We review the results of applying the statistical field-theoretic approach to the problem of fully developed turbulence in nonrelativistic three-dimensional magnetohydrodynamics (MHD), which have been obtained over the past forty years. The review covers both general aspects of the physics of MHD turbulence and the necessary mathematical machinery of statistical field theory, including elements of renormalization theory and the renormalization-group (RG) method. The approach is illustrated using a stochastic model of stationary, locally homogeneous, fully developed three-dimensional MHD turbulence in the general case of a medium with broken spatial parity (helical MHD). In this model, RG techniques make it possible to establish the existence of several infrared-stable scaling regimes and to calculate the critical dimensions of various composite operators, the infrared asymptotics of correlation functions, and the amplitude factors in scaling laws, as well as to incorporate the effects of compressibility, anisotropy, etc.
For an important class of helical MHD systems, the field-theoretic approach provides an elegant formulation of the fundamental problem of large-scale turbulent dynamo action — namely, the generation of a large-scale magnetic field ⟨b⟩ = B (where b denotes magnetic fluctuations) at the expense of the energy of turbulent fluctuations — via the decay of the initial unstable vacuum state ⟨b⟩ = 0 as a result of dynamical spontaneous symmetry breaking in the spirit of the Coleman-Weinberg mechanism, followed by stabilization of the theory in the vicinity of the new ground state ⟨b⟩ = B (the dynamo regime). The field-theoretic formulation we developed, together with a generalization of the standard Feynman diagrammatic technique to the dynamo regime, not only makes it possible to treat within a unified framework the existing theoretical approaches to helical magnetohydrodynamics (kinematic MHD, large-scale dynamo theory), but also extends the RG formalism to the dynamo regime, which — unlike closure procedures still common in dynamo theory — is particularly well suited for studying statistically stationary turbulent states. The richness of MHD physics in the dynamo regime is illustrated both in the emergence of new effects (Goldstone-type corrections to Alfvén waves, anisotropic corrections associated with the transport of the large-scale field) and in the theoretically predicted strong dependence of the magnetic energy-spectrum slope on the degree of mirror-symmetry breaking.
General Relativity (GR) was created in November 1915, and since its creation this theory has undergone many tests. The first realistic cosmological models were proposed in the works of Friedmann, written in the 1920s. For a long time, Friedmann’s cosmological works were actually banned in the Soviet Union due to philosophical reasons, since the models where the birth and evolution of the Universe occurs were considered ideologically unacceptable. Due to great achievements in relativity and cosmology and due to increasing interest in these branches of science in the last decades, we recall the development of relativistic astrophysics and contribution of Russian researchers to these studies. Since one of the world leaders in physical cosmology A. A. Friedmann passed away in September 1925, it is reasonable to outline the main achievements of physical cosmology over the past 100 years. We also discuss observational and theoretical achievements in confirmations of relativistic observational predictions for black holes, including the closest supermassive black hole in our Galactic Center. We outline the evolution of black hole shadow from the purely theoretical concept to observable quantities for supermassive black holes in Sgr A* and M87*.