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Prerequisites for Quantum Electrodynamics

  1. Jun 15, 2006 #1
    Although this is usually a graduate course, I want to know what all the preqrequisites for studying it seriously are. (I've been reading Feynman, Schweber and a bit of Griffiths which pumped my interest).

    I think mathematics should be calculus, group theory and all the stuff needed for QM. As for physics it looks like: CM (Lagrangian and Hamiltonian particularly), QM, some SM, classical electrodynamics. Quantum Field Theory?

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  3. Jun 15, 2006 #2

    George Jones

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    Quantum Electrodynamics is an example of a quantum field theory.

    All the stuff you list could be required as prerequisites, Paul Halmos once wrote something like "The beginner should not be discouraged if he finds that he does not have the prerequisites for the prerequisites." at the beginning of one of his advanced math texts.

    There is a new quantum field theory book that is at about the senior undergraduate level. It has a number of solved problems as examples, and, at the back, it gives short solutions (not just answers) to many of the exercises. This should make it good for self-study.

    I have seen a more detailed Table of Contents for it somewhere on the web, but, unfortunately, I wasn't able to locate it for this post.
  4. Jun 15, 2006 #3
    Hey George,

    Thanks for your encouraging reply. Frankly speaking I did not expect such an encouraging response considering the fact that QFT is taught to senior undergraduates or graduate students in most colleges across the world and most people wait to learn it the step-by-step way (not that I disagree with that outlook).

    Interestingly I have read a bit of the book "Relativity, Gravitation and Cosmology" by Tai-Pei Cheng which is in the same series as the book you have recommended. But I have not been able to locate this QFT book so far.

  5. Jun 15, 2006 #4
    I have actually heard very good things about this book. My advisor likes to use it as a reference when he teaches his QFT course- personally I prefer the book by peskin and schroeder for learning QFT. It has an excellent section on QED. But, it is probably a little more rigorous mathematically than someone outside of graduate school is going to want. Who knows though.

  6. Jun 15, 2006 #5
    But can you list all the mathematics and physics prerequisites which are absolutely must? I have seen neither book yet and I don't think I can get hold of either anytime soon...

    I am guessing that Group Theory, Topology, Real Analysis, Linear Algebra, Differential Geometry are some more mathematical prerequisites? (I don't know much...just gathering data randomly).
    Last edited: Jun 15, 2006
  7. Jun 15, 2006 #6

    George Jones

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    Well, I wouldn't say that the mathematics in Peskin and Schroeder is rigorous, but then there (so far) is no mathematically rigorous formulation of quantum field theory, so this is understandable.

    At this level, I prefer Quantum Field Theory for Mathematicians by Robin Ticciti. This is an excellent quantum field theory book, and, in spite of its title, is not a tome on axiomatic quantum field theory, or a book that emphasizes mathematical rigour. Its presentation is, however, a little less fuzzy than presentations in many other books. It shows how to calculate physical things like cross sections, and is a serious competitor for standard works like Peskin and Schroeder. I wish this book had been available when I was a grad student!

    If you have the time, I am very curious what you (or anybody else) think about this book.

    I suspect that many North American grad students that study Peskin and Schroeder have not taken mathematics courses in topology, real analysis, and differential geometry. All of these courses are important for a rigorous mathematical treatment of Lie groups, but, often, the "physicists' version" of Lie groups and Lie algebras is sufficient for undertsanding the roles Lie groups play in quantum field theory.

    More important is an understanding of Lagrangian and Hamiltonian dynamic, includings, possibly, a Lagrangian treatment of the elctromagnetic field.
  8. Jun 15, 2006 #7


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    If we are talking about the minimum to understand QED, here's my two cents.

    If you are at ease with quantum mechanics, then the only additional physics concepts that are required are
    - Lagrangian mechanics, especially applied to fields (as George mentioned, especially the example of E&M fields).
    - and a good grasp of special relativity (be at ease with four-vector notation, the concepts of spacetime intervals, relativistic dynamics (four-momentum and so on)).

    As for maths, you only need to be at ease doing calculus in the complex plane (more precisely, doing integrals...so using the residue theorem, working with branch cuts, etc).

    This is all you really need. You don't need group theory if you just do QED (it *is* associated to a U(1) symmetry but that's such a simple group that you don't need full-blown group theory). You don't need GR or differential geometry (unless you want to do QED in curved spacetime!).Unless you would do finite temperature field theory, you don't need thermodynamics.

    I think that Griffiths' book is an excellent first step.

    Just my two cents.


    EDIT: to address one of your posts, to do basic QED (i.e. not stuff in curved space-time or getting QED from string theory and that type of stuff) you don't need topology or differential geometry. And you don't need complex analysis, real analysis and linear algebra at deeper levels than what is needed in quantum mechanics. Just focus on intgegration in the complex plane, special relativity and lagrangian dynamics (especially applied to fields)
    Last edited: Jun 15, 2006
  9. Jun 15, 2006 #8

    George Jones

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    Patrick, have you ever looked at either (Maggiore and Ticiatti) of the two very different QFT books that I mentioned. If so, what do you think? I very much like both of them, but that's just my opinion.

    What are your favourite QFT books (at all levels).
  10. Jun 15, 2006 #9
    I would second Patrick's statements about what math is needed to begin in basic QED. The 4-vector notation can usually be picked up pretty quickly and is typically atleast introduced quickly in a QFT book.
    Good luck,
  11. Jun 15, 2006 #10
    Though there isn't much more I can add to the list of prerequisites than has already been given, I will say that teaching yourself QFT isn't as daunting as it may sound. I'm nearly 17 and got Peskin and Schroeder for Christmas. I'd been reading around various aspects of under-grad and graduate QM, electrodynamics and SR for a while and I thought it would be a good move to focus on something that unifies these approaches (in addition to Lagrangian/Hamiltonian mechanics/field theory). If you have enough knowledge about these and the maths that's used in them you can put them together and learn QFT, and QED is a pretty straight forward field theory, you can do a fair amount of QED with just relativistic quantum mechanics ("Quark and Leptons" by Halzen and Martin does this).

    I will say that if you're just interested in QED as a field theory and are also looking for a more comprehensive QM text, Leslie Ballentine's is a good 'un.

    Group theory would be pretty superfluous for QED, it's a pretty simple gauge theory (U(1), as nrged said). If you really wanted to know anything about groups, a few definitions would suffice, you don't need to go into further aspects of group theory to use with QED. You needn't go into performing rigorous analyses of the geometry you're using that one finds in topology and differential geometry as introductory QED texts only deal with pretty trivial geometries (namely Minkowski space-time). Some familiarity with differential geometry though would be useful just to give you a bit more insight into the subtleties of the theory, such as Lie algebras and their relations to gauge theories, the geometry of gauge bundles etc. A very good text for introducing differential geometry to the physicist is Bernard Schutz's "Geometrical Methods of Mathematical Physics", though to be honest it is lacking a little bit in the way of general bundle structures and actual Riemannian geometry, it spends a lot more time studying Lie algebras/groups, differential forms etc. Maybe someone else could recommend a more comprehensive treatment.

    I think probably the only bit of maths that is of importance to QFT and that isn't hugely employed in under-grad QM is complex analysis, and the most relevant part of this is just complex contour interation.
    Last edited: Jun 16, 2006
  12. Jun 17, 2006 #11


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    No, I had never heard of neither. But I am always looking for new books on QFT (hoping that the presentation will be original and significantly different from previous books so that I will feel like I am learning something new).

    I am so interested in low level presentations on QFT (because there are some basic things that I still don't understand) that I have actually ordered Maggiore a few minutes ago! Thanks for mentioning it!
    So I will let you know when I get the book and have a few days to skim through it. I will look for Ticiatti as well. It sounds like it would be a worthy addition to my library!

    My problem with QFT books is that I personally feel that there is too little emphasis on the conceptual basis (on the *meaning* of things). That instead, most book jump as quickly as possible to technical stuff.
    I find that when one stops and starts wondering about the meaning of a lot of expressions encountered in QFT, it is not clear at all what they represent (in contrast with QM where the meaning of any given expression and its relation to possible measurements is always pretty clear).
    So far, I haven't encountered any book that I find satisfying.

    (as just a couple of examples...P&S say that <0|Phi(y) Phi(x)|0> (where Phi is a scalar field) is the amplitude for the propagation of a particle from x to y. And then they show that it does not vanish for spacelike events but then they add that it does not matter! Then, they say that one should show that measurements done at spacelike intervals should not affect each other and they proceed to calculate <0| [Phi(y),Phi(x)]|0> and show that this vanishes for spacelike intervals. But this seems to suggest that Phi should be considered an observable when it is not!
    There are tons of things like this which are glossed over but deserve to be explained very clearly, imho.

    I must say that I am flattered that you might wonder about my opinion!
    I could write several thousand words on this! But I will keep it short. The problem is that it depends a lot on one's background an dthat varies a lot even for a given person (obviously). So some books I like now would have been useless when I was starting. And some books I dislike now may start to make sense in a few months.

    But A brief list:

    For introductory books, I love Griffiths. I also love Aitchison and Hey (Gauge Theories in Particle Physics). I love Greiner's books on QCD and the electroweak theory so I should try to get his books on QM and on QFT and QED.

    I like P&S but the conceptual explanations have left me disappointed. I find myself wishing for a book like this but written by Griffiths (someone who stops often to ask "but what does this really mean?" and "what are we trying to do here?" and so on. Someone who would explain the *idead* behind the calculations and relate them clearly to physics (is this thing an obervable? what does this state represent? What is the meaning of this amplitude? If we square it, we get a probablility which represents what? and so on). I should really get the Greiner book.

    At a more advanced level, I love Hatfield (QFT ofpoint particles and strings, if I recall) because he does discuss the ideas more extensively than the vast majority of books (but not as clearly as I would like..still, it stands out in terms of discussion of the ideas and clarity of the presentation).

    I like the Landau and Lif****z on QED.

    I like Zee for some tidbits that are really neat but the book is unsatisfying. I think it's because when he covers stuff that one is interested in, the presentation is always too short to really learn anything. So one always feels "left on one's appetite" (as we say in French)

    I also like Gross (it's not David Gross..I don't remember the exact title).

    And Weinberg (vol 1).

    I used to dislike Itzykson and Zuber but with a more solid background, I go back and appreciate it more and more. I never liked Kaku, however.

    I know that I have more QFT books but I don't have them with me here so I am surely forgetting a bunch of them.

    What about you? Any comments on the above titles? Any other titles not mentioned?
    I am looking forward to receiving Maggiore.

    Best regards

    EDIT: I forgot Ryder!! I like it as a complement (a bit like Zee but at a lower level).
    Last edited: Jun 17, 2006
  13. Jun 17, 2006 #12


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    Perhaps these are useful.

    Relativity, Gravitation, and Cosmology: A Basic Introduction (Oxford Master Series in Physics) Ta-Pei Cheng (Paperback)

    Table of Contents
    Relativity - metric description of spacetime
    1. Introduction and overview
    2. Special relativity and the flat spacetime
    3. The principle of equivalence
    4. Metric description of a curved space
    5. General relativity (GR) as a geometric theory of gravity - I
    6. Spacetime outside a spherical star
    7. The homogenous and isotropic universe
    8. The expanding universe and thermal relics
    9. Inflation and the accelerating universe
    Relativity - full tensor formulation
    10. Tensors in special relativity
    11. Tensors in general relativity
    12. GR as a geometric theory of gravity - II
    13. Linearized theory and gravitational waves

    A Modern Introduction to Quantum Field Theory (Oxford Master Series in Physics)
    From the site provided by George Jones -
    Table of Contents - http://www.oup.co.uk/pdf/0-19-852073-5.pdf

    Oxford Master Series in Physics

    Quantum Field Theory for Mathematicians Robin Ticciati

    Bibliography in -
    Quantum Field Theory for Mathematicians:
    Background and History
  14. Jun 17, 2006 #13
    Thanks Astronuc. The first 3 chapters of this book are related to our first year course that has Special Relativity, so I am reading this book. As for the others, I have not seen them yet.
  15. Jun 18, 2006 #14

    George Jones

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    My favourite pedagogical treatments of relativity are A Traveler's Guide to Spacetime by Thomas Moore and Gravity: An Introduction to Einstein's General Relativity by James Hartle. I can't say enough good things about these books.

    Back to QED. In post #6, I said "there (so far) is no mathematically rigorous formulation of quantum field theory." If you are reading Schweber's QED and the Men Who Made it, look at section 9.17 to see what I mean. A few more details are given in post #17 (second paragraph), #19, and #24 in this thread.
  16. Jun 18, 2006 #15

    George Jones

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    Me too!

    I am not sure that Maggiore will show you anything new, but I hope you like it.

    I particularly like Ticciati's treatment of symmetries in chapters 3, 6, 14, 15, and 16. He take a little more mathematical care than does a typical QFT book, but I don't think that he takes so much care that physicists will be completely turned off.

    Again, I have similar feelings. I was quite sure that you felt this way, so this is one reason that I am interested in your opinions on QFT books.

    I don't really know enough to comment, but maybe this is relevant. If A and B are self-adjoint operators, then [A , B] =0 ==> [f(A) , f(B)] = 0 for nice functions f and g. Maybe P&S want to apply this to (not necessarily self-adjoint) field operators, so that measurements of obervables built from field observables (via the functions f and g) are guaranteed not affect each other.

    So do I, and I like Halzen and Martin.

    I don't have this, but I am thinking of ordering the new 2-volume edition of Aitchison and Hey. Friday afternoon I downloaded a copy of Aitchison's long pedagogical article "Nothing's plenty: The vacuum in modern quantum field theory," Contemporary Physics, 26(4), 1985. I haven't had a chance to take a look at it yet.

    I have his book Field Quantization. Lots of details.

    If you find such a book, let me know!

    Heard of it - have never looked at it.

    This was used for my graduate quantum mechanics course, and, then, I didn't like it much. Maybe if I take a look at it now, I'll like it more.

    Yes. I find it interesting that the mathematician Roger Penrose in his Road to Reality (saw in another thread that you read and liked this brilliant book) referred to Zee's book more than a dozen times.

    I have seen it. Another book that I should have another look at if I get a chance.

    Well, one can't mention QFT books without mentioning Weinberg. I like that, near the beginning, he talks about infinite-dimensional representations of the Poincare group.

    In grad school, I had a friend who swore by this book, but I have never really looked at it.

    I have Kaku, and I, too, dislike it. However, I know a number of people who really like it, e.g., my supervisor.

    Supposedly, Ryder was the text for my graduate field theory courses, but the prof never made use of it. In fact, in the second semester, he brought photocopies of a few pages from Raymond to each lecture, and transcribed these verbatim onto the board, all the while not letting on what he was doing. Raymond wasn't even on the readings list that he distributed to the class!

    I sincerely hope that the 4-vector that connects the now event on my worldline to the event of me purchasing my favourite QFT book is future-directed. I don't much like the thought that this 4-vector is actually past-directed!
  17. Jun 19, 2006 #16
  18. Jun 19, 2006 #17


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    My favourite with my rating:

    BOGOLIUBOV & SHIRKOV (************** more stars might do justice)

    ITZYKSON & ZUBER (******)

    WEINBERG (V1) (*****)

    HATFIELD (*****)



  19. Jun 20, 2006 #18

    George Jones

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    Is that the green one? Or is it purplish?
  20. Jun 20, 2006 #19


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    I would then highly recommmend buying Hatfield's book, George. The presentation is fairly original and quite thorough. And he makes some effort to discuss things at a conceptual level. He repeats several calculations three times, using canonical quantization, path integral quantization and using a wave functional approach ( I think that when a new formalism is introduced, it is very pedagogical to rederive results already known as a mean to gain familiarity with the new formalism. Unfortunately, books (and teachers) rarely do that. Hatfield is good about repeating a given calculation using different formalisms. And the string theory part is also very good!)

    That's ok. What I want is to see fresh presentations and explanations in order to get a deeper understanding. I don't mind if I don't learn anything new in terms of actual formalism or calculations.

    By the way, I have ordered Greiner also (Field quantization) and will let you know how I find it.

    That sounds very interesting. It's unfortunate that it is so expensive :-(
    That is exactly right. Thanks for reinforcing my interpretation. I have realized that this is the key point. Since observables are constructed out of the fields, the vanishing of their commutator at spacelike separation implies causality of any measurement, even though the Phi's themselves are not observables.

    But you see what I mean...to *me* this specific point (which requires only a few sentences to address) is *key* in understanding the whole issue of causality. And yet it is not explained at all in P&S! (which is meant to be fairly pedagogical). This is what annoys me about QFT books. It feels like the physical interpretation is neglected very badly. Books end up manipulating operators and states and other expressions for pages and pages without ever stopping to discuss what they represent (are these operators observables? is this quantity an amplitude for finding a particle at one position? Is that a measurable quantity? And on and on)

    I do too, I did not mention it because I did not get into Particle Physics books (Granted, Griffiths is really a particle physics book, not a QFT book. But I think undergrads should be introduced to the applications of QFT and to some Feynman diagram calculations before plunging into QFT)

    I have to check this, I did not know they had a two volume edition! (what I have is quite old, maybe there is a lot of new stuff that I would enjoy reading).

    I think Gauge Theories of Particle Physics is *excellent*. It *is* very basic though, but what is covered is presented in a very pedagogical way.
    That's the only book where I have seen a discussion of "old fashioned perturbation theory" (also called "time-ordered perturbation theory") which shows clearly the connection between the language of covariant perturbation theory of Feynman diagrams with the perturbation theory of quantum mechanics (the latter involving several time-orderd diagrams - the so-called "Z" graphs - with particles propagating backward in time which are recovered by doing the sum over the poles of the covariant expression.)
    Is it available for free?

    Btw, I *LOVE* Penrose's book! It is so refreshing to see such an incredible range of concepts discussed almost completely at the conceptual level. It is a brilliant book!! I don't know how many times I have read something and felt like a light bulb just lit up in my head!!

    I need to go back and look at it in more details. I felt that it was getting a bit too heavy at some point and I needed a break.
    I liked that he used as a starting point the need to have a theory allowing for a changing number of particles and that this *leads* to the idea of a quantum field. As opposed to everyone else who start by saying "we must now consider classical fields and quantize them" without explaining where the heck this comes from!! I always disliked that. What is a classical Dirac field anyway? It is totally weird to me that one would find natural to introduce the idea of QFT by saying that we need to quantize all those classical fields that have no observational basis. Even if the Higgs exists, what the heck would be the classical Higgs field?

    yes?! That surprises me.

    I had forgotten about Raymond. I bought it and tried to read it when I was just beginning to learn QFT. And I disliked it. Maybe now I would appreciate it more. I should dig it out of my library.


    Well, maybe *you* should write it!!

    I wish there was a book on QFT at the level of Wheeler and Taylor's book
    "The physics of Black holes". Something intermediary between the little QFT of Griffiths and P&S.

    Thank you for your comments, they are much appreciated.


  21. Jun 21, 2006 #20
    I bought the Maggiore book one week ago and like it very much. As many others I find QFT extremely difficult to learn and kept searching for a good introductory text. I think this one good be it. It is extremely well structured, very compact, has solved problems and feels modern.

    Hi Pertubation

    What!? You are sixteen and reading and understanding Peskin/ Schroeder? That makes me extremely jealous. What are you doing when you are twenty-three? I'm jealous.

    Hi nrqed

    I remember you are learning python, here is a link to a book site of the coolest book ever written (http://www.physics.cornell.edu/sethna/StatMech/). It is about statistical mechanics and happens to be from the same Oxford series that also the Maggiore book is from. But the book is not only exciting, thrilling and lots of fun, the author is also a nice person, so that he puts the book free on the net and programs for its computer models with codes in phyton.
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