We will be approaching the subject of quantum theory from a different direction than the conventional one, starting with the role of symmetry and with the simplest possible finite dimensional quantum systems, systems which are purely quantum mechanical, with no classical analog. This means that the early discussion found in most physics textbooks is rather different from the one here. They will generally include the same fundamental principles described here, but often begin with the theory of motion of a quantized particle, trying to motivate it from classical mechanics. The state space is then a space of wavefunctions, which is infinite dimensional and necessarily brings some analytical difficulties.
Quantum mechanics is inherently a quite different conceptual structure than classical mechanics. The relationship of the two subjects is rather complicated, but it is clear that quantum mechanics cannot be derived from classical mechanics, so attempts to motivate it that way are unconvincing, although they correspond to the very interesting historical story of how the subject evolved. We will come to the topic of the quantized motion of a particle only in chapter 10, at which point it should become much easier to follow the standard books.
There are many good physics quantum mechanics textbooks available, aimed at a wide variety of backgrounds, and a reader of this book should look for one at an appropriate level to supplement the discussions here. One example would be [81], which is not really an introductory text, but it includes the physicist’s version of many of the standard calculations we will also be considering. Some useful textbooks on the subject aimed at mathematicians are [20], [41], [43], [57], and [91]. The first few chapters of [28] provide an excellent while very concise summary of both basic physics and quantum mechanics. One important topic we won’t discuss is that of the application of the representation theory of finite groups in quantum mechanics. For this as well as a discussion that overlaps quite a bit with the point of view of this book while emphasizing different topics, see [85]. For another textbook at the level of this one emphasizing the physicist’s point of view, see [107].
For the difficult issue of how measurements work and how classical physics emerges from quantum theory, an important part of the story is the notion of “decoherence”. Good places to read about this are Wojciech Zurek’s updated version of his 1991 Physics Today article [111], as well as his more recent work on “quantum Darwinism” [112]. There is an excellent book on the subject by Schlosshauer [75] and for the details of what happens in real experimental setups, see the book by Haroche and Raimond [44]. For a review of how classical physics emerges from quantum physics written from the mathematical point of view, see Landsman [54]. Finally, to get an idea of the wide variety of points of view available on the topic of the “interpretation” of quantum mechanics, there’s a volume of interviews [76] with experts on the topic.
The topic of Lie groups and their representation theory is a standard part of the mathematical curriculum at a more advanced level. As we work through examples in later chapters we’ll give references to textbooks covering this material.
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