Best books for mathematical background?

Question

What are the best textbooks to read for the mathematical background you need for modern physics, such as, string theory?

Some subjects off the top of my head that probably need covering:

• Differential geometry, Manifolds, etc.
• Lie groups, Lie algebras and their representation theory.
• Algebraic topology.

This post imported from StackExchange Physics at 2014-05-04 07:44 (UCT), posted by SE-user ahh

Answer 1

The best math book I ever read with respect to being useful for physics is

• Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach (2nd Edition), by Hubbard and Hubbard.

It is an absolute gem. It gets you through linear algebra and differential forms starting from square one, assuming you only know algebra and calculus. The proofs are legitimate and in some cases really creative. The best part is that it's aimed at people who want to use math for applications. Extremization of functions on manifolds is developed really well and the authors give insightful information on how to approach the analytical topics presented in the book numerically. Really useful things like finding Taylor series for implicit functions is done well. I really can't give this book enough endorsement.

After I read that I read

• Analysis On Manifolds by Munkres

This book does integration of differential forms formally. Still, it's amazingly readable, and I never found one single mistake in the entire book. This was a great read and reinforced my understanding, but was not directly relevant to physics.

Then later I read

• Spacetime and Geometry: An Introduction to General Relativity, by Sean Carroll

which is an excellent introduction to curved manifolds. It's nice because he clearly explains the difference between vectors and co-vectors ("up" and "down" indices) and relates it all to real life (ie. physics).

This post imported from StackExchange Physics at 2014-05-04 07:44 (UCT), posted by SE-user DanielSank

Answer 2

The field of operator algebras has a strong connection with quantum theory and certainly is a necessary requirement for studying many literatures in modern Physics I list some of the books relating operator algebras and physics in the following:

S. Attal, A. Joye, C.A. Pillet, Editors, Open Quantum systems 1, the Hamiltonian approach. Springer, Lecture notes in mathematics, vol. 1880, (2006).

B. Blackadar, Operator algebras. Springer, Encyclopaedia of Mathematical Sciences, vol. 122, (2006).

O. Bratteli, D. W. Robinson, Operator algebras and quantum statistical mechanics 1, $C^*$- and $W^*$-algebras, symmetry groups, decomposition of states. Springer, Texts and monographs in physics, 2nd edition, 2nd printing, (2002).

Connes, A., Noncommutative geometry. Academic press, Inc. (1994).

Garcia-Bondia, J.M., Varilly, J.C., Figueroa, H., Elements of noncommutative geometry. Birkhauser Advanced Texts, Birkhauser, (2000).

N. P. Landsman, Mathematical topics between classical and quantum mechanics. Springer, Monographs in mathematics, (1998).

M. Takesaki, Theory of operator algebras I, II, II. Springer, Encyclopaedia of Mathematical Sciences, vol. 124, (2002).

N. Weaver, Mathematical quantization. Studies in advanced mathematics, Chapman and Hall/CRC, (2001).

In addition to the above books, for a more complete list of general references on $C^*$-algebras and operator algebras as well as for an easy reading for beginners see my lecture notes on $C^*$-algebras here.

This post imported from StackExchange Physics at 2014-05-04 07:44 (UCT), posted by SE-user Vahid Shirbisheh

Answer 3

'Modern Mathematical Physics' by Peter Szekeres is the best book I've found for the foundations of mathematical physics. It's extremely clear and conveys deep understanding on the first reading.

There's an amazon preview here: http://www.amazon.com/Course-Modern-Mathematical-Physics-Differential/dp/0521829607

Chapter titles:

1. Sets and structures

2. Groups

3. Vector spaces

4. Linear operators and matrices

5. Inner product spaces

6. Algebras

7. Tensors

8. Exterior algebra

9. Special Relativity

10. Topology

11. Measure Theory and integration

12. Distributions (Fourier transforms, Green's functions)

13. Hilbert Spaces

14. Quantum Mechanics

15. Differential Geometry

16. Differentiable Forms

17. Integration on manifolds

18. Connections and curvature

19. Lie Groups and Lie Algebra

This post imported from StackExchange Physics at 2014-05-04 07:44 (UCT), posted by SE-user Schroeder