Math Prerequisites for Field Theory Course: Wald's Text

[kostas230]

I'm a 3rd year major in physics. I have a very strong background in linear algebra and rigorous calculus, and I'm studying topology from Munkre's book and abstract algebra from Fraleigh. Next year, I'm taking a graduate course called "Field Theory" which covers General Relativity, using Wald's text and QFT using Weinberg's books and some others.

Can you recommend me what books on math (diff geometry for example) should I read for tackling Wald? Thank you. :)

Edit: I accidentally posted it in academic guidance, instead of textbooks' section. So, if an admin can transfer it to the correct section that would be good. :)

 

[WannabeNewton]

You don't really need to do that before starting Wald for a number of reasons. For starters, the majority of the math in Wald consists of abstract index tensor calculus/algebra (take a look at chapters 7, 9, and 10 for example) so only by doing the problems in Wald's book, in this context that is, will you ever get practice with that-a math book on differential geometry won't help with that in the slightest. Secondly, a lot of the concepts you learn from a typical differential geometry text won't show up in Wald's book. The only chapter that even requires much topology is the chapter on global causal structure of space-times. But if you still want, there are a good number of awesome books out there.

https://www.amazon.com/dp/0387954481/?tag=pfamazon01-20
https://www.amazon.com/dp/0387983228/?tag=pfamazon01-20
https://www.amazon.com/dp/0125267401/?tag=pfamazon01-20

https://www.amazon.com/dp/0387979263/?tag=pfamazon01-20
https://www.amazon.com/dp/0387901485/?tag=pfamazon01-20

As a side note, don't use Munkres' book beyond the point set topology half because the algebraic topology half is just plain terrible in that book.

EDIT: Also, if you want a GR book that is much more rigorous mathematically than Wald, check out: https://www.amazon.com/dp/9400754094/?tag=pfamazon01-20

 

[kostas230]

Can't thank you enough WannabeNewton. :)

One question though. Do I have to read Introduction to Topological Manifolds before going on to Introduction to Smooth Manifolds?

 

[WannabeNewton]

Not really, not if you already know basic topological notions such as connectedness/path connectedness, compactness, local compactness, quotient spaces (particularly adjunction spaces), first/second countability, and stuff like that. Munkres would have covered all of that. Still you can skim through Lee's text on topological manifolds if you want because the end of chapter problems contain some really cool counter-examples (line with infinitely many origins, the long line etc.).

 

[George Jones]

Fromm a book by mathematician Paul Halmos:

“…[the reader] should not be discouraged, if on first reading of section 0, he finds that he does not have the prerequisites for reading the prerequisites.”

 

[kostas230]

Unless you are reading GR xD

 

[R136a1]

Not really, not if you already know basic topological notions such as connectedness/path connectedness, compactness, local compactness, quotient spaces (particularly adjunction spaces), first/second countability, and stuff like that. Munkres would have covered all of that. Still you can skim through Lee's text on topological manifolds if you want because the end of chapter problems contain some really cool counter-examples (line with infinitely many origins, the long line etc.).

Lee does need some algebraic topology occasionally though. For example, covering spaces and fundamental groups are sometimes needed in his smooth manifolds book. And if you want to read up on De Rham cohomology, it's best to know a bit of singular homology already.

Not that you can't read smooth manifold without these, since if you only know point-set topology you can still follow 99% of his text.

I do seem to recall that Wald sometimes needs simply connectedness and stuff...