Rigorous Study of the Higgs Boson - Question

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Summary:: I would like to ask about books and other materials to study in order to understand in a really, really rigorous manner the Higgs field, Higgs boson and other related topics.

Answers that are detailed but at the same time precise and to the point would be highly appreciated (please, no "this would take too much effort"). Also if you are an author please do not use this thread to promote your own books (it has happened in the past) unless they are really relevant to what I am asking.

Hello,

I have a degree in physics but it has been focused on the practical applications, rather than on rigorously understanding - and I mean really understanding - the mathematics and the theoretical physics that is behind the theory on which those applications are based.

I have some training in mathematics (derivatives and integrals of one-variable functions, and some vector analysis (though the latter is not by far as in-depth or as rigorous as I would like it to be). I have also studied the "standard" introductory undergraduate courses in quantum mechanics and special relativity (without general relativity), and a few other topics that are typically taught in a BSc degree in physics. But I did not find them satisfactory at all, they were the kind of courses that introduce a few equations and then give you exercises to practice, without going into how they were derived and without going into an in-depth analysis of the mathematics involved in those equations.

Recently, I came across this paper
https://www.rochester.edu/news/hagen/PhysRevLett.13.585.pdf ("Global Conservation Laws of Massless Particles" by T. W. B. Kibble et al., 1964) and I realised I do not understand it at all.

Could you please tell me what books (and / or other materials) I need to study in order to understand the paper?

(Also, somewhat related to this, what books would I need to read in order to understand really well quantum electrodynamics, quantum chromo-dynamics etc.?)

In other words, I am asking for the "ladder" of books (both physics books and mathematics books) that I need to study. Please only provide links to courses if those courses are really well made (the vast majority of the ones I had come across so far aren't, even at world-famous universities), i.e. if they go into the most minute detail of the mathematics and physics used - no "please accept this theorem without proof".)

I am aware that this type of rigorous learning that I am looking for is unlikely to be achieved by studying a single book or single course and that is fine - in other words, it is fine to provide (for example) the title of a book even if that book does not contain the proof for all the theorems used in it, as long as another book of the combination of physics and mathematics books / course materials you provide proves those theorems.

Thank you very much in advance and please keep safe!
 
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  • #2
PeroK
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You probably don't want to hear this but physics isn't about proving theorems. Physics is about finding a mathematical model that predicts the results of experiments. Although a lot of modern mathematics is used in theoretical physics, the emphasis in any physics text is the application of that mathematics, not the proof of the mathematics itself. In physics it's the end product that matters, and whether it stands up to experimental scrutiny. Making a theory fully mathematically rigorous does not in any way encourage nature to adopt that theory.

An excellent standard text in this field is Elementary Particles by Griffiths. It includes QED, QCD and ends with discussions of further topics like the Higgs field. But, that is very much a physics book and would fail all your criteria for a pure mathematical treatment of the subject.
 
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  • #3
vanhees71
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One should also state that all attempts to make relativistic QFT in (1+3) dimensions mathematically rigorous have failed so far. Nevertheless the Standard Model formulated in renormalized perturbation theory gives excellent results.

Imho the best books on the subject are

S. Weinberg, Quantum Theory of Fields, Vols. 1+2 (Vol. 3 is about supersymmetric extensions)
A. Duncan, The Conceptual Framework of Quantum Field Theory

These are complementary. If you are more interested in the formal mathematical structure, maybe Duncan is the right choice, because he explains all the funny obstacles like Haag's theorem, which tells us that in fact the interaction picture doesn't exist though it's used all the time to derive the perturbative expansion leading to results agreeing with experiment to 12 or more digits (##g-2## of the electron, Lamb shift, etc.).
 
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You probably don't want to hear this but physics isn't about proving theorems. Physics is about finding a mathematical model that predicts the results of experiments. Although a lot of modern mathematics is used in theoretical physics, the emphasis in any physics text is the application of that mathematics, not the proof of the mathematics itself. In physics it's the end product that matters, and whether it stands up to experimental scrutiny. Making a theory fully mathematically rigorous does not in any way encourage nature to adopt that theory.

An excellent standard text in this field is Elementary Particles by Griffiths. It includes QED, QCD and ends with discussions of further topics like the Higgs field. But, that is very much a physics book and would fail all your criteria for a pure mathematical treatment of the subject.
Thank you, but I was asking for much more than that. I already have the book Elementary Particle Physics by Griffiths. I asked for the **entire** "ladder" (for lack of a better term) of books, both physics books and mathematics books, that would enable me (after I study them) to understand the paper at the link I provide.
I was definitely not asking just for one book to "help me get started".
Please read the paper at the link I provided and answer accordingly.
 
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One should also state that all attempts to make relativistic QFT in (1+3) dimensions mathematically rigorous have failed so far. Nevertheless the Standard Model formulated in renormalized perturbation theory gives excellent results.

Imho the best books on the subject are

S. Weinberg, Quantum Theory of Fields, Vols. 1+2 (Vol. 3 is about supersymmetric extensions)
A. Duncan, The Conceptual Framework of Quantum Field Theory

These are complementary. If you are more interested in the formal mathematical structure, maybe Duncan is the right choice, because he explains all the funny obstacles like Haag's theorem, which tells us that in fact the interaction picture doesn't exist though it's used all the time to derive the perturbative expansion leading to results agreeing with experiment to 12 or more digits (##g-2## of the electron, Lamb shift, etc.).
vanhees71, thank you very much for your reply and, of course, I was not asking for physics or mathematics books / materials that haven't been written yet.
When I said "rigorous" I meant, of course, "as rigorous as possible under the current circumstances" (i.e. as far as physics / mathematics have progressed to date.)
At any rate, your reply is, at least in my opinion, the best I have received so far so thank you very much again.
Any more books like the two you mentioned would be highly appreciated.

"Introduction to Elementary Particles" by Griffiths summarises or even skips completely quite a bit of mathematics. For example, it introduces Feynman diagrams but it does not present in detail (it would probably take tens or hundreds of pages) the physics and the mathematics that is behind such diagrams. Do you know of any physics or mathematics books that explain these diagrams in detail?
 
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You probably don't want to hear this but physics isn't about proving theorems. Physics is about finding a mathematical model that predicts the results of experiments. Although a lot of modern mathematics is used in theoretical physics, the emphasis in any physics text is the application of that mathematics, not the proof of the mathematics itself. In physics it's the end product that matters, and whether it stands up to experimental scrutiny. Making a theory fully mathematically rigorous does not in any way encourage nature to adopt that theory.

An excellent standard text in this field is Elementary Particles by Griffiths. It includes QED, QCD and ends with discussions of further topics like the Higgs field. But, that is very much a physics book and would fail all your criteria for a pure mathematical treatment of the subject.
Also PeroK, I am sorry to say this but the answer you gave was precisely the type answer I was trying to avoid receiving.
I provided a paper, and I asked what physics and mathematics books do I need to study, given my background, in order to understand that paper.
It is no news that nature does not adapt to fit theories, and I was looking for a precise answer, not for debating the philosophy of science. Nothing of what you said in this regard (what physics is or isn't and so on) is something I heard for the first time, and if I asked that question, I probably did it with a specific purpose in mind. I don't know about you but my time is limited, so I need answers that do not digress unnecessarily. (If, for example, the digression had gone in the direction of suggesting good books, it would have been a welcome digression.)
What I need is an answer like (this, of course, is just an example) "you start with this book (title, author) which will teach you this bit of physics, and you read it in parallel with this book (title, author) of mathematics that will help you understand the mathematics used in the book I mentioned first, then this other book (title, author)....(and so on)...and then you will be able to understand the paper at the link in your original post". I would like a similar "book ladder" for understanding the Higgs boson.

I hope my replies above clarify my question. If what I asked is still not clear enough, please do let me know and thank you again in advance to all who will take the time to answer.
 
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  • #7
Vanadium 50
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Well, you're insisting that the field adapt itself to your way of thinking rather than the other way around. While it's certainly possible to spend "tens or hundreds of pages" on mathematics and proofs before jumping into calculations of, e.g. the muon lifetime, the fact is that textbooks don't do it this way.
 
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  • #8
vanhees71
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In my exprience you understand QFT best when doing the calculations yourself, and you need a real QFT book to learn it. Griffiths is an introductory particle-physics textbook and not a QFT textbook. I also misunderstood obviously the question. It was not about attempts to make QFT mathematically rigorous but simply about learning QFT. I recommended the textbooks by Weinberg and Duncan which are somwhat in between: On the one hand they are physics textbooks and not mathematically rigorous, on the other they give careful derivations of fundamental issues.

Not it seems as if the OP rather needs a QFT texbook to start learning the subject. For that purpose the said textbooks are maybe a bit too detailed. Then I'd rather recommend to start with a more standard introductory textbook. My favorite for this purpose is Schwartz, Quantum field theory and the standard model. Another book is Peskin and Schroeder, but it's full of typos and sometimes also not accurate enough (e.g., to have a dimensionful argument in a logarithm in the chapter about renormalization is a nogo).
 
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  • #9
Delta2
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@vanhees71 I have a specific question about the math required to learn QFT. Is tensor calculus absolutely necessary for someone that wants to learn QFT? Back at the era of my undergraduate studies (mid 90s) tensor calculus was not an obligatory source in my math department and i didnt take the optional courses offered. Due to my occupation as high school (ages 13-15) teacher i never bothered to learn tensor calculus cause it is not needed for my job. Should i learn it if i intend to learn QFT and what book do you recommend for tensor calculus?
 
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vanhees71
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You need some where basic vector calculus for QFT. The main thing you need is to work with operators, path integrals, and some complex function theory to evaluate Feynman diagrams in dimensional regularization.
 
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  • #11
PeroK
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@vanhees71 I have a specific question about the math required to learn QFT. Is tensor calculus absolutely necessary for someone that wants to learn QFT? Back at the era of my undergraduate studies (mid 90s) tensor calculus was not an obligatory source in my math department and i didnt take the optional courses offered. Due to my occupation as high school (ages 13-15) teacher i never bothered to learn tensor calculus cause it is not needed for my job. Should i learn it if i intend to learn QFT and what book do you recommend for tensor calculus?
You could always jump in at the deep end with this video lecture series:


See how much you can follow.
 
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  • #12
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In my exprience you understand QFT best when doing the calculations yourself, and you need a real QFT book to learn it. Griffiths is an introductory particle-physics textbook and not a QFT textbook. I also misunderstood obviously the question. It was not about attempts to make QFT mathematically rigorous but simply about learning QFT. I recommended the textbooks by Weinberg and Duncan which are somwhat in between: On the one hand they are physics textbooks and not mathematically rigorous, on the other they give careful derivations of fundamental issues.

Not it seems as if the OP rather needs a QFT texbook to start learning the subject. For that purpose the said textbooks are maybe a bit too detailed. Then I'd rather recommend to start with a more standard introductory textbook. My favorite for this purpose is Schwartz, Quantum field theory and the standard model. Another book is Peskin and Schroeder, but it's full of typos and sometimes also not accurate enough (e.g., to have a dimensionful argument in a logarithm in the chapter about renormalization is a nogo).
vanhees71 , thank you very much again for this answer too and please never worry about the fact that a book may be too detailed. That is precisely what I need (provided, of course, that the details are relevant).
 
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You need some where basic vector calculus for QFT. The main thing you need is to work with operators, path integrals, and some complex function theory to evaluate Feynman diagrams in dimensional regularization.
vanhees71 , very well - in that case, what books would you recommend for learning these?
 
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An excellent standard text in this field is Elementary Particles by Griffiths. It includes QED, QCD and ends with discussions of further topics like the Higgs field. But, that is very much a physics book and would fail all your criteria for a pure mathematical treatment of the subject.
PeroK, to clarify what I said even more, I did * not * ask *exclusively* for a "pure mathematical treatment" of the subject.
What I asked for are books that do not skip the mathematics, not books that present only the mathematics without having a physics content. Physics content is precisely what I want, except that I do not want the mathematics (that is needed to present those concepts) to be skipped, summarised and so on.
So I want *very detailed physics* and also *very detailed mathematics* too.
(By the way, PeroK, I think it should be clear by now what I am asking for, and I think it is a bit odd that after the several replies I have already written I still have to explain what I asked for the the original post.)
 
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In my exprience you understand QFT best when doing the calculations yourself, and you need a real QFT book to learn it. Griffiths is an introductory particle-physics textbook and not a QFT textbook.
vanhees71, THANK YOU so much for pointing that out! That is precisely what I mean!
And I mean the same regarding quantum chromo-dynamics and the other topics I mentioned.

Yes, you need to do the calculations yourself, which is why I am asked what I asked above regarding textbooks, detailed mathematical treatment and so on. And, of course, also detailed treatment (of the topics presented in such books) from the point of view of the * physics * involved, not just the mathematics.

With regard to what PeroK said, I would like to state again (and even very emphatically!!) that I am not trying to avoid or to skip either the physics or the mathematics involved (quite the contrary!!) and I am fully aware that one needs *both* in order to fully understand this area of physics.
 
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George Jones
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I would like to ask about books and other materials to study in order to understand in a really, really rigorous manner the Higgs field, Higgs boson and other related topics.
It is standard in mathematics and theoretical physics for "rigorous" to be synonymous with "mathematically rigorous", and for "mathematically rigorous" to refer to material that involves mathematical abstractions like epsilon delta proofs, topological neighbourhoods, etc.

Consequently, given your (unkowingly) poor word choice, it is not unexpected that folks post stuff like

But, that is very much a physics book and would fail all your criteria for a pure mathematical treatment of the subject.

One should also state that all attempts to make relativistic QFT in (1+3) dimensions mathematically rigorous have failed so far.
 
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  • #18
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By the way, PeroK, I think it should be clear by now what I am asking for, and I think it is a bit odd that after the several replies I have already written I still have to explain what I asked for the the original post
When there is misunderstanding, some people try and rephrase what they wroite to try and make it clearer. Others blame the people they are talking to. Which do you think is more effective?

For what it's worth, I am more confused rather than less about what you are looking for.
 
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When there is misunderstanding, some people try and rephrase what they wroite to try and make it clearer. Others blame the people they are talking to. Which do you think is more effective?

For what it's worth, I am more confused rather than less about what you are looking for.
Vanadium 50,
1. I am well aware physics and mathematics have limitations, and even great limitations with regard to some problems/ phenomena.
2. Within the given framework of the physics and the mathematics we currently have, I would like for the materials that people are suggesting to be as rigorous as possible - even though in some cases "as rigorous as possible" is pretty far if not very far from rigorous.

Ultimately logic itself is not rigorous, because we take for granted the mechanisms of logical deduction. However, **within this framework of logic** some materials are very rigorous and others end up in "this is the limit of our knowledge/ thinking".

Furthermore, I would like for the books presented to form a "complete" ladder of knowledge so that I can study them. I.e. it is very desirable to send me to study a very advanced book of physics/ mathematics, but at the same time, in case they are too advanced for me (some of them will certainly be) I would also like the books that take me from my current level to the level where can study those advanced books.
However, at the same time, in case you have such very advanced books to suggest but, on the other hand, do not have those other books that that take me from my current level to the level where can study those advanced books, please do suggest those very advanced books. This is because there is a chance that others who will reply to this post will suggest those other books that take me from where i am to the level where I can study those very advanced books.
 
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When there is misunderstanding, some people try and rephrase what they wroite to try and make it clearer. Others blame the people they are talking to. Which do you think is more effective?
Vanadium 50, very well. In line with what you said, I have written another reply (see above) in the hope of clarifying what I am asking for. Hopefully at some point we will reach a clear communication.
 
  • #21
PeroK
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Vanadium 50, very well. In line with what you said, I have written another reply (see above) in the hope of clarifying what I am asking for. Hopefully at some point we will reach a clear communication.
What is your current level, if you don't mind my asking?
 
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Moderator note:

@PeroK is quite obviously not the only one who has difficulties with
What I asked for are books that do not skip the mathematics, not books that present only the mathematics without having a physics content. Physics content is precisely what I want, except that I do not want the mathematics (that is needed to present those concepts) to be skipped, summarised and so on.
So I want *very detailed physics* and also *very detailed mathematics* too.
Just as if good physics textbooks weren't rigorous. They are. You cannot divide physics and mathematics. It is as if you demanded to read Shakespeare, however, in a good English. No wonder people are confused.

I like to remind all participants to remain calm and polite. If someone doesn't like a comment or question, simply do not answer. There is no need for astersikses
I did * not * ask *exclusively* for a "pure mathematical treatment"
 
  • #23
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The book of Folland on QFT has a little bit on symmetry breaking. The book as a whole is as mathematically precise as possible with enough comments on what is and what is not regorous in the field.

There is a book by S. Sternberg "Curvature in mathematics and physics". For the most part it is mostly geometry but it does have sections on the Higgs mechanism from a more geometric point of view. It may be not what you are looking for.
 
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  • #24
George Jones
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The book of Folland on QFT has a little bit on symmetry breaking. The book as a whole is as mathematically precise as possible with enough comments on what is and what is not regorous in the field.
I love Folland's book, but, reading through this thread, I get the impression that this is not what the original poster is looking. You (and some others) might be interested in the nice article "The Higgs Boson for Mathematicians. Lecture Notes on Gauge Theory and Symmetry Breaking"
https://arxiv.org/abs/1512.02632

What I asked for are books that do not skip the mathematics, not books that present only the mathematics without having a physics content. Physics content is precisely what I want, except that I do not want the mathematics (that is needed to present those concepts) to be skipped, summarised and so on.
So I want *very detailed physics* and also *very detailed mathematics* too.
Judging by this, my guess for a first book for @Sunnyocean is "Quantum Field Theory for the Gifted Amateur" by Lancaster and Blundell, a very nice book with (in my opinion) a terrible title,
https://www.amazon.com/Quantum-Fiel...her?_encoding=UTF8&me=&qid=&tag=pfamazon01-20

This introductory book on quantum field give a fair number of steps in the mathematical calculations, and is good for someone who wants something more than Griffiths, but who does not want to jump right into Schwartz or Peskin and Schroeder or Weinberg or ...

On a personal note, a few summers ago, I spent many pleasant hours in coffee shops going through parts of this while avoiding (the hustle and bustle of) my in-laws (' place).
 
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You need some where basic vector calculus for QFT. The main thing you need is to work with operators, path integrals, and some complex function theory to evaluate Feynman diagrams in dimensional regularization.
vanhees71, thank you for your reply. Can you please recommend some books on path integrals? Also on complex function theory? (I don't know what that is).
 
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