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I'm a graduate student in Physics, and on the graduate school I've already taken the QM and QFT courses. I've studied GR by myself. I also have a major in mathematics and mathematical physics, and besides the usual mathematics taught in these courses, I have a reasonable knowledge of differential geometry and theory of fiber bundles.

I'm interested on QFT on curved spacetimes, but I'm actually having a hard time to learn the subject, mainly because it seems to me that there are quite different approaches to the subject, and I don't know where I should start.

First, there's the traditional approach, based on different sets of functions spanning a space of solutions to the classical equations of motion. This is presented, for example, in Carroll's GR book and also in the textbook by Birrell. This approach seems much easier to get started with.

Another textbook following this seems to be Parker's one, but unfortunately I didn't feel good to start with it, because it seems to approach the subject by examples. I prefer treatments that first show the big picture and make a general treatment, and only after give examples.

Then there's the algebraic approach. As far as I know (and I might be wrong), this approach is more rigorous, it allows for a mathematically precise treatment of the free field, and it seems it also allows to deal with interactions via perturbation theory, but more rigorously also.

On the algebraic approach, I've seem different authors tackling the subject in different ways.

Now, I've seem some people saying that Birrell's approach is outdated , on the other hand, I've seem physicists on my department, researching the field with Birrell's approach.

The point is that for a beginner on the subject of QFT on curved spacetime, this is quite overwhelming.

So how can I learn QFT on curved spacetime by selfstudying given my background? Should I start with the traditional approach, then move to the algebraic one? I'm interested here, in how to learn it, as in what approach to pick, what sequence to follow, and what of the lots of resources out there, is better to get started with.

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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.

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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).

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