Fredrik said:
One of the problems with Varadarajan's book is that it attracts interest from physicists and physics students who don't have the prerequisite mathematical knowledge. But there are many other issues. The proofs are difficult to follow, and it's difficult to skim through it to get a "big picture" view or an idea about which parts of the book are important. For example, how much projective geometry do you need to know, and do you have to know everything about systems of imprimitivity or measure theory on simply connected locally compact topological groups to understand the later chapters?
In my case, I learned most of the topics from other sources. Armed with this background, I read Varadarajan and found these topics exposed in a much more detailed (but still clear and in most cases motivated) way and ordered in a beautiful and consistent narrative, which makes the foundations of QM to sound and flow like a Mozart piano concerto.
But it was indispensable to know beforehand most of the topics involved and to some degree of precision.
For example, I learned symplectic geometry in a course I took about the topic (and it was preparing a final monograph for this course that I discovered the book in question, since I wanted to talk about the structural analogies between classical mechanics in the symplectic geometry formulation and QM). So, I could easily follow the first chapter and concentrate in the relevant aspects that are needed later (like symmetries of the configuration space and momenta observables, etc.).
For chapters II,III and IV, I learned these topics for the first time from
this book. I strongly recommend it, it has a very clear and "only the essential" point of view in the exposition of QM as generalized probability measures on the lattice of projectors (it also contains a complete exposition of all the functional analysis needed for QM; it also has material in quantum symmetries, like projective representations, multipliers, extensions, etc., chapter VII in Varadarajan).
For chapters V and VI, definitely Folland's A Course in Abstract Harmonic Analysis. It's of course a math book for mathematicians, but Folland is very clear. A mathematically minded physics student and with the necessary math background shouldn't have any problem with it.
It contains accessible statements and proofs of the Imprimitivity theorem and the Mackey Machine for semidirect products (also all of the stuff about compact groups, like the Peter-Weyl theorem, etc.)
For chapter VIII, Jauch's Foundations of Quantum Mechanics gives an introduction on the application of the Imprimitivity theorem in QM.
For chapter IX, Folland's Quantum field theory, a tourist guide for mathematicians gives a nice and basic introduction on how to use the Mackey Machine to obtain Wigner's classification (supplemented with material from the Moretti book for the galilei group in order to include non-relativistic QM).
From my experience with all this, I think the method that worked for me was: first, read the books that deal with the physical part (in particular, Jauch and Folland's QFT). From these books I was able to understand which are exactly the relevant points for physics. Then I went to Folland's Harmonic analysis to learn more about the math (not so much how to prove the theorems, but about their precise statement and background material, like, e.g., semidirect products, dual groups, characters, induced representations, etc.). Then I went back to the more physical books and learned how these things are used in physics, i.e., concrete things like how to obtain the Dirac equation from the induced representations, etc. Simultaneously, I was studying from the Moretti the lattice approach and all that.
After all that, I went to Varadarajan and read chapters VIII and IX. Since I already knew some of the topics, I finally was able to understand the big picture (which wasn't completely clear to me from the other books). I found the exposition relatively clear, motivated and also I was able to fill many of the details missing from the other more elementary expositions. But yes, I ended using the book more as a reference. In
this post I tried to give a summary of the big picture of these two chapters.
Finally, now that I think I undersatand most of the physical applications and implications, I'm studying in more detail the proofs of the theorems. I'm still studying from the other books rather than Varadarajan. I rely on the latter when details are missing in the others (for example, the Moretti only gives the easy proof about how Gleason's theorem reduces to show that all frame functions are regular, but doesn't prove this, it refers the reader to Varadarajan, but that's fine since the book doesn't want to overwhelm the reader).
I know this post is a little convoluted, but that was my experience.
From the experience I recounted here, I think Varadarajan was difficult for me because it touches on so many different topics and with a lot of detail. From topics in pure math (at the mathematician's level) to their application in physics. From projective geometry to the representation theory of non-compact groups.
