Saturday, August 5, 2017

Quantum Mechanics and the Particles of Nature by Anthony Sudbery

Quantum Mechanics and the Particles of Nature: an outline for mathematicians by Anthony Sudbery has been one of my favorite books on quantum mechanics since I was an undergraduate. This is despite the fact that I probably haven't read it from cover-to-cover ever in my life. It has, however, been a book that I have returned to again and again when looking at quantum mechanics.

The book is a curious amalgam of physics and mathematics. The meat of the book, chapters 2-4, 6 and 7, are formally sound, mathematical beasts proceeding axiomatically though their various topics and become highly algebraic, in the modern sense. Chapter 3, on "Quantum Dynamics" introduces Lie algebras and symmetry groups (SU(2) and all that), and Sudbery never looks back. Most of the book proceeds with a pattern of: (1) description of the physics, (2) a theorem that encapsulates the description, and (3) a rigorous proof of the theorem. This is incredibly abstract for an introduction to the subject -- unless you're an undergraduate mathematics student, which this work was designed for. Sudbery is not trying to teach solution techniques, but rather to express how the artifice of quantum mechanics fits together logically, and this is often as much the province of applied mathematics as it is theoretical physics. I don't know the demarcation between the two, you'll have to get an applied mathematician and a theoretical physicist that are working on similar subjects to draw the line for you.

[It is designed for the British student, not the American one. I went through many undergraduate textbooks from England that I found extraordinarily good that were far too dense for any of my students. This of course may not be true for all colleges, but it was my experience.]

I have a few advantages over the usual reader for a book like this: (1) I survived my graduate Quantum Mechanics courses, (2) I have a degree in "theoretical mathematics" (as opposed to applied mathematics), so I am familiar with proofs and the axiomatic style and have even taken a modern algebra course, (3) I have taught both 300-level modern physics, materials science, and nanotechnology courses to undergraduates, all of which include a reasonable amount of quantum mechanics (even if very basic), and of course, (4) I talk about these sorts of things on a podcast. SO the going this time was easy, even if it is a book that I've owned for twenty-five years.

I originally found this book in the college bookstore -- we had an extraordinary college bookstore when I was an undergrad -- and immediately fell in love with it upon reading the first chapter. I'm sure I got through the next two and the fifth, but I don't know how much of the others. The first chapter, "Particle and Forces" is as lucid a description of particle physics as I've ever read, and it's a description that stayed with me even after I forgot where it came from. I found it so clear and memorable that I thought it came from a popular book, and when in the 2000s I was looking for it, I kept looking in old Scientific American books (Particles and Forces: at the heart of the matter, it think it was; it seems to have disappeared) and Polkinghorne's The Quantum World (which hasn't disappeared, and I'd replace if it did) and similar books that I'd read a little bit earlier or later. I did not go back to a textbook that I was working in my spare time and didn't actually finish. The reason why I wanted to find it again is that I'd attempted this book before I took a quantum mechanics course, and it left a lasting impression on how I thought about the subject. I finally found out that this was the book I had been looking for, off and on, for ten years when I pulled it out to help me prepare lectures for modern physics when I taught it.

But the thing that brought me back to this book most of all was chapter 5, "Quantum Metaphysics." This talks about the quantum theory of measurement, The de Broglie-Bohm Interpretation of Quantum Mechanics and Quantum Interpretations in a rigorous way. Even as an undergraduate, I found this chapter entrancing. When I took a philosophy course on (philosophical) cosmology, I used this general approach to discuss how the various interpretations of probability relate to the interpretations of quantum mechanics (which not think is a lot harder than I thought it was then). You can get an idea about how clear this chapter is if you listen to Randy and I talk about section 5.5 as the intermission in our discussion of Aharanov and Rohrlich's Quantum Paradoxes. It is a touchstone that I come back to every time I think about how to think about quantum mechanics.

That said, I had some rough going with the problems in this book. One reason is that so many of them are proofs, and a problem I've always had with proofs is determining whether or not I've really justified every step. Too often I worry that this or that step is for a special case and isn't extensible to other cases, and so on -- and I'm not as lucky as a student because I don't have a red pen man to tell me what I did wrong or reassure me that I did it right (students may disagree about the desirability of the red pen men). Another is that I think Sudbery expects you to have actually learned things in a mathematics course, meaning that I have to reconstruct the methods for the more application-oriented problems. This makes the problems especially valuable, and I find myself writing up multi-page analyses of what I did to solve a problem to remind myself what I did -- something I haven't done since my graduate work.

The chapters in between these two, 2-4, dealing with "Quantum Statics", "Quantum Dynamics", and "Some Quantum Systems" are all equally rewarding, giving a mathematical picture of the basic ideas of quantum mechanics. The last two chapters are more difficult. "Quantum Numbers" and "Quantum Field Theory" go a little beyond what is presented in most undergraduate courses. They talk about isospin and hypercharge, the weak and strong forces, and grand unification. All very interesting stuff that satisfy the "particles of nature" part of the title more than the "quantum mechanics. The integration of abstract algebra, matrix mechanics, and differential equations is complete here, and if you let your guard down reading about quantum metaphysics, it will bit you in the ass. It did me. I had a harder time with these two chapters than Cohen's An Introduction to Hilbert Space and Quantum Logic (available at Walmart , apparently), which I read a few years ago. I won't look it up to see how many years ago because I don't want to get depressed.

So again, if you're looking for a memorable book on quantum mechanics, one that you'll grow into, pick up Anthony Sudbery's Quantum Mechanics and the Particles of Nature. Even if it did cost me two weeks wages when I was working at the drug store, I never regretted picking this gem up.