Hagen Kleinert, FU BERLIN THIRD QUANTIZATION

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

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

Gauge Transformations Axial Gauge Invariant Field Strength:

Simplest MULTIVALUED FIELD in 2D Solve:

NOTE: Mother of Two Important Green Functions

Application: Magnetostatics Recall:

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

Magnetic Monopoles Magnetic Monopoles

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

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

Multivalued Description of Magnetism Magnetic Field

Action Gauge Invariance

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

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!

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

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

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

Correct:

Villain Model

Relate to Result Confirmed by Monte Carlo (recall )

Double-Gauge QFT of Monopoles

Changing the surface is gauge transformation

Monopole Gauge Invariance Dirac QC:

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

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

Frank Vector  DISCLINATIONS

FUNDAMENTALS: Universality of FREE PARTICLE motion:

Nonholonomic image of is Autoparallel Instead of Geodesic

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

Lattice Defect Theory vs Abelian QED on Lattice Lattice formulation Define

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

Dual Representation

BUT NEED

Modify Elastic Action to and further to FLOPPY CRYSTAL

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

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

General Coordinate Transformation Basis Tetrads Affine Connection

Multivalued Basis Tetrads

INTEGRABILITY CONDITIONS Bianchi Identities

Rewrite as General, then Bianchi Identities Palatini tensor

Gravitational field version of conservation laws

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

Multivalued infinitesimal coordinate transformation

INTEGRABILITY CONDITIONS Bianchi Identities

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

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