I’ve recently noticed that two very good new books on quantum field theory have become available, one aimed more at mathematicians, one purely for physicists.

### What Is a Quantum Field Theory?

Available online now from Cambridge University Press (actual printed books to come soon) is mathematician Michel Talagrand’s *What Is a Quantum Field Theory?*. While it’s subtitled “A First Introduction for Mathematicians” and definitely aimed more at mathematicians than physicists, it’s a wonderful resource for anyone who wants to understand exactly what a quantum field theory is.

Like many mathematicians, Talagrand tried to learn about quantum field theory first from physics textbooks, which tend to avoid any precise definition of even the basics of the subject. He soon found what was the best source for someone looking for more precision, Gerald Folland’s 2008 Quantum Field Theory: A Tourist Guide for Mathematicians. Folland’s book is extremely good, but also extremely terse. In 325 pages it covers more carefully the material of an old-style QFT book such as Schweber’s 900 page or so *An Introduction to Relativistic Quantum Field Theory* from 1961. Talagrand is covering much the same material, but with 742 pages to work with he is able (unlike Folland) to work out many topics in full detail, providing something previously unavailable anywhere else.

Both Folland and Talagrand have written books with much the same goal: to as precisely as possible explain the details of the renormalized perturbative expansion of QED. There is little overlap with the work of mathematical physicists who have aimed at rigorous non-perturbative constructions of quantum field theories. They are using canonical quantization methods and don’t overlap much with many of the more recent physics QFT textbooks, which are based on path integral quantization and aimed at getting to non-abelian gauge theories and non-perturbative techniques as quickly as possible.

When I was learning QFT not that long after the advent of the Standard Model, I had little patience for fat QFT books about perturbative QED and canonical methods. Why not just write down the path integral and start calculating? Over the years I’ve realized that things are not so simple, with canonical quantization and operator fields giving a perspective complementary to that of the path integral. Among the more modern books, volume 1 of Weinberg’s three-volume series is the one that best gives this different perspective, and is most closely related to what Talagrand is covering.

For mathematicians, Talagrand’s book is a great place to start. For physicists, Weinberg’s is an important perspective to get to know. If you’re reading Weinberg and want more detail about precisely what is going on, Talagrand’s new book would be a very good place to turn for help.

### Quantum Field Theory: An Integrated Approach

Over the years I’ve often consulted various parts of Eduardo Fradkin’s notes on quantum field theory on his web pages. On some basic topics I found these to give very clear explanations of things that were done in a confusing way elsewhere. After recently hearing that the notes are now a book from Princeton University Press, I ordered a copy, which recently arrived.

Fradkin’s book has not much overlap with the material in the Talagrand book described above, and is somewhat different than traditional high energy physics-oriented QFT books. It tries as much as possible to integrate the high energy physics point of view with that of condensed matter and statistical mechanics. Path integral methods are then fundamental. Unlike many other modern QFT textbooks that aim at getting to the details of perturbative Standard Model calculations, Fradkin is more oriented towards getting as quickly as possible to non-perturbative techniques and models of interest in statistical mechanics. He gives a good introduction to various of the modern non-perturbative QFT techniques that have been developed in recent decades, often motivated by the so far only partially successful attempt to come to terms with a strongly-interacting gauge theory like QCD.

While most of the book is quite good, the first few pages aren’t, and will immediately drive away mathematicians who might pick it up. The material in these pages about group theory uses bad terminology (for Fradkin, the “rank” of a Lie group is its dimension and the fundamental representation of SU(n) is the “spinor” representation) and sometimes is just completely wrong. On the second page of the first chapter after the introduction, he wants to explain why the Lorentz group is non-compact, in contrast to SO(3). To explain why SO(3) is compact he starts by mistakenly arguing that since it leaves the unit two-sphere invariant the points of SO(3) and of the unit two-sphere are in one-to-one correspondence, showing the volume of SO(3) is $4\pi^2$. This paragraph should be deleted in future editions of the book.

That this kind of thing can make it into a book like this is remarkable, but unfortunately relativistic QFT books and other sources (e.g. here) don’t always get right basic facts about the Lorentz and rotation groups. I once tried to do my part to remedy this, see here.

**Update**: John Collins has here an article that provides a careful discussion of scattering in QFT, starting with the basics, which could be thought of as part of a QFT book. This may be of interest to both physicists and mathematicians who want to see something less superficial than many text book discussions.

One of the things I liked about Folland is that he discusses the full blown representation theory of the Poincaré group on infinite dimensional Hilbert spaces, which allows you to obtain the “wave equations” of the different fields from that classification. An application of Mackey’s Imprimitivity theorem (it’s based on Varadarajan’s Geometry of Quantum Theory.) In the physicists’ treatment, often only finite dimensional representatoions of the Lie algebra are discussed, and one often hears things like “the photon doesn’t have a wave function”. It has! And one can even prove that it doesn’t admit a position operator, thus giving all the correct intuition. Of course, this is all for conceptual understanding and of little use for the hardcore calculations for real applications that physicists like. But I don’t know why physicists’ treatments have moved more and more to a purely pragmatic view in which such conceptual issues are just dismissed almost derogatorily.

As a stat mech-ish person who has been trying to gain a deeper understanding of the methods we borrowed from QFT (although I guess the borrowing has gone in both directions), the Fradkin book sounds intriguing. I’m currently slowly working my way through Blundell and Lancaster’s Quantum Field Theory for the Gifted Amateur. Any idea how it compares with Fradkin?

Alex,

Understanding representations of the Poincare group pretty much completely determines how relativistic free field theories work, and constrain very tightly interacting theories. Interest in this was more widespread during the 1960s, less so after that, partly because it’s to deal with such issues in terms of the path integral. There is a lot of this though behind what Weinberg does in his volume 1.

More recently I see a bit of revival of interest, from the directions of the amplitudes program.

Academic Lurker,

The Blundell/Lancaster book is better for QFT beginners, it does a great job of explaining carefully the basics. Fradkin is better on more advanced topics. His treatment of the stat-mech related stuff is truly integrated with the rest. Much of it is the same material as other modern QFT books that are HEP-oriented (harder to follow than Blundell/Lancaster since less detail, more background assumed) but in many places there is additional material related to stat-mech of a sort you don’t see in other books.

Coincidentally, I was looking for new resources on the topic of quantum fields for mathematicians to prepare for a class next fall (also to read something that is not on another depressing worldly news), and I found out about a new monograph by Albert Schwarz: “Mathematical Foundations of Quantum Field Theory” (2020).

I think this book is a very welcome one for the mathematically-inclined. While for quite understandably reasons most texts of this lot is oriented toward the topological, geometrical and group theoretical aspects, Schwarz’s returns the attention to the algebraic, functional-theoretical issues, which should perhaps receive some renewed attention from time to time.

PS: Perhaps most readers of this site should remember another title by the same author, on topological QFT.

Just wondering if anyone have any recommendations for books on QFT (and for that matter GR) which might be comprehensible to some one with a somewhat ancient PhD in physics who has no ambitions to actually work in the field. A step up from the popular stuff but giving a good conceptual overview rather than just leaping into calculations. Are Zee’s Nutshell books any good? They look interesting, but hellishly expensive, especially to get to NZ where I live, so I don’t want to just take a punt. I see he has another shorter one on QFT in the works.

Jackiw-Teitelboim,

I got a copy of the Schwarz book last years. It’s a nice very detailed treatment of the Fock-space formalism and concentrates on scattering theory. Mostly he is just writing about scalar field theory, and doesn’t get very far into the perturbative calculations and renormalization that Folland and Talagrand get into. So, in some sense a more detailed treatment of a subset of Folland/Talagrand. Very different than Fradkin or other modern QFT books (or his own earlier book on TQFT.

Andrew,

I’d recommend the Blundell/Lancaster QFT for the Gifted Amateur book mentioned earlier. It’s relatively elementary, and works things out completely and clearly. Zee’s QFT book I don’t think is a good place to start, since he’s covering a lot of material without going into details of calculations. So, hard for beginners to make much sense of. If you have gone through something like Blundell/Lancaster, Zee is then great for giving you a deeper conceptual understanding of the subject.

There seems to be a glut of books on QFT in recent years, another one is Quantum Field Theory: From Basics to Modern Topics by Gelis.

Andrew, I have used Student Friendly Quantum Field Theory by Klauber at times and I found it to be quite pedagogical.

Peter,

Thanks for the heads-up on the new books.

I think you’re being slightly unfair to Fradkin’s excellent book. The point you make is valid; similarly, one could find other minor points to complain about, e.g., Fradkin is often not careful with the horizontal placement of tensor indices. (Then again, Schwartz doesn’t care about vertical placement, either!).

However, any major work on such a complicated topic is bound to have minor flaws. The question that I think is important when deciding what to recommend to beginners is if the author exhibits mastery/insight. Fradkin certainly does, whereas the Blundell/Lancaster book cannot be placed in the same category; their text is riddled with conceptual errors (including extensive derivations where they’re forcing the answer to go their way). I’m sure you would spot tons of these if you looked for them.

To offer an alternative recommendation, I think Maggiore’s Modern Introduction to QFT is both beginner-friendly and authoritative. That being said, this is a blog, so you obviously don’t have to do a full bibliographic analysis every time you are asked for a book recommendation 🙂

Alex Gezerlis,

I don’t think I’m being unfair to Fradkin, in trying to make clear that the value of his book lies in the material about non-perturbative QFT and the integration with stat mech. On the standard basics of perturbative QFT, parts are quite good, but parts are not and are done much better elsewhere, especially anything having to do with symmetry arguments. The problem I pointed out with the discussion of SO(3) is not a typo, but getting something very simple and very basic wrong, and it’s not good to see something on the second page of the book that anyone reading it should have seen was a problem.

On QFT books in general, I don’t think there’s an ideal one. Even the best of them have their weaknesses. There’s a lot to be said for Fradkin’s, but others (including Blundell/Lancaster, whatever it gets wrong) are better for someone looking for an introduction to the basics.

Peter and others, thanks for the comments. Blundell & Lancaster looks just the thing for me personally.

I’ve always found lack of physical motivation and consistent terminology an obstacle to understanding. I attended a lecture course on QFT by John Polkinghorne as a graduate student, and he leapt straight into something called “second quantisation” and detailed calculations, confusing the hell out of me. It’s only recently that I saw Steven Weinberg saying that the term “second quantisation” was confusing and should be dropped, so I felt a bit better.

I hope it’s not too far OT to ask if there’s anything similar to Blundell & Lancaster but for GR? There’s a similar abundance of confusing terminology here, especially around covariant vectors, or is it dual vectors or one forms – the last one totally unmentioned as a concept when I was a lad. Again, we leapt straight into index gymnastics, and I just wanted to ask “Why on earth would I want to raise or lower an index? What does that actually mean??”

Anyway, I ramble. Thanks to our host for a most interesting site.

Dear Peter,

thanks for the overview of the interesting new book by M. Talagrand.

As regards quantum field theory books for a mathematical oriented audience, how is Your evaluation of introductory texts like:

E. de Faria, W. de Melo: Mathematical Aspects of Quantum Field Theory ,

J. Dimock: Quantum Mechanics and Quantum Field Theory ?

And of some more specialized texts such as:

R. Haag: Local Quantum Physics ,

J.Derezinski, C.Gerard: Mathematics of Quantization and Quantum Fields ?

Thanks in advance for any comment on the topic.

Paolo

I think it’s also worth mentioning the whole school of QFT in curved spacetimes, whose current torch carrier seems to be Klaus Fredenhagen et al. Not only they do have completely rigorous mathematical constructions, they even extended (and made mathematically rigorous) perturbation theory to curved spacetimes via microlocal analysis! (They use their version of the Epstein-Glaser method)

A short, state of the art review is (details are in the references provided there):

https://link.springer.com/book/10.1007/978-3-642-02780-2

I find quite illuminating to put QFT in a generic curved spacetime background because you realize that many things you thought were essential to QFT… well, they were not! I feel I finally understood some deep conceptuals doubts I had only when I studied QFT in curved spacetimes for the first time. One can start with more welcoming introductions, like Wald’s “QFT in curved spacetimes and BH thermodynamics”. I highly recommend to physicists with a more particle physics background to take a look at that point of view on QFT, too.

Paolo Bertozzini,

Those are all very different books. Some quick comments:

de Faria/de Melo: very short and very superficial. I don’t believe you can usefully cover the wide range of topics they cover in such a short book, unless you’re writing for people who already know the details of much of what you’re writing about.

Dimock: this is a really great book that I highly recommend. It’s from the canonical quantization point of view, and does everything rigorously, but very clearly. This is a different point of view than most modern physics textbooks, well worth understanding both by physicists and great for mathematicians trying to understand the subject.

Haag: This is an exposition of the operator algebra approach to QFT, which is quite different than either canonical quantization or the path integral. Little overlap with other QFT books. I haven’t personally found that very useful or understood it well (but recognize that often once I understand something like this better I appreciate it a lot more)

Derezinski/Gerard: This is a very specialized and detailed treatment of the mathematics of canonical quantization. I think of it more as a reference book for anyone getting deeply into that subject, especially mathematicians with a specialized interest.

And what about the handbooks of:

Iwo Bialynicki-Birula and Zofia Bialynicka-Birula, “Quantum Electrodynamics”

Lewis H. Ryder, “Quantum Field Theory” ?

Andrew,

An anonymous correspondent mentions (new edition about to come out)

https://www.amazon.com/Introducing-Einsteins-Relativity-Deeper-Understanding/dp/0198862032/

and the recently published Coleman notes

https://www.amazon.com/Sidney-Colemans-Lectures-Relativity-Griffiths/dp/1316511723/

Peter,

Re Haag: the operator algebra approach is actually a reformulation of canonical quantization (in the sense of going from the Poisson bracket to the operator bracket), it’s just done at the abstract algebra level to avoid problems related to the failure, in the field case, of the Stone-von Neumann theorem of the CCR. Even LQG is formulated this way today.

Furthermore, a LOT has happened in the field since Haag’s book. The review book I posted before covers all of the newer stuff.

It’s a bit too hardcore math phy, and it requires quite a solid math background (from operator algebras to Hörmander’s theory of wavefront sets for distributions, etc.) I certainly don’t claim myself to have a full understanding of all the required techniques, but it’s definitely worth it to at least give it a try.

Andrew wrote:

Second quantization is a real thing; for example if you take the Fock space of a Hilbert space you get another Hilbert space, so this is a process you can do twice – and this actually has applications in physics. But it’s often not explained very well, so it might be better to ignore it in an introduction to quantum field theory.

I wrote a light-hearted introduction to nth quantization here.

Andrew wrote: ”I attended a lecture course on QFT by John Polkinghorne as a graduate student, and he leapt straight into something called “second quantisation” and detailed calculations, confusing the hell out of me. It’s only recently that I saw Steven Weinberg saying that the term “second quantisation” was confusing and should be dropped, so I felt a bit better.”

Maybe you’ll like the uniform way first and second quantization is cast in my just accepted paper:

A. Neumaier,and A. Ghaani Farashahi, Introduction to coherent quantization,

Analysis and Mathematical Physics, to appear (2022).

https://arxiv.org/abs/1804.01400

I’d like to second (third?) the comments of John Baez and Arnold Neumaier about QFT and second quantization. It took me a while to understand this, but “second quantization” is a very precise and useful way to think about QFT from the canonical quantization point of view.

In brief, Hamiltonian mechanics says phase space is the initial data for an equation of motion, and functions on phase space are a Lie algebra (Lie bracket is Poisson bracket). For a linear phase space one can restrict to the subalgebra of linear functions. This is the Heisenberg Lie algebra, and quantization is just going to its (essentially unique) unitary representation.

If you start with the finite-dim phase space for a single particle theory, this “first quantization” gives you a representation on an infinite dim space of wavefunctions. One gets QFT by doing the same thing, but taking the infinite dim space of wavefunctions as your phase space, for which the term “second quantization” is not bad.

The details of this point of view are in my book

https://www.math.columbia.edu/~woit/QMbook/qmbook-latest.pdf

John Baez wrote “Second quantization is a real thing; for example …”.

Certainly, as John explained it is possible to define the concepts of second (and higher order) quantization. But for me, the physics issue is: What is the reason for using a quantization method?

The standard quantization method is for obtaining a (candidate) formulation of a quantum theory whose classical limit is expected to be some given classical theory. This is useful because before encountering quantum theory one knows about classical systems (including electromagnetic fields). As is indicated in Weinberg’s paper https://arxiv.org/abs/hep-th/9702027 that method is all that is necessary to formulate QFT, and the idea was known from the earliest days of quantum theory. (However he only references the Born-Heisenberg-Jordan paper Z. Phys. 35, 557 (1926). The QFT idea is more clearly visible in the earlier Born-Jordan paper, Z. Phys. 34, 858 (1925).)

So, in agreement with Weinberg, I find it unnecessary to formulate QFT in terms of second quantization. However, when teaching QFT, I find it useful to give an account of non-relativistic Schrödinger QFT. Then there is a nice correspondence between the Schrödinger equation for a single particle wave function and a QFT formulation of the corresponding many-body theory, that can be called second quantization. I find I have to emphasize that its status for theory construction is very different from “first quantization”, which I normally simply call “quantization”. First quantization gets you from an ordinary classical theory to a new type of theory, viz a quantum theory in the Heisenberg picture. But second quantization gets you from one quantum theory (in a Schrödinger formulation) to another quantum theory (in the Heisenberg formulation). I have to emphasize that QFT needs no change in the principles of quantum theory; I also immediately enhance the Schrödinger QFT to include a term for inter particle interactions, which give a non-linear equation for the field.

I find it counterproductive when people treat the formulation of relativistic QFTs as a matter of second quantization. It confuses learners into thinking that it involves going beyond the general principles of quantum theory. The real change is going from situations where the wave function formulation is often the most convenient (for non-relativistic situations) to a Heisenberg-picture formulation with heavy emphasis on its time-dependent operators.

On the topic of “second quantization” I agree with John Collins that the name might confuse some people, but it does reflect a central idea about how the multi-particle formalism works.

When you (first)-quantize the (d-dimensional) harmonic oscillator, with classical phase space $\mathbf R^{2d}$ that you have identified with $\mathbf C^d$, you get a quantum state space that can be identified with polynomials on $\mathbf C^d$, with degree of a monomial the number of quanta. In the Fock space approach to multi-particle systems you are doing exactly the same thing, with classical phase space now the single-particle quantum state space.

This is confusing if you are trying to understand quantum to classical, but it clarifies (at least for me) several things (why particles act like indistinguishable quanta, the relation of symmetries in the classical Hamiltonian formalism to symmetries in the quantum formalism).

Talagrand’s book is beautiful. It a pleasure to read in it. Explanations are clear and detailed. Every chapter has a short summary in a few lines that allows to check one’s progress. The style of writing is friendly to the reader. Content and form are beautiful. It’s a gentleman’s masterpiece.

Peter,

If not too far off topic I’d be interested in your opinion of Adam Marsh’s book “Mathematics for Physics” ? He gives major math definitions and results across what to me is an astoundingly wide range of areas in math, without proofs. Just results. Often very useful for physics, especially when reading a paper depending on one of those areas Marsh covers. I haven’t yet run across any serious error, but someone more involved in math might. If there are few it might deserve to be better known among physicists.

Jim

James Eshelman,

I haven’t seen the book before, just took a quick look. What I looked at seemed pretty accurate. From the table of contents it’s mostly a survey of geometry of use in physics (without discussing any of the physics).

The problem with this kind of thing is that if you cover so much material in 300 pages, there’s nowhere near enough detail for someone who is trying to learn a subject from the basics. And for people who already mostly know the subject, they likely have there own favorite places to use as reference. I suspect this book will mainly be useful to people who already know much or most of the content. Looking through it they’ll find some topics they didn’t know about, and will know enough to be able to follow those topics.

Peter or others what is your opinion of Matthew Schwartz’ book, “Quantum Field Theory and the Standard Model”?

Also, it seems that QFT courses quickly move through background material on Lagrangians and Fields and I haven’t found a book that covers those topics in great depth, is there a good mathematically rigorous treatment out there on those topics?

Interested Lurker,

I haven’t taught a standard sort of QFT class for physicists so haven’t looked closely at the books available. My impression from a quick look is that Srednicki, Peskin/Schroeder and Schwartz cover much the same ground, with different emphases. My main comment about all these books is that they give a deceptively simple idea of how QFT works, one oriented towards computing terms in a perturbative expansion, avoiding engaging with most of the structure you would like to have to really understand a QFT.

There probably are mathematically rigorous treatments of Lagrangian methods out there, but I don’t think that’s actually useful. Physicists writing about QFT now seem to uniformly start with the idea of a general Lagrangian system (for which a rigorous treatment is difficult), but then immediately specialize to very special systems and assuming good properties. They then quickly get into trouble when crucial examples don’t have these properties (e.g. EM fields with their gauge symmetry). To my mind Hamiltonian methods capture much more of the structure one wants, but developing QFT along those lines is a very different project.

“but then immediately specialize to very special systems and assuming good properties. They then quickly get into trouble when crucial examples don’t have these properties”

(e.g. EM fields with their gauge symmetry)

Can you elaborate here? Which properties do you mean?

lun,

In textbooks this shows up as the fact that the Legendre transform from velocity to momentum variables is only one-to-one in special circumstances. Even for free field theories this becomes a problem, typically first seen when you try to quantize the theory of a free photon, and find that the canonical momentum for the time-like component of the vector field is zero.

Peter and Interested Lurker,

I’ve taught QFT (to physicists) quite a number of times, and over the years I’ve become increasingly dissatisfied with the available textbooks, primarily concerning the foundations. (More advanced topics are another matter.) I’ve worked out ways to try to overcome some of the worst problems, and provided some short documents to the students; I plan to make them more public after I’ve re-checked them.

The big problems are about the treatment of scattering theory and the derivation of perturbation theory. Of the many books I’ve examined, only those by Srednicki and by Peskin and Schroeder get reasonably close to what I want to see. Sterman also has some very useful material that helped me. I find the treatments in these books very useful for inspiring better treatments.

It’s a long story to explain all the difficulties. But I see 3 main root causes:

First is the need to treat the infinite time limits in scattering in a physically (and mathematically) sensible way. Any treatment that doesn’t use wave packets is destined to give nonsense. Here, I recommend Peskin and Schroeder’s treatment, including a nice explanation of how one can relate cross sections to calculated amplitudes.

The second problem is the use of the interaction picture to derive perturbation theory. Haag’s theorem says that for a relativistic QFT in an infinite volume of space the interaction picture never exists. As far as I can see, this breaks all derivations that attempt to derive perturbation theory directly for the S-matrix.

The only way I know of that works is slightly indirect, but much better and general I think: First get perturbation theory for the time-ordered Green functions, and then get the S-matrix by the LSZ method. (For some of my approach, see the paper of mine that Peter referred to earlier: https://arxiv.org/abs/1904.10923.)

There’s still the problem of deriving the Gell-Mann-Low formula for Green functions without getting clobbered by Haag’s theorem. You can modify the Peskin-Schroeder derivation to do this: Initially put the theory in a spatial box, and only take the limit later at a suitable point.

Once you take all the necessary limits (and do UV renormalization) you get properly behaved results to the extent that perturbation theory can show them.

The final difficulty is that quantum fields are not operator-valued functions on space time but are operator-valued distributions. It’s easy to get into trouble if one doesn’t allow for this. Here several books can help: Among them are the book by Talagrand that we were discussing here and Peter’s own quantum mechanics book.

John Collins,

“The only way I know of that works is slightly indirect, but much better and general I think: First get perturbation theory for the time-ordered Green functions, and then get the S-matrix by the LSZ method.”

This is sufficient only for theories such as QED that have no bound states. Once bound states are present, even the standard asymptotic treatment is ill-defined. This is the main reason why QCD has unsettled infrared divergences showing up in a Landau pole at (in contrast to QED) physically accessible energies. See the discussion in https://www.physicsoverflow.org/32752/?show=32756#a32756

Arnold Neumaier appears to say that the LSZ method doesn’t apply when there are bound states, and that the the standard asymptotic treatment is then ill-defined.

This is simply not the case. You just use products of fields (not necessarily at the same space-time point) that have the quantum numbers to add or remove one of the bound state particles you are interested in. For example in QCD, if you want to find an S-matrix element or an operator matrix element involving protons, you would use a product of three fields for the up quark for an outgoing proton, and the hermitian conjugate field product for an incoming proton.

Then you apply the LSZ method unchanged.

That is, you examine Green functions that include these fields. In momentum space, corresponding to the bound state there is a pole in the external momentum of the field product. In coordinate space, there is the corresponding asymptotic large-time oscillatory behavior.

Essentially the same idea is used in lattice QCD to calculate matrix elements of operators between hadron states. The important difference is that instead of the oscillating asymptotic behavior in Minkowski coordinate space, one has exponential decay corresponding to the mass of the particle.

John Collins,

“For example in QCD, if you want to find an S-matrix element or an operator matrix element involving protons, you would use a product of three fields for the up quark for an outgoing proton, and the hermitian conjugate field product for an incoming proton.

Then you apply the LSZ method unchanged.”

It is not obvious that this recipe will give the correct perturbative expression. Please point to a paper where it is shown that this indeed yields the correct scattering amplitudes.

Arnold Neumaier:

There are (at least) three separate issues here: One is whether the LSZ method applies to bound states, for which the unambiguous answer is yes (e.g., K. Hepp “On the Connection between the LSZ and Wightman Quantum Field Theory”, Commun. Math. Phys 1, 95 (1965)). The second issue, which is the one you raise, is how to do practical calculations using perturbatively based methods. The third issue is what has been done with the LSZ method in non-perturbative situations.

It is true that in a fixed finite order of perturbation theory for a Green function you don’t see bound states. But you don’t have to restrict yourself to strict fixed order perturbation theory. There are known and successful methods for dealing with bound states in relativistic QFT that involve perturbative methods. You can get an introduction to these from Paul Hoyer’s writings, e.g., https://arxiv.org/abs/1605.01532 One issue he discusses is how bound state poles in amplitudes arise with those methods, even when the poles don’t exist in any finite order of perturbation theory.

One could ask Paul if he knows where these methods are applied to scattering involving bound states.

One other situation where the LSZ idea has been applied to bound states is in current algebra, for example for soft pion theorems. These results necessarily apply to QCD. A suitable current is used as an interpolating field for a pion, with the current having a non-zero matrix element between the vacuum and a state of a single pion. In the momentum space form of a Green function involving the current, there is a pole at the pion mass in the momentum corresponding to the current. The LSZ theorem tells you how to relate a Green function with poles to the corresponding S-matrix element.

John Collins,

I know that bound states are treated with perturbative methods after resummation, but all this is done (even in Paul Hoyer’s work) in a somewhat ad hoc fashion, and not based on first principles.

Your claim was that your 2019 paper provides a more principled derivation, and that it extends trivially to bound states. But it is not straightforward to extend your paper to the bound state case by your recipe

“You just use products of fields (not necessarily at the same space-time point) that have the quantum numbers to add or remove one of the bound state particles you are interested in. For example in QCD, if you want to find an S-matrix element or an operator matrix element involving protons, you would use a product of three fields for the up quark for an outgoing proton, and the hermitian conjugate field product for an incoming proton. Then you apply the LSZ method unchanged.”

The reason is that with this recipe, your free field (5) then becomes a multilocal field with three spacetime arguments, and the analogues of your Green’s functions (6) have three times too many spacetime arguments, too. This is quite different from what Haag/Ruelle/Hepp do! This mismatch affects everything you do later.

In Section VII you briefly allude to states of multiple particles and say that you do not need multiparticle wave functions. But this is an illusion caused by lack of attention to details. If you use 1-particle wave functions to do the smearing you do not get an in-space of bound protons isomorphic to $L^2(R^3,V)$ where $V$ accounts for the discrete indices (as it should be) but an in-space of three quark states isomorphic to a direct product of three $L^2(R^3,V)$. To remedy this you need to do the smearing with some sort of bound state multiparticle wave functions. But there is no simple way to define these.

In particular, when you “put the theory in a spatial box” you destroy any distinction between bound and unbound states, since these are asymptotic notions that need unbounded space to be meaningful.

In Section 4G you mention the need for normalizable (i.e., smeared) states, and appeal indirectly to the cluster decomposition property discussed in Volume 1 of Weinberg’s book. This property is, however, invalid for QCD because of confinement. One has cluster decomposition only for the bound states. Again it is essential to work with bound state multiparticle wave functions to ensure that everything is fine!

In the standard treatment of bound states via field operators one proceeds quite differently: One assumes the validity of 1. the perturbative formalism (ignoring bound states, leading to severe infrared singularies even after UV renormalization) and 2. some resummation techniques. Then one derives from it Schwinger-Dyson equations that, suitably truncated, produce approximate multiparticle wave functions of Bethe-Salpeter form.

In a first principle derivation, one would have to produce the Schwinger-Dyson formalism for bound states simultaneously with the perturbative structure. How this can be done in your setting seems to be a highly nontrivial challenge!

Arnold and I are going to continue our discussion off-line for a bit. We’ll report back when we come to what we hope will be agreement. There are non-trivial issues that we need to sort out at length.