# What is the best place to learn about the mathematical foundations of quantum mechanics?

I'm looking for good references to learn about the mathematical foundations of quantum mechanics. By mathematical foundations, I do not mean rigorous quantum mechanics in general but the axioms behind it from a mathematical point of view: the definition of a state, mean value of operators, representation of states using the spectral theorem and so on. In special, I'd like to see some discussions on the relation between states (in Dirac notation) and wave functions. Most references I know (from the mathematical point of view) discuss the foundations only using wave functions and $$L^{2}$$ spaces and no connection to the actual general picture is provided.

EDIT: Maybe I should express myself a little better to avoid confusion. I know some books on quantum mechanics from a mathematical point of view e.g. Gustafson and Sigal's book. However, these references usually avoid the axioms or discuss them very briefly and the main object of study become wave functions living on $$L^{2}$$ spaces, where the Schrödinger equantion is solved for a bunch of different potentials and so on. I'd like to have some nice presentation on the axioms itself and how quantum mechanics arises from them in a more systematic way.

• The book by Hall was very lovely to me springer.com/gp/book/9781461471158 I think it has a particularly great explanation of spin, which is often quite confusing. That said, I think it's usually best to mix in a few different references to get a holistic perspective. Griffith's QM and "QFT for the gifted amateur" were good for skimming overviews for me. – Finn Lim Jun 11 at 22:29
• Just a word of warning: Different authors can have very different views on what the fundamental underlying mathematical structure is in quantum mechanics, and it can be hard to find a balanced treatment. Part of the issue comes down to the infamous measurement problem, and the associated question of whether mixed states really "exist." – Buzz Jun 11 at 23:46
• I like Fock's "Fundamentals of Quantum Mechanics". I think it complements von Neumann's book nicely. – alvarezpaiva Jun 12 at 9:07
• When you say the "axioms" do you mean the Dirac–von Neumann axioms? Or do you have something else in mind ? – Konstantinos Kanakoglou Jun 12 at 23:10
• @KonstantinosKanakoglou that is a nice question. Actually, I don't know exactly what are these Dirac-von Neumann axioms; all I know is that quantum mechanics can be formulated in terms of postulates as most of physics books do. However, I know that different authors use different postulates, but I though these should be all equivalent. In any case, in my opinion, the most natural set of postulates is the six postulates mentioned in the wikipedia page: en.wikipedia.org/wiki/… – MathMath Jun 12 at 23:29

The question is a little unclear --- you want something axiomatic but not rigorous? Anyway, if you don't care about rigor and you like Dirac deltas, I don't think there's any better place to start than Dirac's Principles of Quantum Mechanics. Then if you want to understand the connection to $$L^2$$ spaces and the spectral theorem, I'd recommend Mathematical Foundations of Quantum Mechanics by von Neumann. The notation is out of date but the exposition is excellent.

• Nik, thanks for the answer! Actually, I want the axiomatic and rigorous formulation. Do you think von Neumann's book is still up-to-date? – MathMath Jun 11 at 23:22
• Oh, I misread it. Thanks! – MathMath Jun 11 at 23:27
• Well, as I said, the notation is dated, but I think the content holds up well. Now if you want QFT that's a different story (and I'm not sure what to recommend). – Nik Weaver Jun 11 at 23:29

The best introductory book that I know is L. D. Faddeev and O. Ya. Yakubovskii, Lectures on quantum mechanics for mathematics students, translated by the AMS in 2009.

I think Strocchi's book An introduction to the Mathematical structure of Quantum Mechanics may do the job. In particular section 1.3 explains what you want (states, observables, expectations, $$C^*$$-algebras and representations). The discussion is carried out in more detail in the first two chapters.

• Note: I mentioned section 1.3 of the first edition. Also another section that may be of interest to you is "1.6 Appendix C: Spectra and States" about the spectral theorem and Stone's theorem. – Roberto Ladu Jun 12 at 15:53

Von Neumann and Dirac are hard to beat, but if you want a more recent perspective you might take a look at Mathematical Foundations of Quantum Mechanics: An Advanced Short Course by Valter Moretti.

This is a review of the formulation of Quantum Mechanics, and quantum theories in general, from a mathematically advanced viewpoint, essentially based on the orthomodular lattice of elementary propositions, discussing some fundamental ideas, mathematical tools and theorems also related to the representation of physical symmetries. The final step consists of an elementary introduction to the C$$^*$$ algebraic formulation of quantum theories.

Just like studying bioinformatics programming is a good way for a non-biologist to get a basic knowledge of the fundamentals of molecular biology, studying quantum computing is a good way to get a basic knowledge of the fundamentals of quantum mechanics. The book Mathematics of Quantum Computing: An Introduction by Wolfgang Sherer (Springer 2019) gives a good introduction to quantum mechanics for computer scientists. It is less advanced than some of the other recommendations and has a different focus, but it might be helpful (especially if your motivation for asking the question is a desire to understand quantum computing).

• I haven't read this book and am no expert on quantum computing, but if using this to learn the topic it's probably good to be aware that (afaik) the standard notation in quantum computing differs a little from that in quantum mechanics, e.g. $Z$ instead of $\sigma^z$, etc, for the Pauli matrices – Jules Lamers Jun 15 at 23:59

Another possibility might be Mackey's Mathematical Foundations of Quantum Mechanics (Mackey obviously being an eminent researcher in functional analysis and noncommutative geometry).

A positive review in Bulletin of the American Mathematical Society can be found here if you need more information on the content of the book.