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Friday, 11 November 2011

Review on our book "Geometric Formulation of Classical and Quantum Mechanics" in Mathematical Reviews

G.Giachetta, L.Mangiarotti, G.Sardanashvily
Geometric formulation of classical and quantum mechanics. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. xii+392 pp. ISBN: 978-981-4313-72-8; 981-4313-72-6

Whereas most textbooks on the differential geometrical approach to classical and quantum mechanics are concerned with the case of autonomous (i.e., time-independent) systems, the present book addresses the case of time-dependent mechanical systems. Except for chapter 10, which explicitly deals with the relativistic case, the treatment is confined to non-relativistic mechanics. The extended configuration space of a time-dependent system is taken to be a fibre bundle Q over R, the time axis, and the corresponding velocity space is the first jet bundle JQ. The resulting description of non-relativistic mechanics becomes covariant, but not invariant under bundle transformations, i.e., time-dependent coordinate and reference frame transformations.
   The first chapter starts with some general preliminaries about fibre bundles, jet bundles, connections and the notions of first- and second-order dynamic equations. After the definition of a reference frame in terms of a connection on the configuration bundle, attention is paid, among other things, to the Newtonian formulation of time-dependent mechanics. Chapters 2 and 3 then deal with the Lagrangian and Hamiltonian description of a time-dependent non-relativistic system, respectively. The Lagrangian formulation is based on the variational bicomplex and the first variational formula and, besides the classical Lagrange equations of motion, the Cartan equations and the Hamilton-De Donder equations are also considered within this framework. A further topic that is discussed is the connection between the conservation laws of Lagrangian systems and variational symmetries, according to Noether's theorem. The Hamiltonian formulation of non-relativistic mechanics is developed on the vertical cotangent bundle V*Q of the configuration bundle Q->R, and it is shown that to any Hamiltonian system on V*Q there corresponds an equivalent autonomous symplectic Hamiltonian system on T*Q. The connections between the Lagrangian and Hamiltonian formulations of time-dependent mechanics are also investigated.
   Chapters 4 to 6 are devoted to the quantization of time-dependent mechanical systems. In chapter 4, a geometric framework for non-relativistic quantum mechanics is presented in terms of Banach and Hilbert manifolds and locally trivial Hilbert and C*-algebra bundles. A quantization scheme in the spirit of geometric quantization is then developed in chapter 5. Chapter 6 studies the geometric quantization of Hamiltonian systems with time-dependent constraints.
   In chapter 7, completely integrable, partially integrable and superintegrable Hamiltonian systems are treated in a general setting of invariant submanifolds which need not be compact. Using appropriate action-angle coordinates, the geometric quantization of completely integrable and superintegrable Hamiltonian systems is discussed. In chapter 8, the vertical extension of a mechanical system is considered from the configuration bundle Q->R to the vertical tangent bundle VQ->R, and the Jacobi fields of the Lagrange and the Hamilton equations of the system are investigated. It is shown, for instance, that the Jacobi fields of a completely integrable Hamiltonian system make up a completely integrable system in twice the number of degrees of freedom, whereby the additional first integrals characterize the relative motion. Chapter 9 deals with mechanical systems with time-dependent parameters. The Lagrangian and Hamiltonian description is analysed, treating the parameters at the same level of the dynamical variables. Next, the geometric quantization of these systems is studied.
   Leaving the non-relativistic setting, chapter 10 is concerned with the description of relativistic mechanics, both Lagrangian and Hamiltonian, and the geometric quantization of a relativistic mechanical system is discussed. Finally, chapter 11 contains several appendices, devoted to various mathematical topics which complement the main treatment, making it somewhat more self-contained (e.g., commutative algebras, geometry of fiber bundles, jet manifolds, connections, differential operators on modules, etc.).
   Although this book is addressed to a wide audience of mathematicians and theoretical physicists, even at an (advanced) undergraduate level, in my opinion it will primarily be appreciated by more experienced researchers who already have some acquaintance with the geometric approach to classical and quantum mechanics.

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (WS 2010)

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