The Theory of Groups and Quantum Mechanics

by Hermann Weyl

This book is devoted to the consistent and systematic application of group theory to quantum mechanics. Beginning with a detailed introduction to the classical theory of groups, Dr. Weyl continues with an account of the fundamental results of quantum physics. There follows a rigorous investigation of the relations holding between the mathematical and physical theories.
Topics covered unitary geometry, quantum theory (Schrödinger's wave equation, transition probabilities, directional quantization, collision phenomena, Zeeman and Stark effects); groups and their representations (sub-groups and conjugate classes, linear transformations, rotation and Lorentz groups, closed continuous groups, invariants and covariants, Lie's theory); applications of group theory to quantum mechanics (simple state and term analysis, the spinning electron, multiplet structure, energy and momentum, Pauli exclusion principle, problem of several bodies, Maxwell-Dirac field equations, etc.); the symmetric permutation group; and algebra of symmetric transformation (invariant sub-spaces in group and tensor space, sub-groups, Young's symmetry operators, spin and valence, group theoretic classification of atomic spectra, branching laws, etc).
Throughout, Dr. Weyl emphasizes the "reciprocity" between representations of the symmetric permutation group and those of the complete linear group. His simplified treatment of "reciprocity," the Clebsch-Gordan series, and the Jordan-Hölder theorem and its analogues, has helped to clarity these and other complex topics.

Genres: Physics, Mathematics, Science, Nonfiction, Quantum Mechanics, Textbooks

464 pages, Paperback

First published January 1, 1931

About the author

Hermann Klaus Hugo Weyl (9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski. His research has had major significance for theoretical physics as well as purely mathematical disciplines including number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years.

Weyl published technical and some general works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. While no mathematician of his generation aspired to the 'universalism' of Henri Poincaré or Hilbert, Weyl came as close as anyone. Michael Atiyah, in particular, has commented that whenever he examined a mathematical topic, he found that Weyl had preceded him (The Mathematical Intelligencer (1984), vol.6 no.1).

Source: http://en.wikipedia.org/wiki/Hermann_...

Reviews

[Manny]

Finding this book extremely challenging, I looked around on the web for background material. After a few minutes, I discovered the following interview, given by the notoriously laconic Dirac to a Wisconsin journalist in 1929:

JOURNALIST: And now I want to ask you something more: They tell me that you and Einstein are the only two real sure-enough high-brows and the only ones who can really understand each other. I won’t ask you if this is straight stuff for I know you are too modest to admit it. But I want to know this – Do you ever run across a fellow that even you can’t understand?

DIRAC: Yes.

JOURNALIST: This will make a great reading for the boys down at the office. Do you mind releasing to me who he is?

DIRAC: Weyl.

Weyl's book was published in 1928. I feel better.

[Erik]

Peter Woit's book Quantum Theory, Groups and Representations (2016) is very helpful in understanding just what Weyl achieved here (in 1928!). Love it or hate it, but once again physics turns out to be just a concrete realization of mathematics. PS, the vocabulary in this edition is a real pain as well.