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I am looking for any good places (preferably textbooks) to study about introductory non-perturbative phenomena in Quantum field theory.
Any suggestion will be appreciated.

knzhou
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TheQuantumMan
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4 Answers4

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The following books assume some acquaintance with perturbative quantum field theory. Together they cover a very wide spectrum of nonperturbative techniques for very different situations.

  • E. Calzetta and B. Hu. Nonequilibrium Quantum Field Theory. Cambridge Univ. Press (2008). A book on nonperturbative quantum field theory at finite time and finite temperature.

  • Y. Frishman and J. Sonnenschein. Non-Perturbative Field Theory. Cambridge Univ. Press (2010). A book on nonperturbative quantum field theory with emphasis on 2-dimensional exactly solvable models.

  • M. Shifman. Advanced Topics in Quantum Field Theory. Cambridge Univ. Press (2012). A book on nonperturbative quantum field theory with emphasis on supersymmetry.

Arnold Neumaier
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If you're interested in non-perturbative aspects of quantum field theory, you should study lattice quantum field theory. In fact, you'd be negligent not to: Lattice QFT is the only reasonably general way of defining non-perturbative quantum field theory, and frequently the only practical way of doing concrete non-perturbative computations. It's also pedagogically worthwhile: Because lattice QFT is oriented towards non-perturbative phenomena, its textbooks present QFT without the usual perturbative haze. You get a much cleaner view of what QFT is.

Decent textbooks include:

  • Smit, Introduction to Quantum Fields on a Lattice
  • Montvay, Muster, Quantum Fields on a Lattice

I also like the following notes:

You should also take some time to study Conformal Field Theory, which is usually concerned with phenomena inaccessible to perturbation theory. There're a number of standard textbooks & lectures notes. For brevity, I'll mention:

  • di Francesco et al, Conformal Field Theory (the bible of 2d CFT)
  • Rychkov, EPFL Lectures on Conformal Field Theory in D>= 3 Dimensions, https://arxiv.org/abs/1601.05000

Lastly, you should spend some time learning about topological aspects of non-perturbative QFT, meaning anomalies, instantons & such.

  • Coleman, Aspects of Symmetry, (the classic introduction, but not much on anomalies)
  • Polyakov, Gauge fields and strings, (eclectic and interesting)
  • Bilal, Lectures on Anomalies, (solid & modern)
user1504
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  • How are the topological aspects visible in lattice QFT, which you claim to be ''a much cleaner view of what QFT is''? It is rather a much poorer view of QFT, comparable to studying fluid mechanics primarily through finite element spaces and simulations. – Arnold Neumaier Jan 16 '18 at 18:37
  • I didn't say the topological aspects are easily visible in lattice QFT. As you know, it's not easy. I said that these three things (lattice QFT, CFT, and topological phenomena) are worth studying if you're interested in non-perturbative QFT. I put the lattice first because I think it's essential material, same as the finite-difference definition of the derivative. – user1504 Jan 16 '18 at 20:28
  • Well, having the derivative already defined in calculus, it is not necessary to do so again in quantum field theory. The lattice gives clean relations to quantum mechanics but is inferior in every other respect, not only the topological features: The lattice has only a finite symmetry group while relativity is governed by the Poincare group. Lattice calculations must use extrapolation to the limit to give results that match experiment. Thus it is only the continuum limit of lattice QFT that has a physical meaning. – Arnold Neumaier Jan 17 '18 at 08:47
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    @ArnoldNeumaier: The continuum QFT is a limit of lattice theories which are a easier to understand conceptually and mathematically. I think it makes perfect sense as suggested by AJ to study the lattice case perhaps first. Of course you are right that the group of symmetries is much bigger in the continuum. In fact this is one of the mathematical mysteries of QFT and CFT in particular and the object of a nice result in this sense, e.g., here: http://annals.math.princeton.edu/2015/181-3/p05 – Abdelmalek Abdesselam Jan 19 '18 at 14:59
  • @AbdelmalekAbdesselam: Even the example actions used to motivate the lattice QFT approach typically come from continuum fields by discretization. I do not want to imply that lattice QFT should not be studied, but certainly that they should not be studied first. In analysis (and classical physics) one also studies differential equations long before difference equations. Though conceptually less simple, the former are simply much more important. The circle is also a much nicer and much more important object than the regular polygons used to approximate it. – Arnold Neumaier Jan 19 '18 at 15:48
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    @Arnold: my point is similar to saying that to learn integrals one needs to have seen finite sums, since the first definition of integrals one sees is via Riemann sums. My point is more about having clear definitions and dealing with well defined quantities rather than exact computation. 2d critical Ising correlations on the lattice are difficult to compute whereas they have a simple formula in the continuum QFT limit. I agree that computing exactly solutions of difference equations are more ugly/difficult than for ODEs. However, the abstract existence/uniqueness is trivial for finite... – Abdelmalek Abdesselam Jan 19 '18 at 19:13
  • ...difference equations and require a proof, e.g., with convergence of Picard iterations in the ODE case. I would say one of the main sources of mathematical difficulty in QFT that people learning it encouter is due to trying to do things in the continuum directly with no discrete/lattice preparation. – Abdelmalek Abdesselam Jan 19 '18 at 19:14
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The OP did not explain what "nonperturbative" means, which can vary. So I will go with beyond perturbation theory, i.e., not just talking of correlations of a QFT as formal power series in $\hbar$ or the renormalized coupling constant. In that case, the literature on constructive quantum field theory deserves to be mentioned (although it might be too mathematical for OP's taste).

The classical reference is the book by Glimm and Jaffe "Quantum Physics: A Functional Integral Point of View". It starts from classical mechanics and statistical mechanics and goes through QM and finally QFT. Not an easy read but it is quite thorough and mathematically rigorous.

Another reference is the book "From Perturbative to Constructive Renormalization" by Rivasseau which is more technical and is a better source for topics like cluster expansions.

Abdelmalek Abdesselam
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For an essentially introductory, well illustrated (!) QFT book for the future, one might do worse than taking a look at Yannick Meurice's

  • Quantum Field Theory: A Quantum Computation Approach (April 2021) (IOP Expanding Physics, IOP Publishing Ltd) ISBN-13 : 978-0750321853

Rather than getting students up to speed for cranking out perturbative cross sections, it dares to bridge the gap to computational physics simulations in finite (small) Hilbert spaces, instead. Focuses on lattice simulation experiments for strongly interacting systems, expressly poised to utilize quantum computing--"an ineluctable modality of the visible (Joyce)", in the author's vision.

Cosmas Zachos
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