Unitary Group Representations in Physics, Probability, and Number Theory (Advanced Book Classics) by George W. MacKey (PDF)

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Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is a classic – someone should reprint it. Quantum theory is all about Hilbert spaces, so when you combine quantum theory with the all-important concept of symmetry, you get the concept of a unitary group representation on a Hilbert space. The fact that we can classify the unitary representations of our favorite symmetry groups is what makes physics what it is today! The periodic table, the classification of elementary particles, and so on – all these involve classifying unitary group representations. Mackey is a mathematician at Harvard who spent his life studying and developing this subject, he knows it inside and out… and in this book, he tells you what he knows.After 100 pages devoted to the general theory of unitary group representations, he tackles applications to ergodic theory, probability theory, the foundations of quantum mechanics, the quantum mechanics of one-particle systems, particle interactions, bound states and the structure of matter, the S-matrix, radiation, special relativity, quantum field theory, temperature, heat, thermodynamics, statistical mechanics and the many-body problem, Fourier analysis, and number theory, including Diophantine equations, algebraic number fields, and automorphic forms.Zounds! As I said, someone had better reprint this book. I learned a lot from it.

⭐This is an outstanding text in group representations and all the application areas mentioned in the title. It builds the subject up from group actions on sets, through semisimple and solvable groups, all the way to factors. Then it applies the subject to an amazing range of subjects. Why this book is out of print is baffling. For a subset of the material in this book, you can get a free PDF of Mackey’s article on the history and applications of harmonic analysis from the American Mathematical Society’s archive of the Bulletin of the AMS, published in 1980 (Vol. 3).

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