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I want to know the math that is required to read Takhtajan's "Quantum Mechanics for Mathematicians". From the book preview on Google, I gather that algebra, topology, (differential) geometry and analysis are needed. What level of real and complex analysis do I need, and some good books for learning them?

PS - Hopefully suitable group :-)

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Don't read it, my advisor said that it sometimes contain errors that could be catched only by experts. I vaguely remember an example where author implicitly supposed that all characters of $\ell_\infty$ are point evaluations. –  Norbert Jan 18 at 0:46

2 Answers 2

I just looked through Lectures on Quantum Mechanics by Leon Takhtajan, and from my skimming, it appears you need to have a good understanding of:

  1. Topology (probably on the order of Munkres)
  2. Calculus of Variation
  3. Lagrangian and Hamiltonian Mechanics
  4. Ability to follow manipulations of vector identities
  5. Real Analysis at the level of Rudin or Apostol.
  6. An understanding of Borel sets
  7. An understanding of manifolds of all kinds so a solid Differential Geometry text
  8. Should know transforms besides Laplace and Fourier

I, also, visited his course website at Stony Brook University and this is what I found. Note that Munkres was listed in the core courses, but since I mentioned it up top, I neglected it in this list.

Prerequisites: The basic core courses curriculum and the basics from MAT 551, MAT 552, MAT 568, MAT 569. This list comes from Takhtajan recommendation of courses needed for the class:

  1. Functional Analysis by Peter Lax
  2. An Introduction to Lie Groups and Lie Algebras by A. Kirillov Jr.
  3. Riemannian Geometry, by Peter Petersen, Springer, 2005.
  4. Lars Ahlfors, Complex Analysis
  5. Dummit and Foote Abstract Algebra
  6. Walter Rudin, Real and complex analysis
  7. Michael Spivak, A Comprehensive introduction to differential geometry (this is a mutli volume set)
  8. Michael Taylor, Partial differential equations (this a two volume set)

He also says: All necessary facts from the theory of operators in Hilbert spaces.

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Hopefully not all volumes of Spivak... –  Hoot Jan 16 at 6:10
@Hoot I just double checked and it doesn't say. –  dustin Jan 16 at 6:11
At least he doesn't require complete understanding of the proofs of Fermat's last theorem and the abc conjecture. –  marty cohen Jan 16 at 19:27
And my second lifetime. The only things I know there is that "identities" and "Apostol" were misspelled. –  marty cohen Jan 16 at 19:31
Even though you skimmed (very superficially, obviously) through only chapters 1 and 2, this does not prevent you from adding an answer that is nothing worth. As I pointed out, 2. and 3. are simply wrong, 8. is wrong also (because you don't understand that they actually introduce the legendre transform, which is not an integral transform (as you thought)). Also what you say here could be said about a generic book –  Laters Jan 18 at 10:19

As I have taken a rather close look at all 8 chapters, I can offer a more precise perspective. I first point out what you don't need to know/have in order to read about, say, 90% of the text:

(i) A prior knowledge of physics. (This is mentioned in the preface).

(ii) $C^\ast$-algebras. (They sometimes use what a $C^\ast$-algebra is, but it is merely used as convenient language, and seldom they invoke deep facts such as the Gelfand-Naimark theorem. I recommend you to look this up when you need it/want to understand it.)

(iii) Complex geometry (Kähler manifolds), Sheaf theory (Ringed spaces). (Same comment as in (ii) applies, Sheaf theory terminology is mainly used in chapter 8.)

(iv) Algebraic topology (Singular cohomology, Stiefel Whitney classes). (If you know nothing about algebraic topology (cohomology theory), it is almost impossible to simply look this up and understand it without serious effort; but I think it is used only at two places in chapter 8.)

(v) Homological algebra (Hochschild cohomology, Lie algebra and group cohomology). (Same comment as in (iv), they sometimes say that something is a cocyle, but explain what this means explicitly. They also introduce the Hochschild complex for their treatment of deformation quantization, so this is basically self-contained.)

(vi) Category theory. (At one point they talk about the category of poisson manifolds and a certain equivalence of categories, but here the comment to (ii) applies.)

(vii) Calculus of variations. (They introduce everything they need, which is hardly anything: the principle of least action and the Euler Lagrange equations. They also (rarely) use the terminology of Frechet manifolds. I was told by T. Shiffrin that this kind of stuff is introduced e.g. Lang's differential geometry book, but "ordinary" DG books don't seem to deal with it.)

That being said, I now point what is quite essential and comes up quite frequently.

(I) Differential geometry (generalities such as e.g. (co-)tangent bundles and induced maps, fiber bundles and vector bundles, differential forms, lie brackets and derivatives, Riemannian manifolds, connections, flows). This is particularly often used in chapter 1. I would recommend to the read the relevant chapters in Lee's introduction smooth manifolds, or in an equivalent source. They also use rather often terminology coming from Lie groups and algebras (such as the enveloping algebra of lie algebra) and rarely deep facts such as Ado's theorem. (For the latter you should consider looking it up in a textbook on the subject, for the former read the relevant chapter in Lee.)

(II) Measure theory (generalities, probability measures, lebesgue stietljes integral, fubini's theorem, riesz theorem, radon nikodym derivatives).

(III) Functional analysis (generalities on hilbert space, adjoint operators, spectrum, Stone Weierstrass theorem, Spectral theorem, Hilbert-Schmidt theorem, Stone's theorem, distribution theory such as e.g. the Schwartz kernel theorem or the N-representation theorem, a bit of Sobolev spaces terminology). This is the most serious part (and one should certainly have some background, even though they mention most of the results they need), and I recommend the books (mainly volume I) of Reed/Simon (modern methods math. physics) for this material (apart from Sobolev spaces, for which I recommend Brezis's functional analysis book).

(IV) A bit of algebra: mainly a bit of representation theory (e.g. Schur's lemma, Young tableaux) and tensor products (e.g. extension of scalars, in particular complexification) which is dealt with in the relevant chapters of Lang's (graduate) algebra text or Fulton/Harris's representation theory book. (My favourite treatment of the tensor product can be found in Atiyah/Macdonald's introduction to commutative algebra.)

(V) A bit of complex analysis: if you have taken any complex analysis course worth the name, you probably have enough background. They use mainly elementary results, such as the fact that an entire function admits an everywhere convergent power series expansion. They also sometimes use very elementary fourier analysis, such as the plancherel theorem.

(VI) A bit of general topology: Mainly elementary results/generalities, at one point they use the one point compactification of $\mathbf{R}^n$.

Generally speaking, the book might look intimidating, but don't be afraid, many things are just terminology which is easy to pick up. I know that "a bit of [...]" is not very precise, but what I mean by this is that if you have taken an introductory course to [...] then you probably have enough background. (In some cases what they actually use is even less, but in general, as Hardy points out, "We must also remember that a reserve of knowledge is always an advantage, and that the most practical of mathematicians may be seriously handicapped if his knowledge is the bare minimum which is essential to him".)

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+1 Very detailed and comprehensive answer. –  Deepak Jan 18 at 0:39
I think this answer is spot on. Also, I don't think it would be the best idea for the OP to try to learn the material ahead of time, but rather they should look it up as needed. –  Olivier Bégassat Jan 18 at 1:12
Exactly what I wanted to know. Thanks. –  fctaylor25 Jan 18 at 4:19
@OlivierBégassat Thanks for your advice, too. I'll get a good foundation in the main subjects and then look up details as necessary. –  fctaylor25 Jan 18 at 4:31
I second what Oliver says; for a lot of the text you merely have to understand the relevant definitions. –  Laters Jan 18 at 10:21

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