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