I am looking for an introductory textbook on QFT in curved space-time via the path integral method. I want to understand the following: How to build a generic perturbative QFT in curved space-time Are there some specific difficulties with normalization How to derive observables / particle states in curved space-time The Unruh effect The Hawking radiation I would appreciate if the author would use path integrals instead of the canonical formalism when possible. The reason for this is purely aesthetic. --- I would suggest one of the (standard) books (though somehow old) on QFT in curved spacetime, Quantum Fields in Curved Space (Birrel & Davies) Relates to path integral formalism, and covers a lot of topics in QFT on curved spacetime Quantum Field Theory in Curved Spacetime (Parker & Toms) Uses DeWitt notation, a lot based on effective action derived from some path integral, very good book treating also black holes [Quantum field theory in curved spacetime and black hole thermodynamics (R. Wald)] Also standard text book, but I feel it might be only the third choice for you If you want to pursue afterwards you can have a look at DeWitt's work (however you have to get used to his style, afterwards its great) Quantum field theory in curved spacetime (DeWitt) Summarizing I would say, for an answer to the first three issues Parker & Toms or DeWitt would be the best choice, however for a general introduction Birrel & Davies is quite nice and sufficient. --------------------------------- I propose that you study the following review article. It is of the algebraic flavor. You will afterwards understand the other approaches fairly easily. Hollands, Stefan, and Robert M. Wald, Quantum fields in curved spacetime, Physics Reports 574 (2015), 1-35. https://arxiv.org/abs/1401.2026 --- I would suggest taking a look at the book "Aspects of Quantum Field Theory in Curved Space-Time" by S.A. Fulling. I am also a physics graduate student who had a strong mathematical physics & applied math (functional analysis & PDE) background as an undergrad. I have had a lot of trouble learning curved spacetime QFT for many of the same reasons you mention above. I sometimes have a lot of trouble with the lack of rigor found in most physics texts, but also the lack of real-world examples in math textbooks. I came across Fulling's book a few weeks ago and it has been the perfect textbook for me to learn from. It goes into a lot of the mathematical details that we tend to gloss over in physics, but takes the time to explain them in context to physicists. Hopefully you will find the book as helpful as I have. In addition to the previously mentioned paper by Hollands & Wald (arXiv:1401.2026), I would also suggest taking a look at the set of lecture notes by Fewster (https://pure.york.ac.uk/portal/en/publications/lectures-on-quantum-field-theory-in-curved-spacetime(5fdd6964-e944-4b32-9b24-af4f49f409c6).html). --- Objectively speaking, the best book on QFT in curved spacetime is DeWitt's The Global Approach to Quantum Field Theory (2003). For one thing, it was written by one of the founding fathers of the subject. In this book you will find the most general and systematic formulation of an arbitrary Quantum Field Theory. The author uses functional methods from the outset, so everything is explicitly covariant. Furthermore, the spacetime manifold is left arbitrary (both its geometry and its topology). Similarly, the fields and their dynamics are also arbitrary: they can be either fermionic or bosonic, have any spin, and be gauge fields (corresponding to an arbitrary algebra, not necessarily closed or irreducible). In this sense, the formulation is as general as possible. In the book you will find a discussion of essentially every topic of QFT and, in particular, of quantum theories in a curved background (dynamical vacuum and its thermal properties, black-holes, etc.). You will also find a (somewhat idiosyncratic but still very informative) discussion of the quantisation of the gravitational field itself. The mathematics are very rigorous (up to physicists standards) and precise. Unfortunately, the non-trivial geometry of the manifold seems to preclude a straightforward implementation of the programme introduced by Epstein and Glaser, so one cannot proceed by a completely rigorous formulation. Therefore, the author anticipates (and finds) UV divergences, as is usual in introductory textbooks. Nevertheless, the analysis of divergences is as general as possible, so that the formulation is rather convincing anyway. If you want generality and completeness, you really can't do better than this book. A must-read indeed! For more mathematically oriented readers, I cannot help but recommend R. Brunetti, C. Dappiaggi, K. Fredenhagen & J. Yngvason's Advances in Algebraic Quantum Field Theory (2015) (with the collaboration of our very own V. Moretti!). In this book you will find a very thorough and up-to-date discussion of AQFT and its applications to, among others, quantum field theory in a curved background. Along the same lines, and as mentioned in the comments, Wald has dedicated several papers to the matter, so make sure to check them out. Finally, the Wikipedia page on QFT in curved spacetime contains a list of many good references that you should check out too. Good luck! --------------- Quantum field theory (QFT) in curved spacetime is nowadays a mature set of theories quite technically advanced from the mathematical point of view. There are several books and reviews one may profitably read depending on his/her own interests. I deal with this research area from a quite mathematical viewpoint, so my suggestions could reflect my attitude (or they are biased in favor of it). First of all, Birrell and Davies' book is the first attempt to present a complete account of the subject. However the approach is quite old both for ideas and for the the presented mathematical technology, you could have a look at some chapters without sticking to it. Parker and Toms' recent textbook should be put in the same level as the classic by Birrel Davis' book in scope, but more up to date. Another interesting book is Fulling's one ("Aspects of QFT in curved spacetime"). That book is more advanced and rigorous than BD's textbook from the theoretical viewpoint, but it deals with a considerably smaller variety of topics. The Physics Report by Kay and Wald on QFT in the presence of bifurcate Killing horizons is a further relevant step towards the modern (especially mathematical) formulation as it profitably takes advantage of the algebraic formulation and presents the first rigorous definition of Hadamard quasifree state. An account of the interplay of Euclidean and Lorentzian QFT in curved spacetime exploiting zeta-function and heat kernel technologies, with many applications can be found in a book I wrote with other authors ("Analytic Aspects of Quantum Fields" 2003) A more advanced approach of Lorentzian QFT in curved spacetime can be found in Wald's book on black hole thermodynamics and QFT in curved spacetime. Therein, the microlocal analysis technology is (briefly) mentioned for the first time. As the last reference I would like to suggest the PhD thesis of T. Hack http://arxiv.org/abs/arXiv:1008.1776 (I was one of the advisors together with K. Fredenhagen and R. Wald). Here, cosmological applications are discussed. ADDENDUM. I forgot to mention the very nice lecture notes by my colleague Chris Fewster! http://www.science.unitn.it/~moretti/Fewsternotes.pdf ADDENDUM2. There is now a quick introductory technical paper, by myself and I.Khavkine, on the algebraic formulation of QFT on curved spacetime: http://arxiv.org/abs/1412.5945 which in fact will be a chapter of a book by Springer.