Book: Quantum Mechanics - Symmetries, 2nd edition, 15 chapters, 496 pages Scope of the book: applications of group theory in elementary particle physic (no field theory!) Reader: PhD student in physics, I am a beginner in that area, this is my first book in symmetries and Lie groups. My evaluation: The math sections in the book give u some basic notion of Lie groups but are NOT sufficient to fully understand the logic behind the scene everywhere. My advice is to read some good book in Lie groups in advance. The strongest feature of the book is its richnes of examples and solved exercises both in group theory and in its application to particle physics. You can learn a lot of analytical 'tricks' from the solutions. At the same time the text is full of small errors (signs, indeces, equation numbers, misprints). They are easy to detect and fun to debug and keep you concentrated while debugging. My main objection is that very often the logic in the text remains hidden, broken or fuzzy. Sometimes they prove some statement but at the end you can't tell what was actually proven or under what conditions that proof is valid, what facts it is derived from, does it rely on implicit assumptions or it's generaly true. As a consequence of that you are not sure if you can apply the statement for a situation that is not exactly the one discussed in the book. Sometimes it's hard to tell if they are talking about a necessary of sufficient condition or both. Or they, having something in mind that you don't know about, make some sudden assumption and you wonder why. Some concepts are not defined sharply from the begining but instead the authors use fussy definitions and define them much later (example: tensor product of multiplets and its reduction is defined understandably in chapter 10 but is used all the time before that). The explanations of the algebra in the examples and exercises is also not the best since in many cases I see a more logical, organized and understandable way to explain it to the reader. Also in some cases the book gives just the algebra without giving the reader the more fundamental cause for some fact(example: in exercise 8.3 page 255 they have two matrices connected by a similarity transformation, they prove with some algebra that the eigenvalues remain the same but don't tell you that's always the case with similarity transformations). To my opinion the authors have to a lot of work to do to make the logic fully explicit and understandable to the reader everywhere in the text. Without that, the book can be regarded as a nice collection of solved examples and exercises in group theory and particle physics. I give that book 3 out of 5 stars and hope that the other volumes of the sequence don't have that flaw. Contents of the book: chap1: symmetries in classical physics, Noether's theorem, symmetries in quantum mechanics and their generators: momentum, angular momentum, energy and spin operators chap2: angular momentum algebra; irreducible representations of SO(3); addition of angular momenta; Clebsh-Gordon coefficients
chap3: Lie groups, generators, Lie algebra; Casimir operators and Racah theorem; multiplets;
chap4: enumeration of the multiplets through eigenvalues of Casimir operators; energy degeneracy within a multiplet; two or more commuting symmety groups
chap5: neutron, proton doublet; isospin SU(2) symmetry; pion triplet; adjoint representation of Lie algebra
chap6: charge Q; hypercharge Y; baryons, antibaryons, baryon resonances; T3-Y diagrams;
chap7: U(n) and SU(n) groups; generators, Lie algebra of SU(3); subalgebras of SU(3) and shift operators; dimensions of SU(3) multiplets D(p,q);
chap8: smallest non-trivial representations of SU(3), quarks; meson multiplets; tensor product of multiplets and their reduction; Gell-Mann-Okubo mass formula; quark models with spin added, SU(6); wave functions construction, proton, neutron, baryon decuplet, baryon octet; mass formula in SU(6);
chap9: permutation group Sn, identical particles; Young diagrams; dimensions of irreducible Sn representations; connection to SU(n) multiplets; dimensions of SU(n); decompositions of SU(n) multiplet into SU(n-1) multiplets; decomposition of tensor product of multiplets with Young diagrams;
chap10: group characters; schur first and second lemma; orthogonality relations of characters of discrete finite groups; reduction of reducible representations; continuous, compact groups, group integration; integration over unitary groups; group characters of U(n); quark-gluon plasma example;
chap11: charm, SU(4), group generators; smallest non-trivial representations of SU(4), [4] and [4bar]; decomposition of tensor products of SU(4) multiplets; OZI rule for suppressing reactions; meson and baryon multiplets, SU(3) content; potential model of charmonium;SU(4)[with spin SU(8)] mass formula;
chap12: weight operators, standard Cartan-Weyl basis of a semi-simple Lie algebra; root vectors; graphic representations of root vectors and Lie algebras; simple roots and Dynkin diagrams;
chap13: space reflection (parity); time reversal; antilinear operators, complex conjugate operator K, antiunitary operator; general form of time reversal operator in coordinate representation for particle with spin;
chap14: classical hygrogen atom constants of motion: energy, angular momentum, Runge-Lenz vector; corresponding quantum constants of motion (operators), their algebra and group SO(4)- dynamical symmetry; decoupling of the SO(4) algebra into two SO(3) algebras and determination of the energy eigenvalues (Pauli method i guess); classical and quantum isotropic oscillator;
chap15: compact and noncompact Lie groups; group SU(p,q); group SO(p,q); generators of SO(2,1), infinitesimal operators, Casimir operators; non-compactness of SO(2,1) and its infinite dimensional irreducible unitary representations; application of SO(2,1) representations to scattering problems; |