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FIBRE BUNDLE FORMULATION OF NONRELATIVISTIC QUANTUM MECHANICS IV: MIXED STATES AND EVOLUTION TRANSPORT'S CURVATURE

Department Mathematical Modeling, Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Boul. Tzarigradsko chaussée 72, 1784 Sofia, Bulgaria
International Journal of Modern Physics A (Impact Factor: 1.13). 01/2012; 17(02). DOI: 10.1142/S0217751X02005669
Source: arXiv

ABSTRACT We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or section along paths in this bundle. The evolution of a pure state is determined through the bundle (analog of the) Schrödinger equation. Now the dynamical variables and density operators are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this work. The present fourth part of this series is devoted mainly to the fibre bundle description of mixed quantum states. We show that to the conventional density operator there corresponds a unique density lifting of paths for which the corresponding equations of motion are derived. It is also investigated the bundle description of mixed quantum states in the different pictures of motion. We calculate the curvature of the evolution transport and prove that it is curvature free iff the values of the Hamiltonian operator at different moments commute.

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    ABSTRACT: 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
    01/2005; World Scientific.
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    ABSTRACT: We point out how some mathematically incorrect passages of Ref. 1 can be formulated in a rigorous way. The fibre bundle approach to quantum mechanics of Refs. 2-6 is compared with the one contained in loc. cit.
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    ABSTRACT: 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 fi�bre 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 fi�eld 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 con�figuration space of non-relativistic mechanics describe reference frames. Holonomic connections on a velocity space defi�ne non-relativistic dynamic equations. Hamiltonian connections in Hamiltonian non-relativistic mechanics defi�ne 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 fi�elds, classical and quantum mechanics with time-dependent parameters, the technique of non-adiabatic holonomy operators, formalism of instantwise quantization and quantization with respect to diff�erent 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 di�fferent 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 fi�bre bundles. 11.2.1 Fibred manifolds. 11.2.2 Fibre bundles. 11.2.3 Vector bundles. 11.2.4 Affi�ne bundles. 11.2.5 Vector �fields. 11.2.6 Multivector fi�elds. 11.2.7 Diff�erential forms. 11.2.8 Distributions and foliations. 11.2.9 Di�fferential 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 Diff�erential operators and diff�erential equations. 11.4 Connections on fi�bre bundles. 11.4.1 Connections. 11.4.2 Flat connections. 11.4.3 Linear connections. 11.4.4 Composite connections. 11.5 Diff�erential operators and connections on modules. 11.6 Diff�erential calculus over a commutative ring. 11.7 In�finite-dimensional topological vector spaces. Bibliography.
    11/2010; World Scientific.

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