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Gravity and Defects. BERLIN ICRANet Pescara. Hagen Kleinert, FU BERLIN & ICRANet Pescara. Imagine : Single- Valued World . Multivalued World. Warm-up: Ginzburg-Landau Theory. set. FALSE!. Chain Rule :. But this is Ising – like !. Jumps!. - PowerPoint PPT Presentation
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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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