### Article

## A FUNCTIONAL INTEGRATION METHOD FOR QUANTUM SPIN SYSTEMS AND ONE-DIMENSIONAL LOCALIZATION

A number of problems in statistical physics can be reformulated in terms of a two-state system evolving in a random field. The corresponding evolution operator can be written in the form of time-ordered operator exponential. Functional formalism allows us to rewrite the latter as a product of usual matrix exponentials using a nonlinear change of functional integration variables. In this review I present this formalism applied to two physical systems the quantum Heisenberg magnet and one-dimensional quantum mechanics in a spatially random potential. First, I derive a representation of the partition function of a quantum Heisenberg ferromagnet as a functional integral over number valued fields (a real one and a complex one) free of constraints. The fields of integration as functions of time obey initial conditions instead of the usual periodic boundary conditions. This is a manifestation of the finite-dimensionality of the space of spin states. In the subsequent sections I study the one-dimensional localization problem. The change of functional integration variables gives simultaneously explicit expressions for the averaging weight and for the Green function of the stationary Schrödinger equation. It allows to compute density correlators of arbitrary orders. The generalization to the case of different energy correlators (Berezinskii-Gor’kov equations) is considered too. In the present review such technical points as regularizations of functional integrals and transformations are discussed in more details than in the original papers.