Geometric and Algebraic Topological Methods in Quantum Mechanics

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

Авторы: Giachetta G., Mangiarotti L., Sardanashvily G.
Год издания: 2005
Издательство: WORLD SCIENTIFIC PUBL CO PTE LTD
Местоположение издательства: 5 TOH TUCK LINK, SINGAPORE,SINGAPORE, 596224
Объём: 703 страниц
ISBN: 981-256-129-3
Аннотация: Preface Contemporary quantum mechanics meets an explosion of diﬀerent types of quantization. Some of these quantization techniques (geometric quantization, deformation quantization, BRST quantization, noncommutative geometry, quantum groups, etc.) call into play advanced geometry and algebraic topology. These techniques possess the following main peculiarities. • Quantum theory deals with inﬁnite-dimensional manifolds and ﬁbre bundles as a rule. • Geometry in quantum theory speaks mainly the algebraic language of rings, modules, sheaves and categories. • Geometric and algebraic topological methods can lead to nonequivalent quantizations of a classical system corresponding to diﬀerent values of topological invariants. Geometry and topology are by no means the primary scope of our book, but they provide the most eﬀective contemporary schemes of quantization. At the same time, we present in a compact way all the necessary up to date mathematical tools to be used in studying quantum problems. Our book addresses to a wide audience of theoreticians and mathematicians, and aims to be a guide to advanced geometric and algebraic topological methods in quantum theory. Leading the reader to these frontiers, we hope to show that geometry and topology underlie many ideas in modern quantum physics. The interested reader is referred to extensive Bibliography spanning mostly the last decade. Many references we quote are duplicated in E-print arXiv (http://xxx.lanl.gov). With respect to mathematical prerequisites, the reader is expected to be familiar with the basics of diﬀerential geometry of ﬁbre bundles. For the sake of convenience, a few relevant mathematical topics are compiled in Appendixes. Contents. Preface. Introduction. 1. Commutative geometry. 1.1 Commutative algebra. 1.2 Diﬀerential operators on modules and rings. 1.3 Connections on modules and rings. 1.4 Homology and cohomology of complexes. 1.5 Homology and cohomology of groups and algebras. 1.6 Diﬀerential calculus over a commutative ring. 1.7 Sheaf cohomology. 1.8 Local-ringed spaces. 1.9 Algebraic varieties. 2. Classical Hamiltonian systems. 2.1 Geometry and cohomology of Poisson manifolds. 2.2 Geometry and cohomology of symplectic foliations. 2.3 Hamiltonian systems. 2.4 Hamiltonian time-dependent mechanics. 2.5 Constrained Hamiltonian systems. 2.6 Geometry and cohomology of K¨ahler manifolds. 2.7 Appendix. Poisson manifolds and groupoids. 3. Algebraic quantization. 3.1 GNS construction I. C∗-algebras of quantum systems. 3.2 GNS construction II. Locally compact groups. 3.3 Coherent states. 3.4 GNS construction III. Groupoids. 3.5 Example. Algebras of inﬁnite qubit systems. 3.6 GNS construction IV. Unbounded operators. 3.7 Example. Inﬁnite canonical commutation relations. 3.8 Automorphisms of quantum systems. 4. Geometry of algebraic quantization. 4.1 Banach and Hilbert manifolds. 4.2 Dequantization. 4.3 Berezin’s quantization. 4.4 Hilbert and C∗-algebra bundles. 4.5 Connections on Hilbert and C∗-algebra bundles. 4.6 Example. Instantwise quantization. 4.7 Example. Berry connection. 5. Geometric quantization. 5.1 Leafwize geometric quantization. 5.2 Example. Quantum completely integrable systems. 5.3 Quantization of time-dependent mechanics. 5.4 Example. Non-adiabatic holonomy operators. 5.5 Geometric quantization of constrained systems. 5.6 Example. Quantum relativistic mechanics. 5.7 Geometric quantization of holomorphic manifolds. 6. Supergeometry. 6.1 Graded tensor calculus. 6.2 Graded diﬀerential calculus and connections. 6.3 Geometry of graded manifolds. 6.4 Lagrangian formalism on graded manifolds. 6.5 Lagrangian supermechanics. 6.6 Graded Poisson manifolds. 6.7 Hamiltonian supermechanics. 6.8 BRST complex of constrained systems. 6.9 Appendix. Supermanifolds. 6.10 Appendix. Graded principal bundles. 6.11 Appendix. The Ne’eman–Quillen superconnection. 7. Deformation quantization. 7.1 Gerstenhaber’s deformation of algebras. 7.2 Star-product. 7.3 Fedosov’s deformation quantization. 7.4 Kontsevich’s deformation quantization. 7.5 Deformation quantization and operads. 7.6 Appendix. Monoidal categories and operads. 8. Non-commutative geometry. 8.1 Modules over C∗-algebras. 8.2 Non-commutative diﬀerential calculus. 8.3 Diﬀerential operators in non-commutative geometry 8.4 Connections in non-commutative geometry. 8.5 Connes’ non-commutative geometry. 8.6 Landsman’s quantization via groupoids. 8.7 Appendix. K-Theory of Banach algebras. 8.8 Appendix. The Morita equivalence of C∗-algebras. 8.9 Appendix. Cyclic cohomology. 8.10 Appendix. KK-Theory. 9. Geometry of quantum groups. 9.1 Quantum groups. 9.2 Diﬀerential calculus over Hopf algebras. 9.3 Quantum principal bundles. 10. Appendixes. 10.1 Categories. 10.2 Hopf algebras. 10.3 Groupoids and Lie algebroids. 10.4 Algebraic Morita equivalence. 10.5 Measures on non-compact spaces. 10.6 Fibre bundles I. Geometry and connections. 10.7 Fibre bundles II. Higher and inﬁnite order jets. 10.8 Fibre bundles III. Lagrangian formalism. 10.9 Fibre bundles IV. Hamiltonian formalism. 10.10 Fibre bundles V. Characteristic classes. 10.11 K-Theory of vector bundles. 10.12 Elliptic complexes and the index theorem. x Geometric and Algebraic Topological Methods in Quantum Mechanics. Bibliography. Index
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