The following assumes you are comfortable with Linear Algebra, Calculus, and Basic Physics (Newtonian Mechanics and Thermodynamics) at an early undergraduate level.
Start with the Lagrangian and Hamiltonian formations of classical mechanics (Landau & Lifshitz - Mechanics). Then, study Quantum Mechanics (R Shankar - Quantum Mechanics). Statistical Physics while not key here, will be useful for building up an intuition (Landau & Lifshitz - Statistical Physics is good). Classical Field Theory, i.e. Electromagnetism should now be learned (J.D. Jackson).
Then, I'd recommend learning Calculus on Manifolds (Spivak) and Differential Geometry (I can't remember a good textbook on this right now).
If you're at this point you will now have a solid understanding of the maths needed to know GR and QFT. Zee has two good pedagogic books for these "Einstein Gravity in a Nutshell" and "QFT in a Nutshell". Once you know QFT, read a textbook on the standard model.
That'll get you most of the way there. Strictly speaking, you could do possibly learn a lot less to understand this paper, but to be able to pick and choose which parts of the above are actually needed would require a significant amount of work in itself.
For the uninitiated, Jackson's EM book is considered the book on E&M. There really is no other book to learn from, if you want to learn EM correctly.
It is also horrifically difficult even for dedicated PhD students at top 50 physics programs.
Story time:
As an undergrad I had a TA that was in the astrophysics program. He told us a story of a final he had to take in his EM class, using Jackson, of course. The final was to present his solution, if any, to the PI. Pass or Fail. He went home and started work on the problem about three weeks before the due date. When his wife would go to work, he would be sitting at the kitchen table, when his wife came home, he was still at the table. For weeks straight. As the deadline got closer, he started sleeping less and working on the problem more. In the last few days, he stopped sleeping entirely. Eventually, he gave up on the problem and took the bus to campus to report on his failure and receive the Fail. Once his deprived mind relaxed on the bus, he had a Eureka moment and was able to solve the problem. Unfortunately, his sleep deprivation caused him to hallucinate. While presenting his hastily put together findings, he was trying to dodge imaginary bats, deal with imaginary blaring car horns, keep from falling asleep while standing, and present very nuanced and complex EM equations. After the presentation the PI said: "Pretty Okay", and passed him.
This is considered a slightly atypical end to a semester with Jackson.
How bad Jackson is depends highly on which problems are attempted and in which context. The content is fine and perfectly understandable. We used it during my third year of undergrad. Homework mostly consisted of the easy-to-medium problems, and self-study could focus on the medium-hard.
Between that and the highly-abstract statistical physics course (it started with an introduction to differential forms), I learned more in my third year of college than in any other.
This is a really great list. I'm currently about midway through a very similar list that I arrived at through trial and error. It would've saved me a lot of time to have seen this a couple of years ago :).
I've got a couple additional suggestions/pieces of advice: First, if Calculus on Manifolds is too advanced Munkres's Analysis on Manifolds is very good and covers mostly the same material. Second if you are shaky on trigonometry it's worth taking a couple of days to relearn it since trig identities and manipulations are used all the time. It makes you feel a bit stupid to review stuff you probably learned in middle school but if you're like me and hadn't used this stuff in years the review will save you lots of time down the road. Third, get used to working problems. It's easy to fool yourself into thinking you understand something because you can follow the worked examples, but you can't actually apply it. I try to do at least a few exercises from each chapter. Shankar's QM book is awesome because the exercises are interspersed with the text, so doing them as you get to them really helps you understand the material. I wish more textbooks had this format.
Munkres is also a great book! It slipped my mind when making the list.
Looking back, maybe I should also have added a group theory textbook.
> This is a really great list. I'm currently about midway through a very similar list that I arrived at through trial and error. It would've saved me a lot of time to have seen this a couple of years ago :).
Thanks, but I wouldn't have had the list a couple of years ago! I finished my undergrad in maths/physics about a year ago, so that's a rough compilation of the books I found most useful. Some were recommended by lecturers, others I found on my own.
I have read part of Zee's Group Theory in a Nutshell for Physicists and it seems pretty good. (Zee is a great writer). It's my only exposure to group theory though so I'm not sure how it compares to other books.
Great suggestions. I would add that Calculus on Manifolds might be sufficient for differential geometry.
One note on expectations - each of the domains you mentioned is typically an upper level undergraduate course (except for the first three). These courses usually progress at a rate of maybe 10 - 15 pages per week (~200 pages of a textbook covered in 16 weeks). It's genuinely hard to absorb this material in that amount of time, and unless you're exceptionally capable you should assume it will take you even longer under self-study.
If you proceed with doing this, make sure you work through at least a few of the exercises at the end of each chapter of each book, and try to compare your solutions against solutions you find online.
This is not to discourage anyone - studying all of this will probably be very rewarding! It's just a lot of work. It takes a massive amount of effort to get to the point where you can understand all the prerequisites needed to follow a research paper in modern math or physics.
> Great suggestions. I would add that Calculus on Manifolds might be sufficient for differential geometry.
Maybe. I'm just going by how all of these were taught during my undergrad. IIRC it doesn't cover metric spaces at all, but in fairness an introductory GR course will cover that; CoM will set up the mathematical context for you.
Start with the Lagrangian and Hamiltonian formations of classical mechanics (Landau & Lifshitz - Mechanics). Then, study Quantum Mechanics (R Shankar - Quantum Mechanics). Statistical Physics while not key here, will be useful for building up an intuition (Landau & Lifshitz - Statistical Physics is good). Classical Field Theory, i.e. Electromagnetism should now be learned (J.D. Jackson).
Then, I'd recommend learning Calculus on Manifolds (Spivak) and Differential Geometry (I can't remember a good textbook on this right now).
If you're at this point you will now have a solid understanding of the maths needed to know GR and QFT. Zee has two good pedagogic books for these "Einstein Gravity in a Nutshell" and "QFT in a Nutshell". Once you know QFT, read a textbook on the standard model.
That'll get you most of the way there. Strictly speaking, you could do possibly learn a lot less to understand this paper, but to be able to pick and choose which parts of the above are actually needed would require a significant amount of work in itself.