Low-temperature quasi-classical approximation for quantum macroscopic phenomena

Quantum systems with two distinct time scales for “fast” (quantum) and “slow” (quasi-classical) degrees of freedom with the respective frequencies $\Omega$ and $\omega\ll\Omega$ are considered. A general path-integral representation for a partition function involves ordinary coherent states for Bose and Fermi degrees of freedom as well as generalized coherent states for a nontrivial algebra of variables. Path-integral averaging over “fast” variables results in a nonlocal (in imaginary time) effective action for a “slow” (quasi-classical) variable, which becomes local in the low-temperature approximation. This low-temperature adiabatic path-integral approach, expressed by the inequality $\beta\Omega\gg1\gg\omega/\Omega$, yields an appropriate basis for a subsequent quasi-classical description. Theories of three complex condensed matter systems are developed with the “slow” and “fast” subsystems represented by localized and band electrons in the lattice Anderson model, by condensate and noncondensate bosons in the Bogoliubov model (with broken translational symmetry), and by soft-mode phonons and electrons induced in large molecules by an electron–phonon optical transition. The corresponding quantum macroscopic phenomena are the Kondo-type reorganization of the spectrum of correlated electrons, Bose condensation in nonuniform media, and chaotization of the phonon spectrum in molecules.