I am taking a course this semester on QFT, which deals much with constructive quantum field theory. Some of its topics so far involve relationships between non-Gaussian probability measures,Feynman path integrals and so on. While I am totally new to the subject of QFT (I only have some basic ideas of classical mechanics, classical field theory and quantum mechanics in terms of physics, with knowledge in geometry, topology and analysis for a 2nd year graduate student), I found myself totally lost in the study of that course. On the other hand, my professor tends to talk about everything in a rather hand-waving way with neither strict definitions nor proofs, which made my struggling even worse. Now I have to come for help: are there any references on constructive quantum field theory that would save me out of these? I expect to see clear definitions and basic introductions so that I can enter this field without the trouble of getting myself familiar with the most fundamental stuff firstly. Thank you!
The standard reference for constructive QFT is the classic book by J. Glimm and A. Jaffe, Quantum Physics: a Functional Integral Point of View (2nd. ed., Springer-Verlag, 1988). It is certainly more than satisfactory from the viewpoint of mathematical rigor, it has a lot of background material (specially the second edition linked above) and parts of it can also be read by theoretical physicists with benefit, since it collects and derives many useful formulae. I have a friend who works on string theory and wanted to have this book badly for this reason.
Other books which deal with more restricted questions and / or methods in constructive QFT include:
R. Fernández, J. Fröhlich, A. D. Sokal, Random Walks, Critical Phenomena and Triviality in Quantum Field Theory (Springer-Verlag, 1992). It has an excellent discussion of renormalization group ideas and continuum limits. Its main goal are the famous triviality results in constructive QFT;
V. Rivasseau, From Perturbative to Constructive Renormalization (Princeton University Press, 1991). It discusses the transition mentioned in the title with more detail than Glimm-Jaffe, albeit in a slightly more informal way. It has a last chapter on the construction of finite-volume Yang-Mills theory in 4 dimensions, but it lacks the later results which comprise Rivasseau's landmark paper together with Magnen and Sénéor on this hard problem;
G. Battle, Wavelets and Renormalization (World Scientific, 1999). This book focuses on multiscale cluster expansion methods, just as Rivasseau's book above, but Battle's approach (developed together with P. Federbush) is based on certain families of wavelets he proposed together with P. G. Lemarié in the 80's;
V. Mastropietro, Non-Perturbative Renormalization (World Scientific, 2008).
See also this related MO question for more references.
$\begingroup$ Thank you so much! I just started to read that book. Hope I can really learn something. $\endgroup$– XuxuJan 22, 2014 at 6:34
Since you are asking your question in a math forum, this book comes to my mind
Gerald B. Folland, Quantum Field Theory: A Tourist Guide for Mathematicians, see http://www.ams.org/bookstore-getitem/item=surv-149
It is not about constructive QFT, though, and concerning definitions the review says "Rigorous definitions and arguments are presented as far as they are available, but the text proceeds on a more informal level when necessary, with due care in identifying the difficulties."
The Wikipedia article on CQFT doesn't say much, but it cites this survey article which has some promising-looking references: