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Can you recommend books or papers that highlight or discuss extensively, or at least more than average, the similarites/differences between phase space and Hilbert space? I am primarily interested in physics, not mathematics, but am willing to look at good math books also.
For all I know, this might be one of the standard intro to QM books that I am unaware of. Or a more advanced text. Or a survey paper.

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This may be a bit too weighted on the mathematical side, rather than the physics side, but if you look up references on Geometric Quantization, you will see how phase space has geometric structure which Hilbert space doesn't have and vice versa. GQ attempts to give explicit instructions for constructing the Hilbert space from a classical phase space, so the characteristics of the spaces involved are brought out very explicitly.

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Concerning phase space you can read some chapters of Arnold : Mathematical Methods of Classical Mechanics. For Hilbert space structure : Quantum Mechanics in Hilbert Space. Now for the comparison of the two I don't know any book but you can read also AN INTRODUCTION TO THE MATHEMATICAL STRUCTURE OF QUANTUM MECHANICS, Strocchi where he compare the structure of classical and quantum mechanics, but it is maybe quite complicate.

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''similarites/differences between phase space and Hilbert space?'' There are no similarities except that they are both infinite-dimensional vector spaces. How to use the two is completely different. Thus there is no book that can answer that.

But if you mean the differences and similarities of classical and quantum physics, my book Classical and Quantum Mechanics via Lie algebras will be the right thing for you. It presents classical and quantum mechanics from a unified point of view, in which the differences (that in typical books are maximized) are minimized.

But if you want an in depth discussion of phase space and Hilbert space in QM, the right book to read is probably the book Harmonic analysis in phase space by Folland. It involves the Heisenberg group, quantization, the Weyl calculus, the metaplectic representation, wave packets, and related concepts, all at the intersection between Hilbert space techniques and phase space techniques.

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Recently, I have rediscovered Mackey: Mathematical Foundations of Quantum Mechanics, which has some very relevant discussion of phase space and Hilbert spaces. More references are still welcome.

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I added another reference to my answer. – Arnold Neumaier Mar 16 '12 at 21:32

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