1. Hagen Kleinert, FU BERLIN THIRD QUANTIZATION

2. Multi-Valued Quantum Field Theory 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 FALSE! Chain Rule: set

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

6. Gauge Transformations Axial Gauge Invariant Field Strength:

7. Simplest MULTIVALUED FIELD in 2D Solve:

8. NOTE: Mother of Two Important Green Functions

9. Application: Magnetostatics Recall:

10. Now: Generate Magnetic Field by Multivalued Gauge Transformations Thin Flux Tube:

11. Magnetic Monopoles Magnetic Monopoles

12. Derive: Minimal Coupling From Non- holonomic Gauge Transformations Then action changes by surface terms only: For nonholonomic Nontrivial

13. Schrödinger Equation Momentum Use nonholonomic then Solved by with nonzero magnetic field

14. Multivalued Description of Magnetism Magnetic Field

15. Action Gauge Invariance

16. Defect Current Conserv.: Integration by parts Integration of Omega Enforced as Bianchi Identity: Double Gauge Theory:

17. Note: Action arises also Note: Action arises also from GL Theory of superfluid He from GL Theory of superfluid He In London (hydrodynamic) Limit Thus same formalism holds for superfluid He!

18. GC Sum Over Lines can be transformed into Disorder QFT Result: Ginzburg-Landau Theory of Superfluid Helium

19. Absorb phase angle (unitary gauge Order of Superconducting Transition in Ginzburg-Landau Theory in Ginzburg-Landau Theory ) )) ) Simple argument:

20. Integrated out cubic term 1st-order transtion: Fluctuations of vector potential

21. Correct:

22. Villain Model

23. Relate to Result Confirmed by Monte Carlo (recall )

24. Double-Gauge QFT of Monopoles

25. Changing the surface is gauge transformation

26. Monopole Gauge Invariance Dirac QC:

27. Quark Confinement add Disorder Theory of magnetic worldlines Exchange electricmagnetic Meissner eff area law

28. Final Examle: Nontrivial Geometry from Nonholonomic Coordinate Transformations Burgers vector b

29. Frank Vector  DISCLINATIONS

32. FUNDAMENTALS: Universality of FREE PARTICLE motion:

33. Nonholonomic image of is Autoparallel Instead of Geodesic

34. QUANTUM THEORY: Trajectory is fat fluctuation sausage!  Tidal forces on wave packet ?

35. Lattice Defect Theory vs Abelian QED on Lattice Lattice formulation Define

36. CURIOSITY: Induced Gravity in `World Crystal´ Elastic Gauge Tfs: Canonical Form Momentum Conservation Enforced as Bianchi Idty: Double Gauge Theory

37. Dual Representation

38. BUT NEED

39. Modify Elastic Action to and further to FLOPPY CRYSTAL

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44. Conservation Laws Define Torsion Linearized Fundamental Identity Define Einstein Tensor Linearized Bianchi Identity Volterra Construction

45. INTEGRABILITY CONDITIONS Define Curvature Tensor: Then above integrability implies: (linearized Biachi identitiy)

46. General Coordinate Transformation Basis Tetrads Affine Connection

47. Multivalued Basis Tetrads

48. INTEGRABILITY CONDITIONS Bianchi Identities

49. Rewrite as General, then Bianchi Identities Palatini tensor

50. Gravitational field version of conservation laws

51. Minimal Coupling from Nonholonomic Coord. Tranfs. Holonomic vierbein transforming to nonholonomic Coordinates

52. Multivalued infinitesimal coordinate transformation

53. INTEGRABILITY CONDITIONS Bianchi Identities

54. Derivation from Nonholonomic Mapping Principle for Dirac Electron Flat Space Local Lorentz Transformations

56. “ EXPERIMENTAL “ SITUATION Hydrogen Atom in Momentum Space Hydrogen Atom in Momentum Space Eliminates candidates