Rigorous Study of the Higgs Boson - Question

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vanhees71, also, what book would you recommend for vector calculus? (Note: I did do some vector calculus during my undergraduate degree in physics, but it was not explicit enough. It basically presented conclusions but it did not prove the theorems taught in the course, and also the exercises in it were lacking. I would like books that present the concepts well and also have solutions to exercises, so that i can check my progress / how well I do.)
 
  • #27
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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.
PeroK, I find it really hard ("hard" as in "impossible) to "trust" and, more importantly, understand the use of a mathematical theorem whose proof I don't understand. That is how my thinking works. I guess this is why I am so "upset" with the situation when I don't understand the mathematics used in all of its details. Or maybe "upset" is not the right word. But I really, really need to understand the mathematics that is used. I am not trying to argue about what physics / mathematics is or should be. I am just trying to present the way I think so that people can give me the help I need.
 
  • #28
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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



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).
George, thank you very much, I ordered it.
Vanhees 71, thank you too. I also ordered Duncan, Weinberg (all three).
martinbn, thank you too. I ordered Folland.
Though somehow I feel these books will not be enough.
 
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What is your current level, if you don't mind my asking?
PeroK, thank you for your question. Please see my original post with regard to my level of education in physics / mathematics.
I feel a dire need to have a robust training in mathematics and in theoretical physics, and also in particle physics and quantum physics, so that I can have a robust understanding of the current knowledge. I am definitely not at the level where I can embark on, say, a PhD in theoretical physics. And probably also not the level of understanding / preparation in physics of someone who has a good BSc degree (one obtained at top universities such as MIT or Oxford).
I was actually offered a place to study physics at Oxford some years ago but I didn't have the funding (I still don't have it) so I could not go (I was told my application at Oxford was one of the best, which in retrospective makes what happened all the more sad). And the physics degree that I did do was, if I can put it this way, not exactly at the level of what they do in Oxford, MIT or other very good universities.
So I am trying to replace that by buying and reading books (and not just reading, but by studying them properly), and by asking for advice from people and so on. I am studying as much as I can in my free time; although the task seems daunting (I realise there is a lot to learn), but I am trying to keep at it as much as I can.
I really want to have a robust understanding, to the point where I can think just like a very good theoretical/ particle / quantum physicist (ultimately I think I need all of these in order to be at the level where I can robustly use the current knowledge with regard to how the universe works).
And at some point maybe even make my own contribution to physics although, at this point, something like that seems very far away. Although, I dare say, I have a certain degree of intelligence - maybe even a lot of it - when I look at the great theoretical physicists they look like giants to me, and I wonder how on earth can I ever reach the volume of knowledge and the depth of understanding that they have in order to actually be at a level where I can make real contributions to science.
Most universities seem like a disappointment to me (and i have been at some that are among the world's top 100, though not the likes of Oxford or MIT). Somehow they fail to do the "it". I.e. to help you get a real training, a real understanding. I wonder if the time of my life will be enough to do all that. So if I can have an "accelerator", i.e. a well-structured learning that would take me faster to where I need to me (not by skipping steps but by making the learning process more efficient, e.g. by studying really good books), that would really help a lot. This is why I have started threads such as this one.
 
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I also wish there were some kind of explicit pathway (manifested as a "ladder" of books, that you mentioned) taking you (leading you by hand) from A to B (state-of-the-art understanding of the cutting edge theoretical physics including all the necessary math), giving you everything that's necessary in an order that makes it comprehensible. Kind of a friendly algorithm to understanding already laid for you so that you could just follow it and don't waste your effort on all these dead-ends, searching for the exit or optimal routes and other pathfinding.

Unfortunately, there's no algorithm for understanding. The knowledge endless generations of very smart people have generated is vast and convoluted, and you have to untangle it for yourself. People are different and require different things to "get it", some books seems like gospel for certain people and like incomprehensible mess for other (and the way people perceive and understand books is constantly changing throughout their whole process of learning). Some people take some things for granted and don't require certain explanations to be happy and use the theory, while other can't get a good sleep until they figure them out. You have to carve your own path through knowledge experimenting, trying and failing, and nobody can do it for you.

I myself am in the process of "getting there", and sometimes dream about this "perfect course" that answers my personal inclinations and ideas about what it should be and how it should work. Maybe when I get there, I can look back (like I'm constantly looking back, or sometimes forward, now, and survey the landscape, trying to make sense of it) and find a way to organize all this knowledge in a coherent comprehensible logical whole (though I'm afraid it will only seem so to me, not to other people, who didn't have my personal experience). But I'm still far away from my goal, and not sure what exactly I want and what theoretical areas I need for that and how deep I should delve into them.

Just keep doing it and enjoy the process.
 
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I also wish there were some kind of explicit pathway (manifested as a "ladder" of books, that you mentioned) taking you (leading you by hand) from A to B (state-of-the-art understanding of the cutting edge theoretical physics including all the necessary math), giving you everything that's necessary in an order that makes it comprehensible. Kind of a friendly algorithm to understanding already laid for you so that you could just follow it and don't waste your effort on all these dead-ends, searching for the exit or optimal routes and other pathfinding.

Unfortunately, there's no algorithm for understanding. The knowledge endless generations of very smart people have generated is vast and convoluted, and you have to untangle it for yourself. People are different and require different things to "get it", some books seems like gospel for certain people and like incomprehensible mess for other (and the way people perceive and understand books is constantly changing throughout their whole process of learning). Some people take some things for granted and don't require certain explanations to be happy and use the theory, while other can't get a good sleep until they figure them out. You have to carve your own path through knowledge experimenting, trying and failing, and nobody can do it for you.

I myself am in the process of "getting there", and sometimes dream about this "perfect course" that answers my personal inclinations and ideas about what it should be and how it should work. Maybe when I get there, I can look back (like I'm constantly looking back, or sometimes forward, now, and survey the landscape, trying to make sense of it) and find a way to organize all this knowledge in a coherent comprehensible logical whole (though I'm afraid it will only seem so to me, not to other people, who didn't have my personal experience). But I'm still far away from my goal, and not sure what exactly I want and what theoretical areas I need for that and how deep I should delve into them.

Just keep doing it and enjoy the process.
Yes, even though there may not be such an algorithm for everyone (although I think there is, although maybe no one that I know of took the time to put it together), there is still a good chance there is at least an algorithm / "ladder" that I can use for myself.
In view of what you said I may have to bring "new strategies" to the table - for example to start asking about specific fragments / equations taken from papers such as the paper in the original post - for example "what does this fragment / equation say and in which book can I read more about it?". This will hopefully help me built at least my own "ladder", even though it may be a ladder good not for everyone else but just for me and /or a specific group of people with specific learning styles / preferences.
 
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Here is a fragment that I read in a paper presented in a related post:
"There is a well-defined Hilbert space, a well-defined
Hamiltonian constructed without any use of perturbation theory,
a well-defined unitary dynamics, well-defined bound states that
are eigenstates of the Hamiltonian, and everything is invariant under
the 2D Poincare group ISO(1,1)"
(taken from https://www.mat.univie.ac.at/~neum/physfaq/topics/different )

I do not know what exactly a Poincare group is, much less a 2D Poincare group, and what group classification is (e.g. "ISO(1,1)").
Could anyone recommend some good / rigorous mathematics books where I can study these?
 
  • #33
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Another question: "Why is the (mass)^2 term of the Higgs Boson negative in the Standard
Model Lagrangian to start with?"
(From https://www.physicsforums.com/threads/why-is-the-higgs-boson-tachyonic.115864/).
In which book can I read about the Standard Model Lagrangian (and also, very importantly, what exactly a Lagrangian is). I have seen it used before, including in my own undergraduate courses, but I need a book (probably also mathematics) that presents in detail, rigorously, what a Lagrangian is and so on. And, of course, following that, another book about the mathematical form of the Standard Model, how the Lagrangian is used in it and so on.
 
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I also wish there were some kind of explicit pathway (manifested as a "ladder" of books, that you mentioned) taking you (leading you by hand) from A to B (state-of-the-art understanding of the cutting edge theoretical physics including all the necessary math), giving you everything that's necessary in an order that makes it comprehensible.
There is. It's called an undergraduate education in physics followed by a graduate education in physics, and it takes a decade or so of full-time study. Unfortunately, if there were a shortcut, everybody would be taking it.
 
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There is. It's called an undergraduate education in physics followed by a graduate education in physics, and it takes a decade or so of full-time study.
In general, yes. Although in my case it would take more than just the physics program (I always disliked the way mathematics is presented in physics textbooks). Full-time study + experienced guides (teachers) is a powerfull combination. If that's an option.

Although I was under impression that TS was thinking of something along the lines of the famous Gerard 't Hooft web-site:
https://webspace.science.uu.nl/~hooft101/theorist.html
https://www.goodtheorist.science/

only more specific
 
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There is. It's called an undergraduate education in physics followed by a graduate education in physics, and it takes a decade or so of full-time study. Unfortunately, if there were a shortcut, everybody would be taking it.
Vanadium 50, I have already been through that (except for the PhD) and sadly I cannot agree with you. Basically now I have to revise even the basics or even replace the "basics" (i.e. the course materials I studied) with proper books / proper materials, because the programme I went through was focused on "getting the grades" rather than on conveying a real understanding of physics to students.
 
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An article by Sergei Winitzki explores somewhat the issue of teaching theoretical physics and the difficulties students encounter in trying to understand it:
Why is theoretical physics hard to learn no matter how much you already know?
This essay is an outgrowth of my ruminations on the problem of learning and teaching theoretical physics. I try to explain why it seems to be very hard to learn a recently discovered branch of physics, while it is comparatively easy to learn and to teach a long-established branch. Another issue is that, among students, even well-known areas of physics have a reputation of being mysterious and incomprehensible. It seems that one source of the problem is the widespread attitude that performing calculations and obtaining specific results is more important than achieving a logical and conceptual understanding of the material. I explore the interdependence of mathematics and physics which is inevitably involved in a learning process and show that this “principle of delayed understanding” has detrimental consequences. Another problem is the mixing of mathematical methods and their application in physics, which frequently confuses students. To illustrate these problems, I quote some examples from widely used physics textbooks (the reader who is not familiar with theoretical physics may skip the examples). I also comment on the basic differences between the contemporary practices of teaching mathematics and physics, in an attempt to propose a better way of presenting physics to students.
 
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Vanadium 50, I have already been through that (except for the PhD) and sadly I cannot agree with you. Basically now I have to revise even the basics or even replace the "basics" (i.e. the course materials I studied) with proper books / proper materials, because the programme I went through was focused on "getting the grades" rather than on conveying a real understanding of physics to students.
Here you go. This is a quotation from Tobias Osborne from this lecture on QFT. About 12 minutes into the video he says:

"What characterises a physicist, I think, is someone who is willing to step outside the comfort zone of rigorous maths."

 
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because the programme I went through was focused on "getting the grades" rather than on conveying a real understanding of physics to students.
The fact that your education was unsatisfactory - and I am sure you bear no portion of the responsibility for that - doesn't change the fact that learning physics at this level takes about a decade, and if there were a shortcut, everyone would take it.
 
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I see the discussion has deviated so I am re-posting these questions:
Can anyone please answer the following two questions regarding books / materials:

Another question: "Why is the (mass)^2 term of the Higgs Boson negative in the Standard
Model Lagrangian to start with?"
(From https://www.physicsforums.com/threads/why-is-the-higgs-boson-tachyonic.115864/).
In which book can I read about the Standard Model Lagrangian (and also, very importantly, what exactly a Lagrangian is). I have seen it used before, including in my own undergraduate courses, but I need a book (probably also mathematics) that presents in detail, rigorously, what a Lagrangian is and so on. And, of course, following that, another book about the mathematical form of the Standard Model, how the Lagrangian is used in it and so on.


Also, the questions below:

Here is a fragment that I read in a paper presented in a related post:
"There is a well-defined Hilbert space, a well-defined
Hamiltonian constructed without any use of perturbation theory,
a well-defined unitary dynamics, well-defined bound states that
are eigenstates of the Hamiltonian, and everything is invariant under
the 2D Poincare group ISO(1,1)"
(taken from https://www.mat.univie.ac.at/~neum/physfaq/topics/different )

I do not know what exactly a Poincare group is, much less a 2D Poincare group, and what group classification is (e.g. "ISO(1,1)").
Could anyone recommend some good / rigorous mathematics books where I can study these?
 
  • #41
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Here you go. This is a quotation from Tobias Osborne from this lecture on QFT. About 12 minutes into the video he says:

"What characterises a physicist, I think, is someone who is willing to step outside the comfort zone of rigorous maths."
This is a great quote. Take it together with Sadri Hassani's motto in the preface of his "Mathematical Physics" textbook: "If math is the language of Nature then physics is its poetry."
 
  • #42
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So a Higgs boson walks into a church. The priest said,"We don't want your kind in here." The Higgs boson replied,"But you can't have Mass without me."
 
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So a Higgs boson walks into a church. The priest said,"We don't want your kind in here." The Higgs boson replied,"But you can't have Mass without me."
I don't get it, why do they not want the God's particle in the church? o_O
 
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Because in reality it's the "god-damn particle". It got its "title" "god particle" only due to the hippocracy of the popular-science-book publishers who feared it might provoke an outcry, if they'd use Lederman's suggestion of this title for the book :oldbiggrin:.
 
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hippocracy
Rule by hippos? :wink:

Getting back to the question at hand, I doubt there is any book that answers both the question "what is a Lagrangian", "what is the SM Lagrangian" and "why is this particular term negative". Especially given all the constraints.
 
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It's negative to get a non-zero vacuum-expectation value for the Higgs field, which "Higgses" the local gauge symmetry from ##\mathrm{SU}(2)_{\text{wiso}} \times \mathrm{U}(1)_{\text{Y}}## to ##\mathrm{U}(1)_{\text{em}}##, such that you end up with only one massive Higgs boson and three massive W and Z bosons and one massless photon, which is exactly what's needed to describe the weak and em. interaction right.
 
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Talking about "understand in a really, really rigorous manner the Higgs field", what happened to the issues with quantum triviality of the Higgs field. I remember proofs about the triviality(in the mathematical sense) of the scalar field ##\phi⁴## in any dimension, how was this circunvented?
 

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