This book originated from a series of lecture notes (prepared for courses given to students of mathematical physics at TU Berlin) and the Habilitationsschrift by the same author.

It develops the basic mathematical tools needed for a description of quantum many-body systems: the method of second quantization (rewriting of the many-body Hamiltonians for electron systems in terms of annihilation and creation operators), perturbation theory (expansion of the exponential of the perturbed Hamiltonian into the corresponding Wick series), and the concept of “Grassmann integration” of anticommuting variables and Gaussian measures (with respect to bosonic or Grassmann, i.e. commuting or anticommuting, variables, therefore giving a unified approach, with the same combinatorics, to determinants, permanents, Pfaffians and unsigned sums over pairings, all these objects appearing as expectations of suitable monomials).

Then some more physical themes are treated in the rest of the book: the bosonic functional integral representation (obtained from the Hubbard-Stratonovich transformation of the Grassmann integral representation), the BCS theory of superconductivity, an introduction to the fractional quantum Hall effect, Feynman diagrams, renormalization group methods, and re-summation of Feynman diagrams. The book concludes with a list of the author's favourite unsolved “millenium” problems. The presentation is mathematically rigorous, where possible. The author’s aim was to create a book containing enough motivation and enough mathematical details for those interested in this advanced and important field of contemporary mathematical physics.

Reviewer:

mzahr