Geometric Formulation of Classical and Quantum Mechanics

Giachetta, G.; Mangiarotti, L.; Sardanashvily, G.

Авторы: Giachetta G., Mangiarotti L., Sardanashvily G.
Год издания: 2010
Издательство: WORLD SCIENTIFIC PUBL CO PTE LTD
Местоположение издательства: 5 TOH TUCK LINK, SINGAPORE,SINGAPORE, 596224
Объём: 392 страниц
ISBN: 10 981-4313-72-6
Аннотация: Preface Geometry of symplectic and Poisson manifolds is well known to provide the adequate mathematical formulation of autonomous Hamiltonian mechanics. The literature on this subject is extensive. This book presents the advanced geometric formulation of classical and quantum non-relativistic mechanics in a general setting of time-dependent coordinate and reference frame transformations. This formulation of mechanics as like as that of classical field theory lies in the framework of general theory of dynamic systems, Lagrangian and Hamiltonian formalism on fibre bundles. Non-autonomous dynamic systems, Newtonian systems, Lagrangian and Hamiltonian non-relativistic mechanics, relativistic mechanics, quantum non-autonomous mechanics are considered. Classical non-relativistic mechanics is formulated as a particular field theory on smooth fibre bundles over the time axis R. Quantum non-relativistic mechanics is phrased in the geometric terms of Banach and Hilbert bundles and connections on these bundles. A quantization scheme speaking this language is geometric quantization. Relativistic mechanics is adequately formulated as particular classical string theory of one-dimensional submanifolds. The concept of a connection is the central link throughout the book. Connections on a configuration space of non-relativistic mechanics describe reference frames. Holonomic connections on a velocity space define non-relativistic dynamic equations. Hamiltonian connections in Hamiltonian non-relativistic mechanics define the Hamilton equations. Evolution of quantum systems is described in terms of algebraic connections. A connection on a prequantization bundle is the main ingredient in geometric quantization. The book provides a detailed exposition of theory of partially integrable and superintegrable systems and their quantization, classical and quantum non-autonomous constraint systems, Lagrangian and Hamiltonian theory of Jacobi fields, classical and quantum mechanics with time-dependent parameters, the technique of non-adiabatic holonomy operators, formalism of instantwise quantization and quantization with respect to different reference frames. Our book addresses to a wide audience of theoreticians and mathematicians of undergraduate, post-graduate and researcher levels. It aims to be a guide to advanced geometric methods in classical and quantum mechanics. For the convenience of the reader, a few relevant mathematical topics are compiled in Appendixes, thus making our exposition self-contained. Contents. Preface. Introduction. 1. Dynamic equations. 1.1 Preliminary. Fibre bundles over R. 1.2 Autonomous dynamic equations. 1.3 Dynamic equations. 1.4 Dynamic connections. 1.5 Non-relativistic geodesic equations. 1.6 Reference frames. 1.7 Free motion equations. 1.8 Relative acceleration. 1.9 Newtonian systems. 1.10 Integrals of motion. 2. Lagrangian mechanics. 2.1 Lagrangian formalism on Q->R. 2.2 Cartan and Hamilton-De Donder equations. 2.3 Quadratic Lagrangians. 2.4 Lagrangian and Newtonian systems. 2.5 Lagrangian conservation laws. 2.5.1 Generalized vector fields. 2.5.2 First Noether theorem. 2.5.3 Noether conservation laws. 2.5.4 Energy conservation laws. 2.6 Gauge symmetries. 3. Hamiltonian mechanics. 3.1 Geometry of Poisson manifolds. 3.1.1 Symplectic manifolds. 3.1.2 Presymplectic manifolds. 3.1.3 Poisson manifolds. 3.1.4 Lichnerowicz-Poisson cohomology. 3.1.5 Symplectic foliations. 3.1.6 Group action on Poisson manifolds. 3.2 Autonomous Hamiltonian systems. 3.2.1 Poisson Hamiltonian systems. 3.2.2 Symplectic Hamiltonian systems. 3.2.3 Presymplectic Hamiltonian systems. 3.3 Hamiltonian formalism on Q->R. 3.4 Homogeneous Hamiltonian formalism. 3.5 Lagrangian form of Hamiltonian formalism. 3.6 Associated Lagrangian and Hamiltonian systems. 3.7 Quadratic Lagrangian and Hamiltonian systems. 3.8 Hamiltonian conservation laws. 3.9 Time-reparametrized mechanics. 4. Algebraic quantization. 4.1 GNS construction. 4.1.1 Involutive algebras. 4.1.2 Hilbert spaces. 4.1.3 Operators in Hilbert spaces. 4.1.4 Representations of involutive algebras. 4.1.5 GNS representation. 4.1.6 Unbounded operators. 4.2 Automorphisms of quantum systems. 4.3 Banach and Hilbert manifolds. 4.3.1 Real Banach spaces. 4.3.2 Banach manifolds. 4.3.3 Banach vector bundles. 4.3.4 Hilbert manifolds. 4.3.5 Projective Hilbert space. 4.4 Hilbert and C*-algebra bundles. 4.5 Connections on Hilbert and C*-algebra bundles. 4.6 Instantwise quantization. 5. Geometric quantization. 5.1 Geometric quantization of symplectic manifolds. 5.2 Geometric quantization of a cotangent bundle. 5.3 Leafwise geometric quantization. 5.3.1 Prequantization. 5.3.2 Polarization. 5.3.3 Quantization. 5.4 Quantization of non-relativistic mechanics. 5.4.1 Prequantization of T*Q and V*Q. 5.4.2 Quantization of T*Q and V*Q. 5.4.3 Instantwise quantization of V*Q. 5.4.4 Quantization of the evolution equation. 5.5 Quantization with respect to different reference frames. 6. Constraint Hamiltonian systems. 6.1 Autonomous Hamiltonian systems with constraints. 6.2 Dirac constraints. 6.3 Time-dependent constraints. 6.4 Lagrangian constraints. 6.5 Geometric quantization of constraint systems. 7. Integrable Hamiltonian systems. 7.1 Partially integrable systems with non-compact invariant submanifolds. 7.1.1 Partially integrable systems on a Poisson manifold. 7.1.2 Bi-Hamiltonian partially integrable systems. 7.1.3 Partial action-angle coordinates. 7.1.4 Partially integrable system on a symplectic manifold. 7.1.5 Global partially integrable systems. 7.2 KAM theorem for partially integrable systems. 7.3 Superintegrable systems with non-compact invariant submanifolds. 7.4 Globally superintegrable systems. 7.5 Superintegrable Hamiltonian systems. 7.6 Example. Global Kepler system. 7.7 Non-autonomous integrable systems. 7.8 Quantization of superintegrable systems. 8.1 The vertical extension of Lagrangian mechanics. 8.2 The vertical extension of Hamiltonian mechanics. 8.3 Jacobi fields of completely integrable systems. 9. Mechanics with time-dependent parameters. 9.1 Lagrangian mechanics with parameters. 9.2 Hamiltonian mechanics with parameters. 9.3 Quantum mechanics with classical parameters. 9.4 Berry geometric factor. 9.5 Non-adiabatic holonomy operator. 10. Relativistic mechanics. 10.1 Jets of submanifolds. 10.2 Lagrangian relativistic mechanics. 10.3 Relativistic geodesic equations. 10.4 Hamiltonian relativistic mechanics. 10.5 Geometric quantization of relativistic mechanics. 11. Appendices. 11.1 Commutative algebra. 11.2 Geometry of fibre bundles. 11.2.1 Fibred manifolds. 11.2.2 Fibre bundles. 11.2.3 Vector bundles. 11.2.4 Affine bundles. 11.2.5 Vector fields. 11.2.6 Multivector fields. 11.2.7 Differential forms. 11.2.8 Distributions and foliations. 11.2.9 Differential geometry of Lie groups. 11.3 Jet manifolds. 11.3.1 First order jet manifolds. 11.3.2 Second order jet manifolds. 11.3.3 Higher order jet manifolds. 11.3.4 Differential operators and differential equations. 11.4 Connections on fibre bundles. 11.4.1 Connections. 11.4.2 Flat connections. 11.4.3 Linear connections. 11.4.4 Composite connections. 11.5 Differential operators and connections on modules. 11.6 Differential calculus over a commutative ring. 11.7 Infinite-dimensional topological vector spaces. Bibliography. Index
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