''Perturbative Algebraic Quantum Field Theory. An Introduction for Mathematicians'' is the title of a book by
Kasia Rejzner. She is like me both a mathematician and a theoretical physicist, and works like me in a mathematics department. The book makes excellent reading for those who have some mathematical maturity and don't mind a fast pace without too much prior physical motivation.
She writes in the introduction,
As opposed to some other textbooks on the subject, I will not use the excuse that “physicists often do something that is not well defined”, so as mathematicians we don’t need to bother and just turn around for a while, until it’s over. Instead, I will jump straight into the lion’s den and will try to make mathematical sense of perturbative QFT all the way from the initial definition of the model to the interpretation of the results.
In the book, she features an introduction to classical mechanics in terms of modern differential geometry, the Haag-Kaster axioms for local quantum field theory in terms of nets of algebras, deformation quantization, Kaehler geometry and its quantization, Moeller operators and the S-matrix, the (Bogoliubov-)Epstein-Glaser axioms for the causal approach to quantum field theory, the renormalization group, and distribution splitting. Then she goes on to introduce classical gauge theories, their Batalin-Vilkovisky quantization, and she ends the book with an outlook on quantum field theory in curved space.
All in all, an introduction to the state of the art, featuring a rich variety of algebraic quantum physics. With its emphasis on the algebraically rigorous side of perturbative quantum field theory it complements other books on algebraic field theory that generally have a fully nonperturbative nature. Many of the latter are based on the Wightman axioms (figuring also in the Clay millennium problem on Yang-Mills quantum fields), about which the present book is silent.