This book deals with an original contribution to the hypothetical missing link unifying the two fundamental branches of physics born in the twentieth century, General Relativity and Quantum Mechanics. Namely, the book is devoted to a review of a "covariant approach" to Quantum Mechanics, along with several improvements and new results with respect to the previous related literature. The first part of the book deals with a covariant formulation of Galilean Classical Mechanics, which stands as a suitable background for covariant Quantum Mechanics. The second part deals with an introduction to covariant Quantum Mechanics. Further, in order to show how the presented covariant approach works in the framework of standard Classical Mechanics and standard Quantum Mechanics, the third part provides a detailed analysis of the standard Galilean space-time, along with three dynamical classical and quantum examples. The appendix accounts for several non-standard mathematical methods widely used in the body of the book.

Written in a quite formal mathematical way, by using the language of modern differential geometry Contains a genuine learning program including modern differential geometry, classical mechanics, and quantum mechanics Provides an appendix devoting to the exposition of some geometric methods

Autorentext
Josef Janyka received his PhD in Geometry in 1984 from Charles University in Prague. From 1983 to 1991 he completed several study stays at the St. Petersburg University, the Lomonosov Moscow State University and the University of Florence. Since 1978 he worked at Masaryk University in Brno, rst as an assistant professor and from 1988 as an associate professor. Since 2007 he is full Professor of Mathematics-Geometry at the Masaryk University in Brno. His primary research interests include the theory of natural and gauge-natural bundles and applications of di erential geometric methods in theoretical physics. Marco Modugno received his degree in Physics at University of Florence in 1966. Full professor of Mathematical Physics at Lecce University (1975-1979) and at Florence University (1979-2013). Currently, emeritus professor. Chief Editor of the "Journal of Geometry and Physics" (1982-2003), Editor of the Journal "Di erential Geometry and its Applications"(1992-2016). Coordinator of the research group "Geometry and Physics" at University of Florence (1980-2013). His primary research interests deal with geometric approach to Mathematical Physics including General Relativity and Quantum Mechanics.

Inhalt
Introduction.- Spacetime.- Galileian metric field.- Galileian gravitational field.- Galileian electromagnetic field.- Joined spacetime connection.- Classical dynamics.- Sources of gravitational and electromagnetic fields.- Fundamental fields of phase space.- Geometric structures of phase space.- Hamiltonian formalism.- Lie algebra of special phase functions.- Classical symmetries.- Quantum bundle.- Galileian upper quantum connection.- Quantum differentials.- Quantum dynamics.- Hydrodynamical picture of QM.- Quantum symmetries.- Quantum differential operators.- Quantum currents and expectation forms.- Sectional quantum bundle.- Feynman path integral.- Conclusions and further developments.- Examples.