Hagen Kleinert, FU BERLIN

THIRD QUANTIZATION

Multi-Valued Quantum Field TheoryIn 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

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

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 Theoryof Superfluid Helium

Absorb phase angle (unitary gauge

Order of Superconducting 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

add Disorder Theory of magnetic worldlines

Exchange electric magnetic

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 Theoryvs 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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THIS IS ALL FOR TODAY

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 Electron

Flat Space

Local Lorentz Transformations

“ EXPERIMENTAL “ SITUATION

Hydrogen Atom in Momentum Space

Eliminates candidates