Table of contents

The present state of the gauge theory of high-temperature superconductivity in strongly-correlated two-dimensional spin systems is reviewed. Basic ideas on the statistics of elementary excitations in spatially two-dimensional systems are presented and are used in an analysis of the structure of the energy spectrum and the form of the wave function of a set of anyon quasiparticles in both the long-wave continuous and lattice limits. The continuous and lattice theories are used to classify the phase states, and the hierarchy of phase transitions is described in terms of the topological quantum field theory. Thermodynamic and electrodynamic properties of anyon systems are described, and the experimental consequences of the Chern–Simons theory of high-T_c superconductivity are discussed.

The theoretical and experimental studies on the transition to dynamical chaos in magnetic systems are reviewed. Both dissipative and Hamiltonian systems are discussed, with their fundamental scenarios of the transition to chaos. We discuss the models and characteristics of chaotic oscillations in magnetic materials upon parametric excitation of spin waves, in NMR masers, in the dynamics of spin chains, in the motion of a spin in an external alternating magnetic field, and in nonlinear NMR with dynamic shift of the precession frequency. An analysis of the problems of quantum chaos in spin systems is presented.

The physics of gaseous metals in the vicinity of the critical point is analyzed. A theory is presented for mixed states of valence electrons in percolation clusters of overlapping atoms, and is used for the interpretation of experiments. It is shown that the condensation of gaseous metals near the critical point has a plasma-like nature. The electrical and electrodynamic properties, as well as plasma oscillations and optical properties are analyzed in the region of the broadened metal-nonmetal transition. The cases of partial ionization at lower densities and multiple ionization at very high temperatures are discussed.

The informal aspects, arising in the interpretation of physical experiments, of the theory of probability and mathematical statistics are discussed. The conditions that verifying experiments must satisfy are presented and the role of heuristic (extralogical) assertions is analyzed using the example of mathematical expectation. The principal hypotheses implicit in experiments are enumerated: the principle of reproducibility ("the past will be repeated in the future"); the principle of reasonable sufficiency; and, the statistical principle ("better to predict something rather than nothing"). Considerable attention is devoted to Fisher and multisample confidence intervals. It is noted that Fisher confidence intervals are inconsistent. The arguments for introducing contrivances into practical calculations of probabilities are enumerated: incompleteness of any system of hypotheses; subjective estimates of probabilities; adjoining of statistical ensembles; nonstationariness and instability; rare phenomena; and, the use of classical probabilities and the law of large numbers. It is concluded that the relative frequency of appearance (empirical probability) is a "normal" physical quantity in the sense that it admits physical measurement. Its "abnormality" is manifested in the fact that it is burdened, more than other physical quantities, with conventions and hypotheses which must be specially checked (verified).