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Books for the foundations of quantum mechanics?

  1. Jun 20, 2013 #1
    I finished reading "Linear Operators for QM" by Jordan a couple of weeks ago, and it was extremely interesting for me. It introduced various topics in functional analysis that are relevant to the foundations of QM (Hilbert spaces, spectral theory, von Neumann algebras, etc.).

    I was wondering if anyone could recommend some pure maths books that go into these topics in more detail, or any other physics books that explain the mathematical foundations of QM written for advanced undergraduate/early graduate students. If you know about any books on an introduction to functional analysis in general, then that would be great too.

    Also, how much measure theory and Lebesgue theory would I need? My only exposure to those things are from Rudin's mathematical analysis and the QM book I've mentioned.
     
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  3. Jun 20, 2013 #2

    marcusl

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    Edmonds, Angular Momentum in Quantum Mechanics

    Many people like Tinkham, Group Theory and Quantum Mechanics, but I have no direct experience
     
  4. Jun 20, 2013 #3
    "Functional Analysis" (Walter Rudin), "Fundamentals of the Theory of Operator Algebras" (Kadison and Ringrose, two volumes), "Methods of Modern Mathematical-Physics" (Reed and Simon, four volumes), "Mathematical Methods in Quantum Mechanics (with applications to Schrodinger Operators)" (Gerald Teschl), "Mathematical Structure of Quantum Mechanics (a short course for mathematicians)" ( F. Strocchi), "Mathematical Foundations of Quantum Mechanics" (John Von Neumann, 1932 ).


    For the mathematics behind big parts of QFT:

    "Local Quantum Physics" (Haag), "Quantum Physics: A Functional Integral Point of View" (Glimm and Jaffe), "Introduction to Algebraic and Constructive Quantum Field Theory" (Baez, Segal...), "Quantum Field Theory for Mathematicians" (Robin Ticciati), "Quantum Field Theory (a tourist guide for mathematicians)" (Folland).

    To be able to really understand the physical side, you'll need to study at the same time books like:

    For QM: "Quantum Mechanics, A Modern Development" (Ballentine), "Mecanica Cuantica" (Galindo y Pascual)

    For QFT: "The Quantum Theory of Fields" (Weinberg, 3 volumes), "Quantum Field Theory" (Eberhard Zeidler, 3 volumes already and 3 more are coming).


    There are many more important textbooks, but with these ones you'll be busy for 20 or 30 years (no kidding).
     
  5. Jun 20, 2013 #4

    bhobba

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    Yea - Ballentine's book is the one you want. He explains with reasonably good rigor how QM is the result of two axioms. And one of those axioms to a large extent follows from the other via Gleason's Theorem (not completely - which is why you cant ditch it - but it is seen to apply with the very mathematically intuitive idea of non contextuality):
    http://kof.physto.se/cond_mat_page/theses/helena-master.pdf

    You can learn Lebesque integration from many sources - Ruden is fine. For Hilbert spaces and its relation to QM get Von Neumann's Classic - Mathematical Foundations of QM. But, while Von Neumans method, based on the spectral theorem, is mathematically rigorous, it's not what is used by physicists. To understand the way they look at it you need to come to grips with Rigged Hilbert Spaces, the Generalized Spectral Theorem, and Distribution Theory. Ballentine gives a reasonable introduction to that but I actually reccomend the following little gem:
    https://www.amazon.com/books/dp/0521558905

    Thanks
    Bill
     
    Last edited: Jun 20, 2013
  6. Jun 20, 2013 #5
    Simon & Reed Functional Analysis is a great book for this purpose.
     
  7. Jun 20, 2013 #6

    micromass

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  8. Jun 20, 2013 #7
    Thanks for the suggestions everyone, I'll take a look at them.
     
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