Mathematical foundations of quantum statistical mechanics. Continuous systems. Transl. from the Russian by P. V. Malyshev and D. V. Malyshev (Q1900726)

This book one should consider as the second volume of the author's series on Statistical Mechanics of continuous systems. The first volume [\textit{V. I. Gerasimenko}, \textit{P. V. Malyshev} and \textit{D. Ya. Petrina} ``Mathematical foundations of classical statistical mechanics'' Gordon Breach (1989; Zbl 0728.46053)] was dedicated to classical continuous systems while the second one to their quantum counterpart. By the author's idea these two volumes should give a complete presentation of Statistical Mechanics of continuous systems from a common point of view. In chapter 1 the author proposes a brief, but sufficient for what follows, survey of Quantum Mechanics which gives a basis for introduction of the finite-volume reduced density matrices, Bogoliubov's chain equations for them and finally their representation in terms of Wiener integrals (Chapter 2). Using this representation the cluster properties of the reduced density matrices at low densities are considered in chapter 3. Chapter 4 gives the first acquaintance with two fundamental models of Quantum Statistical Mechanics (QSM): the BCS-Bogoliubov model of superconductivity (including the solution of this model by the Approximating Hamiltonian Method) and the Bogoliubov model of superfluidity. Then the author switches to the Green functions formalism (Chapter 5) illustrating it by solving the above models as well as by the proof of existence of the infinite-volume limit of Green functions at low densities. The most original is chapter 6 devoted to exact solution of the some fundamental models of QSM (BCS-Bogoliubov model, Bogoliubov model of superfluidity, Huang-Yang-Lattinger interacting Boson model and one-dimensional Frolich-Peierls electron-photon model) by a method invented by the author. It attempts to give a mathematical status to formal infinite-volume Hamiltonian as an operator in the space of translation-invariant functions well-adjusted for description of quantum clusters like, e.g., Cooper's pairs. This formalism reproduces the known solutions of the exactly solvable models mentioned above and gives an insight into the spectral analysis of the corresponding clustering Hamiltonians. The final chapter 7 is devoted to Bogoliubov's quasi-averages and the Bogoliubov inequality related to the famous \(1/q\) (square) singularities theorem which initiated a number of the Mermin-Wagner-type ``no-go'' theorem about the absence of the order parameter in low dimensional systems with continuous symmetries. Each chapter has as extensions references, historical and bibliographical remarks as well as mathematical appendices. The presentation in the book oscillates between paragraphs which are more rigorous than usual textbooks on QSM and parts which are written only on a formal mathematical level. As a consequence, in spite of motivation, not all of the messages in the book are properly formulated. In summary, this book gives a point of view of the author on how one would present QSM based essentially on Bogoliubov's contributions into the subject. In this sense it fills the existing gap in the textbook literature on QSM.