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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.
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13 Answers 13

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).

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'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

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A very good introduction to Lie groups from a physicist point of view is

H. Lipkin, Lie Group for Pedestrians

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The last book I read on "background in math for physicists" was "Mathematics for Physics" by Stone and Goldbart, and I enjoyed it quite a bit. (Since then I've tended to hit the pure math books, but that's a different story).

Even better, a version of the book is available online at Paul Goldbart's webpage.

Here's a list of topics:

* Calculus of Variations
* Function Spaces
* Linear Ordinary Differential Equations
* Linear Differential Operators
* Green Functions
* Partial Differential Equations
* The Mathematics of Real Waves
* Special Functions
* Integral Equations
* Vectors and Tensors
* Differential Calculus on Manifolds
* Integration on Manifolds
* An Introduction to Differential Topology
* Groups and Group Representations
* Lie Groups
* The Geometry of Fibre Bundles
* Complex Analysis I
* Complex Analysis II
* Special Functions and Complex Variables
      o Appendix A: Linear Algebra Review
      o Appendix B: Fourier Series and Integrals 
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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.

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  • One, pretty mathy, but classic book about Riemannain manifolds is: Semi-Riemannian Geometry by O'Neill.

  • Some approachable Lie Algebra notes are available here, they are designed to require little background: Lecture Notes on Lie Algebra.

  • My personal favorite book about Algebraic/Differential Topology is : Calculus to Cohomology. This book is extremely approachable, requiring only multivariable calculus and linear algebra to completely understand it. I cannot recommend it enough, particularly for physics.

Also I third Road to Reality. It is a very fun/interesting book!

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Are you asking for an intro level book or a more advanced book for someone who already has some background in those topics?

For an introductory level, I second the Schutz and Spivak recommended above. Penrose and Frankel are suitable only if you already have had an introductory course in those subjects, in my opinion. Frankel's introduction to manifolds is very condensed, and Penrose is really providing a bird's eye view while skipping over many details beginners would need to build basic intuitions.

The best introductory notes I've come across for manifolds as used in GR are David Malament's, which you can download here.

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Physics Reports 66: Gravitation, Gauge Theories, and Geometry (Eguchi, Gilkey and Hanson).

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Nice question. I don't know much about either differential geometry or algebraic topology, but having studied groups a little, I think I can provide some references for Lie groups. So here are the books I found useful

  • Samelson, Notes on Lie Algebras written in a Definition, Theorem, Proof style, so it's little hard to grasp (I recommend mutliple rereading) but gives a good overview of the structure, classification (root systems and Dynkin diagrams) and representations (highest weight theory) of Lie algebras.

  • Humphreys, Introduction to Lie Algebras and Representation Theory less theorem-heavy and more talkative than Samelson and contains huge number of great exercises.

  • Fulton, Harris, Representation Theory A First Course discusses more or less everything a physicist needs to know about groups (also mentions some finite groups). Lacks the systematic theorem based approach of the two books above, but boasts great explanations and nice pictures. I'd suggest it as a nice first reading about groups it if weren't for its length.

  • Goodman, Wallach, Representations and Invariants of the Classical Groups this is an ultimate bible on groups. Authors take an algebraic geometrical approach to the Lie groups (instead of the usual differential geometrical) which makes the book somewhat hard to read for a regular physicist. But besides this the book provides an in-depth look at lots of concrete representations (e.g. tensor representations and connection with symmetric group; this is often omitted elsewhere), discusses highest weight theory at great length, provides a nice introduction to spinors and also mentions branching rules. And lots of other stuff. Definitely recommended.

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I found Mathematical Methods in the Physical Sciences by Mary Boas to be a very good broad book covering the basics. You'll need other books obviously but if you are looking for one book for a solid review of the basics, this book is excellent.

Here are the chapter titles:

  1. Infinite series, power series
  2. Complex numbers
  3. Linear algebra
  4. Partial differentiation
  5. Multiple integrals
  6. Vector analysis
  7. Fourier series and transforms
  8. Ordinary differential equations
  9. Calculus of variations
  10. Tensor analysis
  11. Special functions
  12. Series solutions of differential equations, legendre, bessel, hermite, and laguerre functions
  13. Partial differential equations
  14. Functions of a complex variable
  15. Probability and statistics

I also second Roger Penrose's The Road to Reality as a good book with a broad scope of math with a more theoretical slant.

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For a general approach to the maths involved in both classical and quantum physics, one of my favourite books is:

-"Mathematics of classical and quantum physics", Byron & Fuller.

In the more geometrical side, besides the already mentioned books, you can try:

-"The geometry of physics. An introduction", Theodore Frankel.

And, as a general reference, the usual text is Arfken's "Mathematical methods for physicists".

But, IMHO, if you want to thoroughly understand the mathematical tools of physics, you should use "Methods of Theoretical Physics", by Morse & Feshbach. It is an old book, but essential if you want to understand Jackson's Classical Electrodynamics or Messiah's Quantum Mechanics.

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Sean Carroll's Lecture Notes on General Relativity contain a superb introduction to the mathematics of GR (differential geometry on Riemann manifolds). These also also published in modified form in his book, Spacetime and Geometry.

Spivak's Calculus on Manifolds is a gem.

Bishop's Tensor Analysis on Manifolds is a great introduction to the subject, and published by Dover, is very cheap (less than $10 on amazon).

Georgi's Lie Algebras In Particle Physics is enjoyable and fast-paced, but probably skips around too much to be used as an adequate first exposure.

Shutz's Geometrical Methods of Mathematical Physics and A First Course in General Relativity.

Despite it's incredibly pompous title, Penrose's The Road to Reality: A Complete Guide to the Laws of the Universe provides an enjoyable high-level view of a vast expanse of mathematical physics.

As mentioned by Cedric, I am a huge fan of Sussman and Wisdom's Structure and Interpretation of Classical Mechanics and the associated Functional Differential Geometry memo. The citations in those publications will also point to towards a lot of good material and there's more goodies if you dig around in the source code.

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@Schutz's "Geometrical methods ..." is what I turned to when Lie derivatives were causing me great headaches. The best pedagogical explanation of diff. geom. for noobs, IMO. –  user346 Jan 7 '11 at 3:57
    
Schutz'z "A First Course in GR" has a good mix of both physical AND mathematical principles in the early chapters defining concepts. There's also an unusual primer at the end on the measurement theory and technology of gravitational wave detection. –  WetSavannaAnimal aka Rod Vance Jul 27 '13 at 13:02
    
Spivak's "Physics for Mathematicians. Mechanics I", see mathpop.com/mechanics1.htm. Even though this is not just on math background, but rather a very math minded physics text book. –  UwF Sep 21 '13 at 10:01

Here you have a book from Sussman and Wisdom from MIT about functional differential geometry covering your first query about "manifolds".

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What a fascinatingly unusual book! Differential Geometry explained as computer algorithms. –  WetSavannaAnimal aka Rod Vance Jul 27 '13 at 13:06

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