Quantum mechanics for mathematicians
I'm looking for books about quantum mechanics (or related fields) that are written for mathematicians or are more mathematically inclined.
Of course, the field is very big so I'm in particular looking for books that explain how C*-algebras and von Neumann-algebras are used, I want to see "practical" applications of those.
What I know is not much more than Griffiths' book (I don't like that book!) contains. I have in my possession the books by Claude Cohen-Tannoudji but they don't seem to contain this.
It may not have the specific topics you want, but I like Leon Takhtajan's book entitled, coincidentally, Quantum Mechanics for Mathematicians.
Books like Dirac tended to frustrate me with drawn-out developments of "this is a ket, this is a bra, they have such-and-such properties and we combine them in the following ways." Takhtajan is kind enough to come out and say "take a Hilbert space and a self-adjoint operator."
I was recently looking for such a book and I found two that look promising:
- An Introduction to the Mathematical Structure of Quantum Mechanics by F. Strocchi. This is a small book and assumes that you have prior knowlegde of functional analysis (in particular C* algebras).
- Quantum Mechanics in Hilbert Space by Eduard Prugovecki. This book is large as the first three parts are functional analysis and the last two parts deal with quantum mechanics. No C* algebras here though, only Hilbert spaces of operators.
If you want to look at a book that centralizes the role of an algebras of observables (operator algebras) in quantum theory, you should look at the book written by one of the founders of this approach. I recommend "Local Quantum Physics" by Rudolf Haag.
Furthermore, I think Reed and Simon's Vol.1 "Functional Analysis" will be very helpful here. Read the first two, and the last three chapters. While you are doing this, I strongly recommend you use Vaughan Jones' notes on Von Neumann Algebras (http://www.math.berkeley.edu/~vfr/MATH20909/VonNeumann2009.pdf). There is a section in there about Bosons and Fermions.