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Hagen Kleinert, FU BERLIN

THIRD QUANTIZATION. Hagen Kleinert, FU BERLIN. Multi- Valued Quantum Field Theory In Condensed Matter, Electromagnetism , Quark Confinement , and Gravitation. Hagen Kleinert, FU BERLIN. Imagine : Single- Valued World. Multivalued World. Warm-up: Ginzburg-Landau Theory.

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Hagen Kleinert, FU BERLIN

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  1. THIRD QUANTIZATION Hagen Kleinert, FU BERLIN

  2. Multi-Valued Quantum Field Theory In Condensed Matter, Electromagnetism, Quark Confinement, and Gravitation Hagen Kleinert, FU BERLIN

  3. Imagine: Single-Valued World Multivalued World

  4. Warm-up: Ginzburg-Landau Theory • set FALSE! Chain Rule:

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

  6. Gauge Transformations Invariant Field Strength: Axial Gauge

  7. Simplest MULTIVALUED FIELD in 2D Solve:

  8. NOTE: Mother ofTwoImportant Green Functions

  9. Application: Magnetostatics Recall:

  10. Now: GenerateMagnetic FieldbyMultivalued Gauge Transformations ThinFlux Tube:

  11. Magnetic Monopoles

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

  13. SchrödingerEquation 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. Note: 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. Order ofSuperconducting Transition in Ginzburg-Landau Theory Simple argument: ) Absorbphase angle (unitarygauge

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

  21. Correct:

  22. Villain Model

  23. Relateto Result ) (recall Confirmedby Monte Carlo

  24. Double-Gauge QFT of Monopoles

  25. Changingthesurfaceisgaugetransformation

  26. Monopole Gauge Invariance Dirac QC:

  27. Quark Confinement Exchange electric magnetic addDisorderTheoryofmagneticworldlines Meissner eff arealaw

  28. Final Examle: NontrivialGeometry fromNonholonomicCoordinateTransformations Burgers vectorb

  29. DISCLINATIONS  Frank Vector

  30. FUNDAMENTALS:Universalityof FREE PARTICLE motion:

  31. Nonholonomicimageof isAutoparallel InsteadofGeodesic

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

  33. LatticeDefectTheoryvsAbelian QED on Lattice Latticeformulation Define

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

  35. Dual Representation

  36. BUT NEED

  37. Modify Elastic Action to andfurtherto FLOPPY CRYSTAL

  38. Ifyouwanttoknowmore, readmynewbook (only 25 EU) THIS IS ALL FOR TODAY

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

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

  41. General Coordinate Transformation Basis Tetrads Affine Connection

  42. Multivalued Basis Tetrads

  43. INTEGRABILITY CONDITIONS Bianchi Identities

  44. General , then Bianchi Identities Rewrite as Palatinitensor

  45. Gravitationalfieldversionofconservationlaws

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