Literature on lattice quantum field theory

Question

Could you please list some books on lattice quantum field theory? Hardcopy books or electronic books. I'm interested in quantum field theory with only discrete impulses, with impulse cut-offs, both infra-red and ultra-violet, as well as with position cut-offs. So, in short, I'm interested in finite lattice QFT.

Answer 1

There are many textbooks or reviews representing a good introduction to Lattice QFTs. Here I list some of them I found very useful in my studies (sure I'm missing something).

• Lattice Gauge Theories by H.J.Rothe; This book provide a very good introduction to the argument, starting from the very beginning of the Path Integral formulation and then explaining how to discretize scalar and fermion fields and construct gauge theories on the lattice and use them with Monte-Carlo methods with Markov Chains (MCMC). Later in the chapters there is also the part on the finite temperature and density, described in details. Being focused on Gauge theories, the rest of the book touches QED but it is dedicated especially on LQCD and hence the study of non-perturbative strong phenomena (with many physical results and how to get them). However, if you want you can drop these chapters and concentrate on the more general about the general results provided about LQFTs.

• Quark, Gluons and Lattices by M.Creutz; This book represents a milestone in LQFT. It covers essentially the same arguments of the one before, but many times it allows to have a more in-depth view of some details about the argument. Again, later in the book it focuses on LQCD, but the general methods and theory is available in the other chapters. Notice also that this book is about 10 years older than the others in this list, so some arguments may not be updated in the methods or the results. I would recommend to use it at least as a complement to an other reading, because some parts are very interesting and clear.

• Quantum Fields on a Lattice by I. Montvay and G.Münster; As before, this book coves also the arguments presented above. Personally, I did not used this book expect to take a look to some details about something in order to have a slightly different point of view. However, I put it in here in the list because some colleagues of mine said that they found it a good read and found them better with it with respect to the other above. Consider it as a same-level alternative to the other.

• Quantum Chromodynamics on the Lattice by C.Gattringer and C.B.Lang; This book is specialized in LQCD, but the introduction chapters are more general. It is also an alternative to the others above, but if you are not interested only in LQCD you may miss something. Actually, I used it often later in my studies because I found it very useful and clear on concept you already get but you want to refine. If you are interest, in addition (and this is missing in other books) there is also a section about methods of analysis of the data obtained by lattice simulations.

• The Monte Carlo method in quantum field theory by C.Morningstar; This is not a textbook but a series of lectures about MC methods in LQFT. You didn't specified that your are interested in it but maybe it can be useful if you want to perform simulations. Here you can find more details about all the methods which sometimes are described only quickly on other resources. There are also a lot of examples which allows to get closer to the argument outside the textbooks.

I do not know if some of them have a free version on the web and in the list I just indicated the link to the publisher, but try to check. Let me also add that these are the most general textbooks on the argument. If you are interested in more detailed topics in LQFT, depending on your interests, probably you may find a lot of reviews (for example on arXiv) about them.

Answer 2

Disclaimer: I use lattice QFT for intuition, for proofs, and occasionaly for manual computation (e.g., strong-coupling expansion), but not for number-crunching computer calculations. In other words, I care more about mathematical clarity than I do about computational efficiency. That bias influences my recommendations.

By the way, even though the question was posted in 2018, I posted this answer much later (in 2021), so when I refer to other "recent" posts, I'm referring to 2021.

Montvay and Münster

Over the years, I've consulted several books on lattice QFT. The one I use most by far is this one:

• Montvay and Münster (1994), Quantum Fields on a Lattice, Cambridge University Press

I showed some extensive excerpts in my answer to another question. Those excerpts highlights one of the things I like about the book: it reviewes lots of results from lattice QFT. But the book also does a good job of introducing concepts and techniques of lattice QFT. It includes chapters about scalar fields (30 pages), gauge fields (70 pages), fermion fields (80 pages), QCD (80 pages), Higgs and Yukawa models (50 pages), and simulation algorithms (50 pages). Altogether, it's about 480 pages long, including a 40-page bibliography. The book's approach to defining the fermion path-integral measure is not as clear and concise as I would like, but otherwise it's pretty good.

Smit

Although I haven't been using it as often recently, this one is also pretty good:

• Jan Smit (2002), Introduction to Quantum Fields on a Lattice, Cambridge University Press

Compared to M&M, it's more concise (about 260 pages), it's slightly more recent, and the format/notation is easier to parse (in my opinion). Like M&M, it emphasizes concepts more than number-crunching techniques, which for my interests is a good thing.

Creutz (a review paper, not the book)

The book by Creutz is well-known, but here I want to recommend this 120-page review article by the same author:

• Cruetz (2011), Confinement, chiral symmetry, and the lattice (https://arxiv.org/abs/1103.3304)

Again, this recommendation reflects my own interests — in this case my interest in chiral anomalies, which you might have guessed from my username :). The review revolves around QCD, particlarly the roles of chiral (a)symmetry. The paper not a rant (it's inviting and expository), but it is apparently at least partly motivated by a desire to clarify some specific issues that were a little controversial for a while. It's a good source of insights about nonperturbative QCD that are different than the usual how-does-confinement-work theme.

Harlow and Ooguri

I've recommended this paper several times on this site, and I even just posted another resource-recommendation answer about it yesterday:

• Symmetries in Quantum Field Theory and Quantum Gravity by Harlow and Ooguri (https://arxiv.org/abs/1810.05338)

It's relevant in this context because it is one of the best introductions I've found to Hamiltonian lattice gauge theory, especially for discrete gauge groups. My other post already commented on the writing style, so here I'll just list the relevant sections:

• Section 3.2: Hamiltonian lattice gauge theory for general compact groups

• Appendix F: Hamiltonian for lattice gauge theory with discrete gauge group