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Multi- Valued Fields In Condensed Matter, Electromagnetism , and Gravitation

Multi- Valued Fields In Condensed Matter, Electromagnetism , and Gravitation. Hagen Kleinert, FU BERLIN. Why Multivalued Fields ? Example : Ginzburg-Landau Theory. set. FALSE!. Chain Rule :. Indistinguishability. Correct Chain Rule :.

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Multi- Valued Fields In Condensed Matter, Electromagnetism , and Gravitation

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  1. Multi-Valued Fields In Condensed Matter, Electromagnetism, and Gravitation Hagen Kleinert, FU BERLIN

  2. WhyMultivalued Fields ?Example: Ginzburg-Landau Theory • set FALSE! Chain Rule:

  3. Indistinguishability Correct Chain Rule: In 1D, can be removed by going to covering group U(1) In >1D impossible

  4. Gauge Transformations Invariant: Axial Gauge No Go

  5. EXAMPLE FOR MULTIVALUED FIELD in 2D Solve:

  6. NOTE: Mother of Two Green Functions

  7. Example: Magnetostatics Recall:

  8. Now: GenerateMagnetic FieldbyMultivalued Gauge Transformations

  9. Magnetic Monopoles

  10. Minimal CouplingFrom Non- holonomic Gauge Transformations Thenactionchangesbysurfacetermsonly: Fornonholonomic Nontrivial

  11. SchrödingerEquation Momentum Solved by Use nonholonomic then with nonzero magnetic field

  12. Multivalued Description ofMagnetism Magnetic Field

  13. Action Gauge Invariance

  14. Integration byparts Integration of Omega Enforcedas Bianchi Idty: Double Gauge Theory: DefectCurrentConserv.:

  15. actionarisesfrom New Chain Rule In London (hydrodynamic) Limit ThusFormalismholdsfor superfluid helium!

  16. GC Sum Over LinescanbetransformedintoDisorder QFT ResultGinzburg-Landau Theoryof Superfluid Helium

  17. Order ofSupercontucting Transition in Ginzburg-Landau Theory Simple argument: ) Absorbphase angle (unitarygauge

  18. Fluctuationsofvector potential Integrated out cubicterm 1st-order transtion:

  19. Correct:

  20. Villain Model

  21. Relateto Result ) (recall Confirmedby Monte Carlo

  22. Double-Gauge QFT of Monopoles

  23. Changingthesurfaceisgaugetransformation

  24. Monopole Gauge Invariance Dirac QC:

  25. Quark Confinement Exchange electric magnetic DisorderTheoryofmagneticworldlines

  26. NontrivialGeometry fromNonholonomicCoordinateTransformations Burgers vectorb

  27. DISCLINATIONS  Frank Vector

  28. FUNDAMETALS:Universalityof FREE PARTICLE motion:

  29. Nonholonomic image of is Autoparallel InsteadofGeodesic

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

  31. LatticeDefectTheoryvsAbelian QED on Lattice Latticeformulation Define

  32. CURIOSITY: Induced Gravity Elastic Gauge Tfs: Canonical Form MomentumConservation Enforcedas Bianchi Idty: Double Gauge Theory

  33. Dual Representation

  34. BUT NEED

  35. Modify Elastic Action to andfurtherto FLOPPY CRYSTAL

  36. THE END ThankstoAxel and Wolfhard Ifyouwantotknowmore, readmynewbook MULTIVALUED FIELDS

  37. Volterra Construction Conservation Laws DefineEinstein Tensor DefineTorsion LinearizedBianchi Identity LinearizedFundamental Identity

  38. INTEGRABILITY CONDITIONS DefineCurvature Tensor: Thenaboveintegrabilityimplies: (linearizedBiachiidentitiy)

  39. General Coordinate Transformation Basis Tetrads Affine Connection

  40. Multivalued Basis Tetrads

  41. INTEGRABILITY CONDITIONS Bianchi Identities

  42. General , then Bianchi Identities Rewrite as Palatinitensor

  43. Gravitationalfieldversionofconservationlaws

  44. Minimal Coupling from Nonholonomic Coord. Tranfs. Holonomicvierbein transformingto nonholonomic Coordinates

  45. Multivalued infinitesimal coordinatetransformation

  46. INTEGRABILITY CONDITIONS Bianchi Identities

  47. Derivation from Nonholonomic Mapping Principlefor Dirac Electron Flat Space Local Lorentz Transformations

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