I didn't mean one to exclude the other :wink:
I was just returning in my head to that slight feeling of unease at getting good grades in an exam yet feeling like I didn't really understand.
John Baez (May 21 2020 at 21:45):Okay, yes. Sometimes it feels like any course is just an appetizer for all the things one could learn.
John van de Wetering (May 29 2020 at 11:38):John van de Wetering said:
John Baez said:
Hmm, I haven't seen Folland's book Quantum Field Theory: a Tourist Guide for Mathematicians.
Looking at the review, this seems to be the book for me! Thanks!
I have been reading this book the last week and it's great!
I'm amazed at the kind of semi-correct numbers you can get out of a model that stacks approximation on approximation on mathematical shifty sands.
I just started the chapter on using perturbation theory to study interacting fields. My mind is currently boggling at the idea of "renormalising" the "bare mass" due to interactions.
John van de Wetering (May 29 2020 at 11:39):I will probably have an increasing stack of questions as I progress through this book. Would anybody be willing to help answer these?
John Baez (May 29 2020 at 19:24):Sure, I'd be happy to try to answer questions!
Renormalization sounds weird at first but it makes complete physical sense. I tried to explain why it's a reasonable thing to do, with a minimum of technicalities, here:
The essence is this:
... virtual particles affect the observed charge and mass of a particle. Heuristically, at least, we should think of an electron as surrounded by a cloud of virtual particles. These contribute to its mass and 'shield' its electric field, reducing its observed charge. It takes some work to translate between this heuristic story and actual Feynman diagram calculations, but it can be done.
Thus, there are two different concepts of mass and charge for the electron. The numbers we put into the QED calculations are called the 'bare' charge and mass, and Poetically speaking, these are the charge and mass we would see if we could strip the electron of its virtual particle cloud and see it in its naked splendor. The numbers we get out of the QED calculations are called the 'renormalized' charge and mass, and These are computed by doing a sum over Feynman diagrams. So, they take virtual particles into account. These are the charge and mass of the electron clothed in its cloud of virtual particles. It is these quantities, not the bare quantities, that should agree with experiment.
I was being a bit cautious by sticking in words like "heuristic" and "poetically". It's basically true; you just need to translate it into math correctly.
Pastel Raschke (May 31 2020 at 09:44):a physicist friend i asked about renormalization sent me this set of notes, which starts with the metaphysical justification of (effective) irreducibility, then gets into the math with scaling invariance and fractals, then random walks, and only then gets to ising models and further
the general idea of renormalization reminds me of
John van de Wetering (Jun 01 2020 at 11:56):John Baez said:
Thus, there are two different concepts of mass and charge for the electron. The numbers we put into the QED calculations are called the 'bare' charge and mass, and Poetically speaking, these are the charge and mass we would see if we could strip the electron of its virtual particle cloud and see it in its naked splendor. The numbers we get out of the QED calculations are called the 'renormalized' charge and mass, and These are computed by doing a sum over Feynman diagrams. So, they take virtual particles into account. These are the charge and mass of the electron clothed in its cloud of virtual particles. It is these quantities, not the bare quantities, that should agree with experiment.
So how do we know the value of the bare mass? As in, how do we know which value to put into the equations? Is that as simple as picking a symbolic bare mass and calculating the renormalized mass , comparing it to the real mass and then setting ?
John van de Wetering (Jun 12 2020 at 13:41):A thing that is baffling me, is that you can take a classical field theory, like that of electromagnetism, and "quantize" it by replacing some symbols formally with more quantum like things, and that you just happen to end up with the correct theory
John van de Wetering (Jun 12 2020 at 13:43):Is there some kind of deeper reason why this would be?
Is it that Lorentz invariance is so restrictive that there is really only a few possibilities for good QFTs, and hence if you want something that reduces in the classical limit to established classical behaviour you need to make minimal changes to the field
John van de Wetering (Jun 12 2020 at 13:44):Unrelated question: In the book I'm reading it says that the phi^4 theory is the simplest interacting quantum field theory. Why does it need to be a power of 4? Why not 3? (As far as I understand a power of 2 would just contribute to the mass term, so I get why that is not interesting)
John Baez (Jun 12 2020 at 20:26):John van de Wetering said:
A thing that is baffling me, is that you can take a classical field theory, like that of electromagnetism, and "quantize" it by replacing some symbols formally with more quantum like things, and that you just happen to end up with the correct theory.
Electromagnetism was always quantized, we just didn't know it. Maybe the answer your question is that if we neglect interactions with charged particles, electromagnetism is a free quantum field theory - i.e. its Hamiltonian and Lagrangian are quadratic, so its equations of motion are linear.
There's a very beautiful simple "classical limit" for free quantum fields - that is, an limit - which goes along with very nice "coherent states" - i.e. states of the quantum field that mimic states of the classical field. So, not only are the equations of motion of the quantum Maxwell equations formally the same as for the classical Maxwell equations, the behavior of the quantum theory closely mimics that of the classical theory in many situations.
Given that electromagnetism is actually a quantum field theory, something like this had to be true, or nobody would have taken the classical theory seriously in the first place.
John Baez (Jun 12 2020 at 20:29):The story is quite different as soon as we do something so simple as throw charged spin-1/2 particles into QED. People didn't first think of a classical version of Dirac's equation coupled to classical electromagnetism and then later decide quantize it. The theory was first invented in work on quantum physics. (It was, however, first invented at the level of relativistic quantum mechanics, before a full-fledged quantum field theoretic treatment.)
John Baez (Jun 12 2020 at 20:33):John van de Wetering said:
Unrelated question: In the book I'm reading it says that the phi^4 theory is the simplest interacting quantum field theory. Why does it need to be a power of 4? Why not 3? (As far as I understand a power of 2 would just contribute to the mass term, so I get why that is not interesting).
You're right, a term would just give a mass: since it's quadratic the equations of motion are linear.
The problem with a term is that a term like this makes the Hamiltonian unbounded below (just because the function is unbounded below). So, there's no state of least energy - no "vacuum". Such theories are, apparently, very badly behaved.
In axiomatic quantum field theory we typically impose as an axiom that there's a state of least energy, the "vacuum".
John Baez (Jun 12 2020 at 20:56):At some point you might want to look up the Wightman axioms of quantum field theory, so you can see what people are shooting for when they do quantum field theory. (There are newer, more general axioms but the Wightman axioms are closer to textbook QFT.)
John van de Wetering (Jun 13 2020 at 09:13):The book I'm reading did define the Wightman axioms, but I don't understand this material well enough yet that I could infer from them that phi^3 would be forbidden. Although I did suspect that the "oddness" of a cubic function might have something to do with it
John van de Wetering (Jun 13 2020 at 09:15):A thing I'm pleasantly surprised by is that I am less bothered by renormalization and "removing infinities" than I thought I would be. It happens in a much more structured way than I would have suspected from popular accounts. It really does feel as if there should be some rigorous mathematical justification for why this should all be allowed.
John Baez (Jun 13 2020 at 15:41):Yes, perturbative QFT, where you write out answers as formal power series in the relevant coupling constants (e.g. the fine structure constant) is completely rigorously understood now, and you can see in this framework why renormalization is not only "allowed" but absolutely required.
John Baez (Jun 13 2020 at 15:41):The only problem is that we don't know if the power series converge - and they probably don't.
John Baez (Jun 13 2020 at 15:42):But that's not as bad as it sounds, because all sorts of simpler problems can be solved perturbatively, and you get power series that don't converge, but are merely "asymptotic", which is a weaker but still useful condition.
John van de Wetering (Jun 16 2020 at 13:00):Do you have a nice example of such a simpler problem where you need to solve it perturbatively even though the power series doesn't converge?
John Baez (Jun 16 2020 at 17:01):What do you mean, "need to" solve it perturbatively?
John Baez (Jun 16 2020 at 17:02):Do I need to have a proof that no other method can work? That's very hard.
John van de Wetering (Jun 24 2020 at 18:42):Yeah, I guess "need" is too strong a word. Just a problem where doing it perturbatively is preferable to doing it in another way, would already be helpful
John van de Wetering (Jun 24 2020 at 18:43):So I have finished the book "The mathematicians tourist guide to QFT", and I definitely need to reread some chapters, to get some stuff more clear in my head
John van de Wetering (Jun 24 2020 at 18:45):Regardless, there are a few things I'm already curious about: the book gives a brief explanation of the Higgs mechanism, and it says it is "needed" to give the force bosons mass. I don't understand why you need it for that. Couldn't you just write down a Lagrangian where those particles happen to have mass? Why do you need to go through all these hoops of adding a new field, and breaking the symmetry (other than that it apparently happens to give the right physics of course)?
John van de Wetering (Jun 24 2020 at 18:46):Also, it demonstrates the Higgs mechanicsm for the electroweak theory, and the book says that the gauge group you need to pick there is SU(2)xU(1). Is there some way that could have been known a priori, or is this just a case of trying different gauge groups, until you've found one that works
John van de Wetering (Jun 24 2020 at 18:54):Also also, I'm intrigued by the fact the Higgs mechanism symmetry breaking mass-giving only works when the higgs field is close to a minimum, if I'm understanding correctly. So in the high-energy limit you have different particles and they are massless?
John Baez (Jun 25 2020 at 05:26):John van de Wetering said:
Yeah, I guess "need" is too strong a word. Just a problem where doing it perturbatively is preferable to doing it in another way, would already be helpful
A good example is finding the ground state energy of the anharmonic oscillator - or in math language, the lowest eigenvalue of the operator
on .
John Baez (Jun 25 2020 at 05:29):You can compute this using perturbation theory as a power series in , and you get a series that diverges, but is asymptotic, meaning that it takes more and more terms to diverge as gets smaller and smaller, and it gets closer and closer to the right answer before it diverges.
John Baez (Jun 25 2020 at 05:32):There are papers about this:
This is a baby version of the theory - you can think of it as the theory in 0+1 dimensions (no space dimensions and one time dimension).
John Baez (Jun 25 2020 at 05:36):John van de Wetering said:
Regardless, there are a few things I'm already curious about: the book gives a brief explanation of the Higgs mechanism, and it says it is "needed" to give the force bosons mass. I don't understand why you need it for that. Couldn't you just write down a Lagrangian where those particles happen to have mass?
If you tried to give the W and Z their mass by just adding the usual sort of mass term like it would break gauge invariance. It wouldn't be a gauge theory anymore. And this in turn would cause other problems.
John Baez (Jun 25 2020 at 05:39):Also, it demonstrates the Higgs mechanicsm for the electroweak theory, and the book says that the gauge group you need to pick there is SU(2)xU(1). Is there some way that could have been known a priori, or is this just a case of trying different gauge groups, until you've found one that works.
It's not an a priori thing. If you believe in electromagnetism you know you want a group with a U(1) subgroup and you know you'll need a Higgs that gives mass to all the other gauge bosons. But there was a huge amount of experimental work that first revealed the existence of "weak isospin" (the SU(2) factor) and then "hypercharge" (the U(1) factor, which is not the same as the electromagnetic U(1) subgroup).
John Baez (Jun 25 2020 at 05:43):It's fun to read the history of particle physics to see how this was worked out,but you can read a very simplified "mathematician's account" here:
John Baez (Jun 25 2020 at 05:44):This does not explain the Higgs or spontaneous symmetry breaking. But it does explain a bit about SU(2) U(1).
John Baez (Jun 25 2020 at 05:50):John van de Wetering said:
Also also, I'm intrigued by the fact the Higgs mechanism symmetry breaking mass-giving only works when the higgs field is close to a minimum, if I'm understanding correctly. So in the high-energy limit you have different particles and they are massless?
Yes, this kicks in at energies that we're starting to see at the Large Hadron Collider. In terms of history, 10 picoseconds after the Big Bang, when the universe cooled down to a temperature about 1 quadrillion Kelvin, there was a phase transition. Cooler regions, expanding bubbles, formed in which the SU(2) U(1) symmetry broke. The concept of "photon" only made sense inside these bubbles, which eventually took over and became our universe.
John van de Wetering (Jun 26 2020 at 11:51):John Baez said:
There are papers about this:
- S. Graffi, V. Grecchi, and B. Simon, Borel summability: application to the anharmonic oscillator, Phys. Lett. B32 (1970), 631-634.
This is a baby version of the theory - you can think of it as the theory in 0+1 dimensions (no space dimensions and one time dimension).
Cool! I will check it. Theoretical physics papers were so intense in those days, hardly any introduction or establishing of background, and just straight into the maths...
John van de Wetering (Jun 26 2020 at 11:54):John Baez said:
John van de Wetering said:
Regardless, there are a few things I'm already curious about: the book gives a brief explanation of the Higgs mechanism, and it says it is "needed" to give the force bosons mass. I don't understand why you need it for that. Couldn't you just write down a Lagrangian where those particles happen to have mass?
If you tried to give the W and Z their mass by just adding the usual sort of mass term like it would break gauge invariance. It wouldn't be a gauge theory anymore. And this in turn would cause other problems.
What kinda problems would you run into if you try to build a Lagrangian without caring about gauge fields?
Sorry I'm asking so many questions, but it is really hard to find suitable answers online, as many resources regarding quantum field theory are either popular accounts, or very in-depth.
For instance, I tried searching for "Why is the Higgs mechanism needed?" and I only found news articles that basically said "because it gives particles mass!"
Javier Prieto (Jun 26 2020 at 16:55):John van de Wetering said:
What kinda problems would you run into if you try to build a Lagrangian without caring about gauge fields?
I don't know too much about the quantum case, but in classical electromagnetism gauge invariance amounts to saying that all relevant physical information is encoded in the electromagnetic tensor, which is an exact 2-form, $F=d A$. Since closed 1-forms are in the kernel of $d$, you have a whole class of $A+\eta \in \Lambda^1 M$ with $\eta$ closed corresponding to the same $F$. Gauge invariance says physics shouldn't depend on the choice of $\eta$.
Javier Prieto (Jun 26 2020 at 16:56):John van de Wetering said:
What kinda problems would you run into if you try to build a Lagrangian without caring about gauge fields?
I don't know too much about the quantum case, but in classical electromagnetism gauge invariance amounts to saying that all relevant physical information is encoded in the electromagnetic tensor, which is an exact 2-form, . Since closed 1-forms are in the kernel of , you have a whole class of with closed corresponding to the same . Gauge invariance says physics shouldn't depend on the choice of .
John Baez (Jun 26 2020 at 18:34):What kinda problems would you run into if you try to build a Lagrangian without caring about gauge fields?
The main problem is that in 3+1 dimensions it's very hard to find theories involving spin-1 particles that are "fully consistent" that aren't gauge theories!
John Baez (Jun 26 2020 at 18:35):Here "fully consistent" is my own made-up term for a not very precise concept. Quantum field theory in 3+1 dimensions is very hard to make completely rigorous, so nobody really knows for sure which theories exist, except for free theories (which describe worlds where nothing interacts).
John Baez (Jun 26 2020 at 18:36):So, people have rules of thumb: for example, non-renormalizable theories are considered bad, and so are theories with Landau poles, etc. etc.
John Baez (Jun 26 2020 at 18:37):There's an enormous literature on this stuff, and it's not math so it's hard to find clearly stated theorems, even though it's possible to prove theorems about these things.
John Baez (Jun 26 2020 at 18:38):So, all I can say is that "the general consensus" is that there are no quantum field theories in 3+1 dimensions that are
1) not free,
2) involve spin-1 particles,
3) are renormalizable, and
4) are not gauge theories.
John Baez (Jun 26 2020 at 18:39):This consensus (which is based on a huge amount of work, and perhaps even some rigorous theorems I haven't seen), makes physicists use gauge theories. After all, we live in 3+1 dimensions and we see spin-1 particles!
John Baez (Jun 26 2020 at 18:44):For instance, I tried searching for "Why is the Higgs mechanism needed?" and I only found news articles that basically said "because it gives particles mass!"
Instead of Google searches it's probably better to get a stock of QFT books like Itzykson and Zuber, Weinberg, and so on. Even if you can't understand all the details, there are some very nice paragraphs here and there.
Easier books, which I find to be full of wisdom, are these:
Kerson Huang, Quarks, Leptons & Gauge Fields, World Scientific, Singapore, 1982.
L. B. Okun, Leptons and Quarks, translated from Russian by V. I. Kisin, North-Holland, 1982. (Huang's book is better on mathematical aspects of gauge theory and topology; Okun's book is better on what we actually observe particles to do.)
T. D. Lee, Particle Physics and Introduction to Field Theory, Harwood, 1981.
John van de Wetering (Jun 26 2020 at 19:07):John Baez said:
So, all I can say is that "the general consensus" is that there are no quantum field theories in 3+1 dimensions that are
1) not free,
2) involve spin-1 particles,
3) are renormalizable, and
4) are not gauge theories.
Interesting. So then the necessity of the Higgs mechanism could be explained as follows.
John van de Wetering (Jun 26 2020 at 19:09):Related question: If I give an expert QFT person a Lagrangian, would this person be "easily" able to tell whether it is a bad QFT? I.e. if it is non-renormalisable or not? If so, would it be possible to let a computer programme generate candidate Lagrangians and simply check if there are any that are renormalisable and contain bosons with mass? If this would be possible, and the program wouldn't find any, then I guess this would be a neat way to see why you need to do more fancy stuff to get bosons with mass
John Baez (Jun 26 2020 at 19:40):John van de Wetering said:
Interesting. So then the necessity of the Higgs mechanism could be explained as follows.
- We need certain bosons to have mass.
- Every attempt to put this mass in manually has lead to bad theories.
- Gauge theories more easily lead to good theories, but unfortunately have massless bosons.
- So we need to add a Higgs field to break certain symmetries which does funky stuff to the fields, so that the boson fields combine into different fields that now have mass.
Yes, that's the idea - except when you write "bosons" you meant "spin-1 particles". There's a whole different story for spin-0 and spin-2 bosons (and higher-spin bosons, which we don't see).
John Baez (Jun 26 2020 at 19:43):John van de Wetering said:
Related question: If I give an expert QFT person a Lagrangian, would this person be "easily" able to tell whether it is a bad QFT?
It probably depends a lot on how many terms there are in the Lagrangian, and whether you've written it down in a way that makes gauge symmetries clearly visible. If you deliberately try to obfuscate the gauge symmetries, you could make it very difficult.
Consider this, for example:
The Standard Model Lagrangian, written in a suboptimal way
John Baez (Jun 26 2020 at 19:44):If you leave out one of the terms here, you could easily make it nonrenormalizable!
But this is a stupid way to organize the terms in the Lagrangian of the Standard Model.
John Baez (Jun 26 2020 at 19:46):would it be possible to let a computer programme generate candidate Lagrangians and simply check if there are any that are renormalisable and contain bosons with mass?
It can be a lot of work to tell if a Lagrangian is renormalizable. On top of that, renormalizability is neither a necessary nor a sufficient condition for a theory to be "completely consistent" (in the vague sense to which I'm alluding).
John Baez (Jun 26 2020 at 19:47):It would be great if computer programs could help physicists generate useful Lagrangians, but it would definitely take very good physicists to write such programs.
John van de Wetering (Jun 26 2020 at 20:26):John Baez said:
It's fun to read the history of particle physics to see how this was worked out,but you can read a very simplified "mathematician's account" here:
- John Baez and John Huerta, The algebra of grand unified theories.
I'm reading this paper now (and got to page 26 so far). I don't think I've seen this B Boson before that is discussed in the section on the weak interaction. If I understand correctly, it is what we should see instead of photons at high energy. Has this particle been observed?
John Baez (Jun 26 2020 at 20:29):The photon, and are just a different basis of than the basis involving the .
John Baez (Jun 26 2020 at 20:30):I guess you know that, but I wanted to say it.
John Baez (Jun 26 2020 at 20:31):The only particle we actually "see" is the photon; the rest only show up in intermediate steps of reactions.
John Baez (Jun 26 2020 at 20:32):It might make some sense in certain circumstances to say we've "seen" a B.
John Baez (Jun 26 2020 at 20:33):But as you note, this would mainly happen at high energies, maybe roughly > 150 GeV.
John Baez (Jun 26 2020 at 20:34):I haven't actually heard of people saying "we've seen a B!"
John Baez (Jun 26 2020 at 20:35):What they mainly saw, that indicates this electroweak symmetry breaking works as expected, is the Higgs. This has a mass of 125 GeV.
John Baez (Jun 26 2020 at 20:36):So, in some sense we're just getting to the energies where we could see (as opposed to predict) what things are like when the electroweak symmetry is unbroken.
John van de Wetering (Jun 26 2020 at 20:56):That is really exciting! I imagine the particle physics community would blow up if this energy regime where electroweak symmetry is unbroken behaves differently than predicted. Or are there alternative theories (supersymmetry or the like) that says that this unbroken high energy regime should behave significantly differently?