Operator algebras in statistical mechanics and noncommutative probability theory

The fundamental notions of statistical mechanics of quantum spin systems are introduced. A survey of the main properties of the states satisfying the Kubo–Martin–Schwinger boundary conditions is given. The problem of describing the invariant states and the first integrals for the multidimensional Heisenberg model is solved. A central limit theorem of noncommutative probability theory and a noncommutative analog of the individual ergodic theorem are formulated and proved. The asymptotics of the distribution of the eigenvalues of the multiparticle Schrödinger operator is studied.