For the last few years most of my time has been spent working on writing a textbook, with the current title Quantum Theory, Groups and Representations: An Introduction. The book is based on a year-long course that I’ve taught twice, based on the concept of starting out assuming little but calculus and linear algebra, and developing simultaneously basic ideas about quantum mechanics and representation theory. The first half of the course stuck to basic non-relativistic quantum mechanics, while the second introduced free quantum field theories and the relativistic case. By the end, the idea is to bring the reader to the point of having some appreciation of the main elements of the Standard Model, from a perspective emphasizing the representation theory structures that appear.
The discussion of quantum field theory I think is rather different than that of other textbooks, taking a Hamiltonian point of view, rather than the Lagrangian/path integral one in which most physicists are now trained (myself included). One basic idea was to try and work out very carefully the quantization of a finite-dimensional phase space in all its representation-theoretic glory, with the idea that free quantum field theories could then be developed as a straightforward extension to the case of taking solutions of a field equation as phase space. While this point of view on quantum field theory is fairly well-known, writing up the details turned out to be a lot more challenging than I expected.
As part of this, the book attempts to carefully distinguish mathematical objects that usually get identified by physicist’s calculational methods. In particular, phase space and its dual space are distinguished, and the role of complex numbers and complexification of real vector spaces receives a lot of attention.
At the same time, the book is based on a relatively simple philosophical take on what the fundamental structures are and how they are grounded in representation theory. From this point of view, free relativistic quantum field theories are based on starting with an identification of irreducible representations of the space-time symmetry group using the Casimir operator to get a wave-equation. This provides a single-particle theory, with the quantum field theory then appearing as its “second quantization”, which is a metaplectic (bosonic case) or spinor (fermionic case) representation. These are some of the specifics behind the grandiose point of view on how mathematics and physics are related that I described here. For some indications of further ideas needed to capture other aspects of the Standard Model, there’s this that I wrote long ago, but which now seems to me hopelessly naive, in need of a complete rethinking in light of much of what I’ve learned since then.
The manuscript is still not quite finished, and comments are extremely welcome. While several people have already been very helpful with this, few have been willing to face the latter chapters, which I fear are quite challenging and in need of advice about how to make them less so. The current state of things is that what remains to be done is
- A bit more work on the last chapters.
- A rereading from the start, bringing earlier chapters in line with choices that I made later, and addressing a long list of comments that a few people have given me.
- An old list of problems (see here) needs to be edited, with more problems added.
- I need to find someone to make professional drawings (there are some funds for this).
- An index is needed.
Optimistically I’m hoping to have this mostly done by the end of the summer. Springer will be publishing the book (my contract with them specifies that I can make a draft version of the book freely available on my web-site), and I assume it will appear next year. To be honest, I’m getting very tired of this project, and looking forward to pursuing new ideas and thinking about something different this fall.
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