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Now, I've seem some people saying that Birrell's approach is outdated , on the other hand, I've seem physicists on my department, researching the field with Birrell's approach.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