This book couples a theory of multivalued fields with different applications of theoretical physics. The book is divided into twenty two chapters. First three chapters review some basic concepts of classical mechanics and the theory of single-valued fields, and also give a perceptive and condensed account of basic mathematical and physical tools of modern classical and quantum physics. Ch. 4 presents ideas for development of gravitation theory based on that the physical laws in Euclidean space can be transformed directly into spaces with curvature and torsion. For this, a field-theoretic technique invented by Dirac is generalized geometrically to introduce magnetic monopoles into electrodynamics. Ch. 5 discusses two examples of phase transitions in superfluids and superconductors, and also a role of multivalued fields in these phenomena. Dynamics of superfluids is discussed in Ch. 6 on the basis of their hydrodynamic description and rotation. Ch. 7 presents a mathematical description of electromagnetic effects on the dynamics of charged superfluids and superconductors in the framework of hydrodynamics and the corresponding physical theories by determining the main critical parameters for above two phenomena. Ch. 8 extents the theory of multivalued fields for magnetism developed in Ch. 4 to a full relativistic theory of charges and monopoles. Before than using the multivalued coordinate transformations for transformation theories in flat space and in spaces with curvature and torsion, Ch. 9 reviews multivalued mapping from ideal crystals to crystals with defects. In Ch. 10, the author considers the proliferation mechanism of dislocation and disclination lines leading to the melting of crystals, which is similar to the mechanism of vortex line proliferation at the critical temperature of phase transitions in superfluids and superconductors. Ch. 11 is devoted to the presentation of tensor fields of relativistic mechanics in curvilinear coordinates. This allows in Ch. 12 to state a quantum equivalence principle by using multivalued gauge transformations which violate Schwarz integrability conditions, as being plastic distortions of a world crystal. The spaces with torsion and curvature are introduced by employing the non-integrability of coordinate transformations and non-integrability of their derivatives. Ch. 13 discusses how to avoid the second-type non-integrability by using an embedding procedure. This procedure consists in the consideration of a higher-dimensional spacetime and mapping it into a sub-spacetime of the desired dimension. The presence of torsion in spacetime and antiparallel trajectories in these spaces are considered in Ch. 14. The obtained equations of motion for a particle, subjected to a gravitational field, depend on certain properties of the metric. Ch. 15 studies the effect of a gravitational massive object on the metric. For this, the components of the metric tensor are considered as dynamical variables, and an action principle is formulated to determine them. Ch. 16 discusses the coupling of relativistic fields to gravity on the basis of the multivalued mapping principle developed in Ch. 14. Ch. 17 shows how electrons and other particles of half-integer spin are coupled to gravity. In Ch. 18, it is shown that the canonical energy-momentum tensor and spin current density, introduced by the functional derivative of action, satisfy covariant conservation laws. Ch. 19 is devoted to study of gravitation of spinning matter as a gauge theory. The properties of torsion in gravity are theoretically investigated in Ch. 20. However, they are difficult to be verified experimentally due to the weakness of the coupling of torsion to the intrinsic spins of the fundamental constituents of celestial body. Ch. 21 considers the theory of teleparallelism proposed by Einstein by using a Bianchi identity for teleparallel spacetime. Finally, Ch. 22 presents an entirely different scenario in which the most important features of string theories are connected with Lorentz invariance at all energies in trans-Planckian regime.
In total, this excellent book on multivalued fields opens a new page in general study of the problems of condensed matter physics, electromagnetism and gravitation on the basis of a single approach. This comprehensive book may be very useful for students, graduate students and specialists working in the above scientific directions, and also to prepare special courses on phase transitions, quantum field theory, gravitational physics and differential geometry.