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Multivalued Fields and Ginzburg-Landau Theory: Insights into Magnetostatics and Quantum Theory

Explore the interplay of multivalued fields in the context of Ginzburg-Landau theory and its applications to magnetostatics. This work delves into the foundational aspects of gauge invariance, defect currents, and the implications of nonholonomic transformations. By revisiting key contributions from Thomson, Maxwell, Dirac, and Schrodinger, the author sheds light on the complexities of gauge transformations and their role in modern quantum field theory. The study connects superfluid helium dynamics with lattice defect theory, offering a unique perspective on induced gravity and elastic gauge transformations.

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Multivalued Fields and Ginzburg-Landau Theory: Insights into Magnetostatics and Quantum Theory

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  1. Gravity andDefects BERLIN ICRANet Pescara Hagen Kleinert, FU BERLIN & ICRANet Pescara

  2. Imagine: Single-Valued World Multivalued World

  3. Warm-up: Ginzburg-Landau Theory • set FALSE! Chain Rule: But thisisIsing –like!

  4. Jumps! Correct Chain Rule: In 1D, canberemovedbygoingtocoveringgroup U(1) In >1D impossible

  5. Vortex Gauge Field Invariant Field Strength: Axial Gauge

  6. Simplest MULTIVALUED FIELD in 2D Solve:

  7. NOTE: Mother ofTwoImportant Green Functions

  8. Correct Treatment:Ginzburg-Landau Theory • set Chain Rule: XY–like!

  9. ApplicationofMultivaluedFields toMagnetostatics Recall Thomson and Maxwell: W. Thomson (1824-1907) J.C.Maxwell (1831-1879)

  10. GenerateMagnetic FieldbyMultivalued Gauge Transformations ThinFlux Tube:

  11. Also: Magnetic Monopoles Paul Dirac (1902-1984)

  12. Derive: Minimal CouplingFrom Non-holonomicGauge Transformations J.L.Lagrange (1736-1819) Thenactionchangesbysurfacetermsonly: Fornonholonomic Nontrivial

  13. SchrödingerTheory E.Schrödinger (1887-1961) Momentum Solved by Use nonholonomic then with nonzero magnetic field

  14. Multivalued Description ofMagnetism Magnetic Field

  15. Action Gauge Invariance

  16. Integration byparts Integration of Omega Enforcedas Bianchi Identity: Double Gauge Theory: DefectCurrentConserv.:

  17. Recall: Action arises also from GL Theoryof superfluid He In London (hydrodynamic) Limit Thus same formalismholdsfor superfluid He!

  18. GC Sum Over LinescanbetransformedintoDisorder QFT Result: Ginzburg-Landau Theoryof Superfluid Helium

  19. Double-Gauge QFT of Monopoles

  20. Double-Gauge QFT of Monopoles

  21. Now : NontrivialGeometry fromNonholonomicCoordinateTransformations Burgers vectorb

  22. DISCLINATIONS  Frank Vector

  23. FUNDAMENTALS:Universalityof FREE PARTICLE motion:

  24. Nonholonomicimageof isAutoparallel E.Schrödinger (1887-1961) InsteadofGeodesic A.Einstein (1879-1955)

  25. QUANTUM THEORY:Trajectoryisfatfluctuationsausage! Tidalforces on wave packet ?

  26. LatticeDefectTheoryvsAbelian QED on Lattice Latticeformulation Define

  27. CURIOSITY: InducedGravity in `World Crystal´ Elastic Gauge Tfs: Canonical Form MomentumConservation Enforcedas Bianchi Idty: Double Gauge Theory

  28. Dual Representation

  29. BUT NEED

  30. Modify Elastic Action to andfurtherto FLOPPY CRYSTAL

  31. Ifyouwanttoknowmore, readmynewbook Only 38 Euro

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