Is Ch. 6 of Birrell & Davies book on QFT in curved space and in particular the 1-loop effective action that they derive up-to-date with the current state of the art in (effective) quantum gravity?
I have no background in string theory and only in condensed-matter and I'm quite liking the approach from the book. Is there something important that I should be aware that appeared since the book was published?
Best Answer
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!
Exotic objects known as "negative branes" can be constructed (at least perturbatively) within string theory. Once they are introduced, many exotic aspects of timelike compactification, closed timelike curves or even "emergent timelike directions" can be studied. The relevant paper is Negative Branes, Supergroups and the Signature of Spacetime. For an overview see this talk.
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
Abstract:
We review the theory of quantum fields propagating in an arbitrary, classical,
globally hyperbolic spacetime. Our review emphasizes the conceptual
issues arising in the formulation of the theory and presents known results
in a mathematically precise way. Particular attention is paid to the distributional nature of quantum fields, to their local and covariant character, and to microlocal spectrum conditions satisfied by physically reasonable states.
We review the Unruh and Hawking effects for free fields, as well as the behavior
of free fields in deSitter spacetime and FLRW spacetimes with an
exponential phase of expansion. We review how nonlinear observables of
a free field, such as the stress-energy tensor, are defined, as well as timeordered-products. The “renormalization ambiguities” involved in the definition of time-ordered products are fully characterized. Interacting fields are then perturbatively constructed. Our main focus is on the theory of a scalar
field, but a brief discussion of gauge fields is included. We conclude with a
brief discussion of a possible approach towards a nonperturbative formulation
of quantum field theory in curved spacetime and some remarks on the
formulation of quantum gravity.
Best Answer
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!