# QFT axioms

1. Oct 22, 2008

### newbee

Hello Physics Forum

I will be taking a quantum field theory course next semester. I bought Mandl's book and Zee's
book and looked them a bit. I have also been talking to others that have taken the class in
previous semesters. I have a general idea of the failings of nonrelativistic quantum mechanics
and therefore the need for QFT.

Here's my comment and question. I have asked those that have taken the class previously to
summarize QFT for me. I encountered huge resistance to doing so. So in dismay I pointed out
to my fellow students of physcis that the fundamental principals of classical mechanics,
electrodynamics, equilibrium statistical, special relativity and nonrelativistic QM can each be
given concisely in a page or two. So what are the axioms or basic premises of QFT? OK, I
have heard online that it has not, so far, been possible to give such axioms but why? Can
somebody clear this up for me. Why does QFT appear to be such a morass?

2. Oct 22, 2008

### Haelfix

That depends what you mean by QFT axioms. In a sense, they are the same as the ones you learned from quantum mechanics and special relativity. The hard part is to disentangle the mess that ensues.

The best you will find, is probably chapter 1 and a bit of 2 of Zee's book. Thats more or less the whole thing --in a nutshell =). Unfortunately learning to calculate is 95% of the rest, and there exists no quicky summary.

If you want the more esoteric mathematical statements and formulations of QFT, well thats an entirely different, controversial and extremely complex can of worms that I strongly suggest you dont deal with for now.

Theres a lot of different ways you can come at the problem, perhaps its best if you identified the frameworks you are currently most comfortable with, and then go from there. So for instance if you already know parts of particle physics and scattering theory, you could peruse Griffiths 'Introduction to Elementary Particles'. Otoh if you are more comfortable with the pure Heisenberg commutator stuff, then go directly into Canonical quantization (see Zee). Again many paths...

3. Oct 22, 2008

### atyy

I'm a biologist who's tried to learn QFT on my own, so I'm not sure how applicable my experience is for real physicists. The most puzzling thing about most QFT books is that they "second-quantize" classical fields - which don't exist! Why would anyone think of second-quantizing them?

The best motivated approach for me was from condensed matter physics where the field description in terms of modes that can be created and destroyed is exactly equivalent to the usual quantum mechanics of particles whose numbers remain fixed. It was more helpful for me to think that the main problem solved by QFT was not relativity, but developing a quantum mechanics formalism consistent with particle creation. I liked Mattuck's "A guide to Feynman diagrams in the many body problem" which has really cute pictures, but also contains the full details of the dual descriptions in his later chapters.

The disadvantage of that approach is that in high energy physics, there are no underlying particles which are fixed in number, only the fields and their modes (the so-called elementary particles) exist. If you want to go straight to that, V.P Nair's book is the text that motivates it best for me. Aitchison and Hey (good for qualitative stuff, but I can't understand their calculations) and Kerson Huang are two books that go between those approaches. I also liked Veltman's "Facts and Mysteries in Elementary Particle Physics" because it has some description of the experimental apparatus.

A very logical approach which does second quantize classical fields is Warren Siegel's free text on arXiv (unfortunately one has to read every single word of 200 pages before understanding why it is logical, so it is lacking in motivation, but not in logic).

4. Oct 22, 2008

### Peter Morgan

The best book on the axiomatic approach to recommend is Rudolph Haag's "Local Quantum Physics", Springer, 1992, 2nd edition 1996. Don't expect easy mathematics, but it won't lead you far astray. I take it you're a graduate student, and that how far you go from the mainstream is on your own responsibility, but it's the uniqueness of how far astray you go, what directions you choose, and, crucially, how well you can tell people what you found that will make your research interesting or everyday.

The Wightman axioms (from the 50s) are largely uncontroversial, BUT the only known models of these axioms in Minkowski space are free fields. The mathematics of renormalization, even of the much-touted renormalization group, is not controlled enough for anyone to be able to prove that any renormalized interacting quantum field satisfies the Wightman axioms (in Minkowski space; in lower dimensions interacting quantum fields can be constructed, but not in ways that are understood well enough to prove that there are or are not models in Minkowski space).

You should recognize the stuff you see in your QFT classes when you look at the Wightman axioms. The more recent Haag-Kastler axioms (from the 60s), however, are considerably more abstract. This is where most of the mathematical work is done these days. Equally, however, only free field models are known in Minkowski space.

Both the Wightman and the Haag-Kastler axioms are decribed pretty well by Haag. Good luck.

5. Oct 22, 2008

### Haelfix

Yea a student probably doesn't deserve to be subjected to the Haag Kastler axioms and constructive field theory when he/she is first learning the subject. The material is confusing enough without worrying about test functions, compact support and operator valued distributions in their full glory.

Stick to physics level material for the time being (actually learning via condensed matter is not such a bad idea, as its actually easier and more intuitive, just remember that its sort of like particle physics done upside down)

6. Oct 22, 2008

### George Jones

Staff Emeritus
As an introduction, overview, and appetite whetter, you might want to take a look at chapter 24 Dirac's electron and antipartilcles, chapter 25 The standard model of particle physics, and chapter 26 Quantum field theory in Penrose's book The Road to Reality. See if your library has a copy. Penrose trained as a pure mathematician, so I was surprised, amused, and delighted to find that Penrose references Zee thirteen times in the end notes for these three chapters.

7. Oct 22, 2008

### Peter Morgan

Haelfix is right. In a few months, you have to learn QFT to the point that you can answer specific types of questions quickly. Getting involved in thinking about deep questions will get in the way. After almost 20 years of heterodox thinking, I possibly have some sense of how QFT works, etc., but I only just got my MSc in Particle Physics because I went off on tangents, and I was never able to enroll in a PhD program because I was clearly not a good bet for finishing more-or-less on time, I didn't share interests with any of the people I was interviewed by, and I couldn't pay them enough for them to ignore how off their map I was. I'm happy enough in this approach, but it can be an unhappy fate to be bitter that you couldn't just do what you wanted: you have to decide for yourself what risks you are willing to take, and what you think the rewards are.

8. Oct 22, 2008

### newbee

I really appreciate the advice that all of you have given - especially Peters. But I am getting here the same type of answers to my question that I have gotten from students of QFT. What I am looking for is a list of premises that fully define QFT. I am interested in taking a look at the axiomatic approaches that Peter mentioned but it sounds like there exist problems with axiomatizing QFT in some complete sense. So given that these problems exist with an axiomatic approach then in what sense are these problems dealt with in a non-axiomatic approach? Once again I am just looking for a core set of premises similar to the five axioms of QM (depending on how you count them) or Newtons 3 laws. I hope that I am not frustrating you with my insistence upon answering this question.

9. Oct 22, 2008

### atyy

Newton's 3 laws would not be 3 laws if you wanted to list all the possible forces (gravity, electromagnetism, friction, normal force, ....). The Heisenberg picture or the Feynman path integral picture of QM are in QFT exactly Newton's 3 laws. Most of QFT is devoted to finding the "forces", which I suppose our best attempt at axiomatising would be the standard model Lagrangian (a much more systematic catalogue of QFT forces than any available for Newtonian forces, unless you exclude friction, throw away Newton's laws and use only general relativity).

10. Oct 22, 2008

### Peter Morgan

It doesn't exist. You will hear Mathematical Physicists say that the renormalization group is a beautiful mathematical structure, and that it's good enough that you shouldn't worry about the infinities in interacting quantum field theory, but it's messy, messy, messy.

Haelfix will leap on me for pointing you to my approach, which I've most recently posted about on PF here. This comes with a much fiercer health warning than the Haag-Kastler axioms, and it is far from axiomatized. Anyone who tries to tell you that you can understand QFT better than the standard textbooks (which give or take a few details all tell more-or-less the same story), should be rejected as snake oil. That includes me. Everything interesting on QFT is at the level of an unproven, unconventional research discussion.

So, where else can you go? I suggest trying the two books on QFT by Tian Yu Cao, "Conceptual Foundations of Quantum Field Theory", an edited volume that contains many papers by very serious Physicists on their understandings of QFT; and "Conceptual Developments of 20th Century Field Theories", which is Cao's own monograph on field theories more generally, however the book's intention is to lay the ground for QFT, which he decides to understand in an effective field theory way.

To see how other Philosophers of Physics have followed the standard Physics story almost verbatim, you could try Sunny Auyang's "How is Quantum Field Theory Possible?" and Paul Teller's "An interpretative Introduction to Quantum Field Theory". Even though these will not lead to any big aha moments, they illustrate the standard Physics stories in enough of a different way from the standard textbooks that they will probably usefully enrich your knowledge of QFT.

Finally, I've been moderately impressed by two articles by Art Hobson recently, "Teaching Quantum Physics Without Paradoxes", The Physics Teacher 45, p96(2007), DOI: 10.1119/1.2432086, (and in an Am. J. Phys. paper he cites there), where he takes the view that ordinary QM should be taught by showing undergraduates how to think about experiments in terms of QFT. There are points where I part company with him, but he's managed an interesting read (also, it's deliberately accessible enough for undergraduates, so they should be relatively quick reads for you -- always a merit). Ultimately, I would argue that an axiomatic system has to make contact with experiments pragmatically, models in Physics are always idealizations and approximations, so I would back off the insistence on axioms a little.

Even more finally, I agree with Haelfix's comment that a condensed matter approach is a good idea. Indeed, I extoll it, for it was by asking what the differences are between quantum fields and field models for condensed matter at non-zero temperature, a dozen years ago and since then with increasing precision, that I have come to whatever understanding of QFT I may have. A textbook that I've always liked, only partly because it deliberately brings QFT and condensed matter together, is J.J.Binney, Dowrick, Fisher, and Newman, "The theory of critical phenomena: an introduction to the renormalization group", Oxford University Press, 1992.

If you read all this, you may have almost no hope of passing your course. Good luck again.

11. Oct 22, 2008

### atyy

Actually, what I like about the renormalization group approach is the view that QFT is only an effective theory, not a fundamental theory.

I liked the books by Cao and Teller you suggested. I cannot understand Auyang (not as a whole, but I understood lots of good points on the way). But one has to at least know how to calculate the results for one interacting scattering experiment and the Lamb shift to appreciate their work.

http://www.mth.kcl.ac.uk/~streater/lostcauses.html#IX [Broken]
http://www.mth.kcl.ac.uk/~streater/lostcauses.html#V [Broken]

Last edited by a moderator: May 3, 2017
12. Oct 22, 2008

### Peter Morgan

I agree, more-or-less, with that view of it.
I think Auyang's neo-Kantianism is a little too much. I did intend it more for the good points along the way.

Agreed also that one's understanding of any of these books is enhanced if one already knows how to do the calculations. Alternatively, however, I'd say that reading these or similar books before gaining facility can yield motivations for doing the calculations.
A math slightly too far for me, without a lot of effort, but Streater is always a good reminder of the realities. I guess he's famous for these pages, or at least I've been pointed there a number of times over the years.

Last edited by a moderator: May 3, 2017
13. Oct 22, 2008

### newbee

Thanks atyy. This is the type of response I was hoping I would receive.

So could you and others please elaborate upon how fields become part of the path integral approach?
Could you also elaborate upon your force analogy by telling me for which types of systems these
"forces" are presently known and possibly something about the form of these "forces"?

Thank you!

14. Oct 22, 2008

### atyy

Very roughly and heuristically, particles with definite momenta that are detected in scattering experiments are plane waves. So they must be excitations of an underlying field, and we postulate a field for each type of fundamental particle - a photon field (also called the electromagnetic field), an electron field, etc. As we know, plane waves bumping into each other don't interact (think of standing electromagnetic waves in a box), so we have to build some sort of coupling between these different fields. The tricky thing is to figure out the mathematical forms of these couplings so as to describe experimental data. The Lagrangians containing the right couplings are called the QED Lagrangian, the electroweak Lagrangian, the QCD Lagrangian, and are collected together as the Standard Model Lagrangian.

One unclear point (to me) is that after coupling fields, we no longer expect plane waves to be solutions of the equations, so why would free particles with definite momenta still exist after coupling the fields? Regardless, the experimental data indicate that they still exist to a truly excellent approximation.

15. Oct 22, 2008

### newbee

Thanks again atty

So the standard model Lagrangian is just the collection (a list) of Lagrangians?

16. Oct 22, 2008

### newbee

atty
Here's a possibly related comment and questions. Classically an electron is the "source" of an EM field. If the EM field has its own free Lagrangian and the electron field has its own free Lagrangian then does the coupling term in the Lagrangian of QED make clear what is meant by the term "source"? Or is the term "source" without meaning in QFT?

17. Oct 23, 2008

### newbee

Could it be that "free particles" must exist in an approximate sense only - a superposition of plane waves that satisfy the coupled field equations but behave rather free-like if the coupling is not too strong. What parameters do the couplings depend upon?