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Multi- Valued Fields In Condensed Matter, Electromagnetism , and Gravitation. Hagen Kleinert, FU BERLIN. Why Multivalued Fields ? Example : Ginzburg-Landau Theory. set. FALSE!. Chain Rule :. Indistinguishability. Correct Chain Rule :. - PowerPoint PPT Presentation
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Multi-Valued FieldsIn Condensed Matter, Electromagnetism,
and Gravitation
Hagen Kleinert, FU BERLIN
Why Multivalued Fields ?Example: Ginzburg-Landau
Theory
FALSE!Chain Rule:
set
Indistinguishability
In 1D, can be removed by going to covering group U(1)
In >1D impossible
Correct Chain Rule:
Gauge Transformations
Axial Gauge
No Go
Invariant:
EXAMPLE FOR MULTIVALUED FIELD in 2D
Solve:
NOTE:
Mother of Two Green Functions
Example: Magnetostatics Recall:
Now: Generate Magnetic Field
by Multivalued Gauge Transformations
Magnetic Monopoles
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 Idty:
Double Gauge Theory:
New Chain Rule
action arises from
In London (hydrodynamic) Limit
Thus Formalism holds for superfluid helium!
GC Sum Over Lines can be transformed into
Disorder QFT
Result Ginzburg-Landau Theoryof Superfluid Helium
Absorb phase angle (unitary gauge
Order of Supercontucting Transition 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 surfaceis gauge transformation
Monopole Gauge Invariance
Dirac QC:
Quark Confinement
Disorder Theory of magnetic worldlinesExchange electric magne
tic
Nontrivial Geometry from Nonholonomic Coordinate Transformations
Burgers vector b
Frank Vector
DISCLINATIONS
FUNDAMETALS: 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 Theoryvs Abelian QED on Lattice
Lattice formulation
Define
CURIOSITY: Induced Gravity
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
THE END Thanks to Axel and Wolfhard
If you want ot know more, read my new book MULTIVALUED FIELDS
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 vierbeintransforming tononholonomic Coordinates
Multivalued infinitesimal coordinate transformation
INTEGRABILITY CONDITIONS
Bianchi Identities
Derivation from Nonholonomic Mapping Principle for Dirac ElectronFlat Space
Local Lorentz Transformations
“ EXPERIMENTAL “ SITUATION
Hydrogen Atom in Momentum Space
Eliminates candidates
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