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BERLIN ICRANet Pescara Gravity and Defects

Hagen Kleinert, FU BERLIN & ICRANet Pescara

Imagine: Single-Valued World

Multivalued World

Warm-up: Ginzburg-Landau TheoryFALSE!

Chain Rule:

set

But this is Ising like!

Jumps!

In 1D, can be removed by going to covering group U(1)

In >1D impossibleCorrect Chain Rule:

Vortex Gauge Field

Axial Gauge

Invariant Field Strength:Simplest MULTIVALUED FIELD in 2D

Solve:NOTE:Mother of Two Important Green Functions

Correct Treatment: Ginzburg-Landau Theory

Chain Rule:

set

XYlike!

Application of Multivalued Fields to Magnetostatics

Recall Thomson and Maxwell:

W. Thomson(1824-1907)J.C.Maxwell(1831-1879)

Generate Magnetic Fieldby Multivalued Gauge Transformations

Thin Flux Tube: Also: Magnetic Monopoles

Paul Dirac(1902-1984) Derive: Minimal Coupling From Non-holonomic Gauge Transformations

Then action changes by surface terms only:For nonholonomic Nontrivial

J.L.Lagrange(1736-1819) Schrdinger Theory

MomentumUse nonholonomic

then

Solved by

with nonzero magnetic field

E.Schrdinger (1887-1961)Multivalued Description of Magnetism

Magnetic Field Action

Gauge Invariance

Defect Current Conserv.:Integration by partsIntegration of OmegaEnforced as Bianchi Identity:Double Gauge Theory:

Recall: Action arises also

from GL Theory of superfluid He

In London (hydrodynamic) LimitThus same formalism holds for superfluid He!

GC Sum Over Lines can be transformed intoDisorder QFT

Result: Ginzburg-Landau Theoryof Superfluid Helium

Double-Gauge QFT of Monopoles

Double-Gauge QFT of Monopoles

Now : 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

E.Schrdinger (1887-1961) A.Einstein (1879-1955)QUANTUM THEORY:Trajectory is fat fluctuation sausage! Tidal forces on wave packet

?

Lattice Defect Theoryvs Abelian QED on Lattice

Lattice formulation

Define

CURIOSITY: Induced Gravity in `World Crystal

Elastic Gauge Tfs:Canonical FormMomentum ConservationEnforced as Bianchi Idty:Double Gauge Theory

Dual Representation

BUT NEED

Modify Elastic Action toand further to

FLOPPY CRYSTAL

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