A gauge invariant notion of a strong connection is presented and characterized. It is then used to justify the way in which a global curvature form is defined. Strong connections are interpreted as those that are induced from the base space of a quantum bundle. Examples of both strong and non-strong connections are provided. In particular, such connections are constructed on a quantum deformation of the two-sphere fibrationS
 2→RP
 2. A certain class of strongU
 q
 (2)-connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with theq-dependent hermitian metric. A particular form of the Yang-Mills action on a trivialU
 q
 (2)-bundle is investigated. It is proved to coincide with the Yang-Mills action constructed by A. Connes and M. Rieffel. Furthermore, it is shown that the moduli space of critical points of this action functional is independent ofq.

/pdf/strong-connections-on-quantum-principal-bundles-w2essk22bp.pdf

Strong connections on quantum principal bundles

A gauge invariant notion of a strong connection is presented and characterized. It is then used to justify the way in which a global curvature form is defined. Strong connections are interpreted as those that are induced from the base space of a quantum bundle. Examples of both strong and non-strong connections are provided. In particular, such connections are constructed on a quantum deformation of the fibration $S^2 -> RP^2$. A certain class of strong $U_q(2)$-connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with the q-dependent hermitian metric. A particular form of the Yang-Mills action on a trivial $U\sb q(2)$-bundle is investigated. It is proved to coincide with the Yang-Mills action constructed by A.Connes and M.Rieffel. Furthermore, it is shown that the moduli space of critical points of this action functional is independent of q.

Strong Connections on Quantum Principal Bundles

Preface 1. Introduction: a role for history Part I. Human and Artificial Mathematicians: 2. Communicating with automated theorem provers 3. Automated conjecture formation 4. The role of analogy in mathematics Part II. Plausibility, Uncertainty and Probability: 5. Bayesianism in mathematics 6. Uncertainty in mathematics and science Part III. The Growth of Mathematics: 7. Lakatos's philosophy of mathematics 8. Beyond the methodology of mathematical research programmes 9. The importance of mathematical conceptualisation Part IV. The Interpretation of Mathematics: 10. Higher dimensional algebra Appendix Bibliography Index.

/pdf/towards-a-philosophy-of-real-mathematics-55g4j0zop6.pdf

Towards a Philosophy of Real Mathematics

The category of Banach Lie-Poisson spaces is introduced and studied. It is shown that the category of W*-algebras can be considered as one of its subcategories. Examples and applications of Banach Lie-Poisson spaces to quantization and integration of Hamiltonian systems are given. The relationship between classical and quantum reduction is discussed.

Banach Lie-Poisson Spaces and Reduction

Acknowledgements List symbols, conventions, and formulary Part I. The Physical and Mathematical Environment: 1. The physical and mathematical environment Part II. Quantum Mechanics: 2. First lesson: gaussian integrals 3. Selected examples 4. Semiclassical expansion: WKB 5. Semiclassical expansion: beyond WKB 6. Quantum dynamics: path integrals and operator formalism Part III. Methods from Differential Geometry: 7. Symmetries 8. Homotopy 9. Grassmann analysis: basics 10. Grassmann analysis: applications 11. Volume elements, divergences, gradients Part IV. Non-Gaussian Applications: 12. Poisson processes in physics 13. A mathematical theory of Poisson processes 14. First exit time: energy problems Part V. Problems in Quantum Field Theory: 15. Renormalization 1: an introduction 16. Renormalization 2: scaling 17. Renormalization 3: combinatorics 18. Volume elements in quantum field theory Bryce DeWitt Part VI. Projects: 19. Projects Appendix A. Forward and backward integrals: spaces of pointed paths Appendix B. Product integrals Appendix C. A compendium of gaussian integrals Appendix D. Wick calculus Alexander Wurm Appendix E. The Jacobi operator Appendix F. Change of variables of integration Appendix G. Analytic properties of covariances Appendix H. Feynman's checkerboard Bibliography Index.

/pdf/functional-integration-action-and-symmetries-59sq0vx0yj.pdf

Functional Integration: Action and Symmetries

I. Operators with Index.- 1. Fredholm Operators.- A. Hierarchy of Mathematical Objects.- B. Concept of Fredholm Operator.- 2. Algebraic Properties. Operators of Finite Rank.- A. The Snake Lemma.- B. Operators of Finite Rank and Fredholm Integral Equations.- 3. Analytic Methods. Compact Operators.- A. Analytic Methods.- B. The Adjoint Operator.- C. Compact Operators.- D. The Classical Integral Operators.- 4. The Fredholm Alternative.- A. The Riesz Lemma.- B. Sturm-Liouville Boundary-Value Problem.- 5. The Main Theorems.- A. The Calkin Algebra.- B. Perturbation Theory.- C. Homotopy-Invariance of the Index.- 6. Families of Invertible Operators. Kuiper's Theorem.- A. Homotopies of Operator-Valued Functions.- B. The Theorem of Kuiper.- 7. Families of Fredholm Operators. Index Bundles.- A. The Topology of F.- B. The Construction of Index Bundles.- C. The Theorem of Atiyah-Janich.- D. Homotopy and Unitary Equivalence.- 8. Fourier Series and Integrals (Fundamental Principles).- A. Fourier Series.- B. The Fourier Integral.- C. Higher Dimensional Fourier Integrals.- 9. Wiener-Hopf Operators.- A. The Reservoir of Examples of Fredholm Operators.- B. Origin and Fundamental Significance of Wiener-Hopf Operators.- C. The Characteristic Curve of a Wiener-Hopf Operator.- D. Wiener-Hopf Operators and Harmonic Analysis.- E. The Discrete Index Formula.- F. The Case of Systems.- G. The Continuous Analogue.- II. Analysis on Manifolds.- 1. Partial Differential Equations.- A. Linear Partial Differential Equations.- B. Elliptic Differential Equations.- C. Where Do Elliptic Differential Operators Arise?.- D. Boundary-Value Conditions.- E. Main Problems of Analysis and the Index Problem.- F. Numerical Aspects.- G. Elementary Examples.- 2. Differential Operators over Manifolds.- A. Motivation.- B. Differentiable Manifolds - Foundations.- C. Geometry of C? Mappings.- D. Integration on Manifolds.- E. Differential Operators on Manifolds.- F. Manifolds with Boundary.- 3. Pseudo-Differential Operators.- A. Motivation.- B. "Canonical" Pseudo-Differential Operators.- C. Pseudo-Differential Operators on Manifolds.- D. Approximation Theory for Pseudo-Differential Operators.- 4. Sobolev Spaces (Crash Course).- A. Motivation.- B. Definition.- C. The Main Theorems on Sobolev Spaces.- D. Case Studies.- 5. Elliptic Operators over Closed Manifolds.- A. Continuity of Pseudo-Differential Operators.- B. Elliptic Operators.- 6. Elliptic Boundary-Value Systems I (Differential Operators).- A. Differential Equations with Constant Coefficients.- B. Systems of Differential Equations with Constant Coefficients.- C. Variable Coefficients.- 7. Elliptic Differential Operators of First Order with Boundary Conditions.- A. The Topological Interpretation of Boundary-Value Conditions (Case Study).- B. Generalizations (Heuristic).- 8. Elliptic Boundary-Value Systems II (Survey).- A. The Poisson Principle.- B. The Green Algebra.- C. The Elliptic Case.- III. The Atiyah-Singer Index Formula.- 1. Introduction to Algebraic Topology.- A. Winding Numbers.- B. The Topology of the General Linear Group.- C. The Ring of Vector Bundles.- D. K-Theory with Compact Support.- E. Proof of the Periodicity Theorem of R. Bott.- 2. The Index Formula in the Euclidean Case.- A. Index Formula and Bott Periodicity.- B. The Difference Bundle of an Elliptic Operator.- C. The Index Formula.- 3. The Index Theorem for Closed Manifolds.- A. The Index Formula.- B. Comparison of the Proofs: The Cobordism Proof.- C. Comparison of the Proofs: The Imbedding Proof.- D. Comparison of the Proofs: The Heat Equation Proof.- 4. Applications (Survey).- A. Cohomological Formulation of the Index Formula.- B. The Case of Systems (Trivial Bundles).- C. Examples of Vanishing Index.- D. Euler Number and Signature.- E. Vector Fields on Manifolds.- F. Abelian Integrals and Riemann Surfaces.- G. The Theorem of Riemann-Roch-Hirzebruch.- H. The Index of Elliptic Boundary-Value Problems.- J. Real Operators.- K. The Lefsehetz Fixed-Point Formula.- L. Analysis on Symmetric Spaces.- M. Further Applications.- IV. The Index Formula and Gauge-Theoretical Physics.- 1. Physical Motivation and Overview.- A. Classical Field Theory.- B. Quantum Theory.- 2. Geometric Preliminaries.- A. Principal G-Bundles.- B. Connections and Curvature.- C. Equivariant Forms and Associated Bundles.- D. Gauge Transformations.- E. Curvature in Riemannian Geometry.- F. Bochner-Weitzenbock Formulas.- G. Chern Classes as Curvature Forms.- H. Holonomy.- 3. Gauge-Theoretic Instantons.- A. The Yang-Mills Functional.- B. Instantons on Euclidean 4-Space.- C. Linearization of the "Manifold" of Moduli of Self-Dual Connections.- D. Manifold Structure for Moduli of Self-Dual Connections.- E. Gauge-Theoretic Topology in Dimension Four.- Appendix: What are Vector Bundles?.- Literature.- Index of Notation Parts I, II, III.- IV.- Index of Names/Authors.

Topology and analysis : the Atiyah-Singer index formula and gauge-theoretic physics

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Topology and analysis : the Atiyah-Singer index formula and gauge-theoretic physics