Stara Lesna 2002

between Sine-Gordon Model and Thirring Model

in the chirally broken phase

**Andrei Ivanov, Manfried Faber**

- Equivalence in chiral symmetric phase
- ThM Nambu-Jona-Lasinio
- Path integral approach to Thirring model
- Operator formalism and BCS-model
- Chirally broken phase populated by solitons
- Solitons in 3+1 dimensions

Sine-Gordon model

- Lagrangian
- Invariance
- Most interesting property: Solitons
stable classical solutions with finite energy,

Solitons can annihilate with anti-solitons

Many-soliton solutions obey Pauli's exclusion principle

Fermion-like behaviour

*Skyrme, 1961*

Thirring model

- Lagrangian

parameters: fermion mass , dimensionless coupling metric tensor: and antisymmetric tensor: - Invariances:

Sine-Gordon Thirring model equivalence

- Skyrme, 1961: Conjecture of equivalence
Thirring fermions Sine-Gordon solitons

- Mandelstam's proof, 1975: at operator level
appropriate functionals of the SG field possess the properties of the fermionic fields of Thirring model

- Coleman's proof, 1975: at level of Green functions
Abelian bosonization rules:

Coleman's proof of equivalence

- SG:
of free bosons of mass

- massless ThM: Klaiber (1964): by canonical transformation massless Thirring model reducible to a quantum field theory of a massless free fermion field

ThM
Nambu-Jona-Lasinio

in 1+1 dimensions

Thirring model describes attractive
-interaction

Fiertz transformation

Path integral approach to massless ThM

introduce collective fields

integrate out fermions

evaluate path integral

neglect and effective potential

polar coordinates and

integrals of type

with cut-offs and

Effective potential

for and neglecting

(-70,60)k=1
(-150,110)k=2
(-200,140)k=3
(-300,200)

Non-trivial solution minimum for non-trivial solution

mexican hat potential

(-400,150)
(-15,65)
(-55,80)(1,0)40
(-195,170)
(-180,140)(1,4)5

equations of motion

kinetic term

path integral, up to constant

transform functional determinant

with

appropriate regularisation scheme

1-loop = leading contribution for

Bosonization rules

ThM:

SG:

bosonisation rules

1-loop:

Coleman in massive Thirring model

Massive Thirring model

(-400,150)
(-400,30)
(-15,65)
(-55,80)(1,0)40
(-195,170)
(-180,140)(1,4)5

bosonized massive Thirring model

Sine-Gordon model

equations of motion:

generalized bosonization rules:

Operator formalism normal ordering

two-body term

Trivial vacuum:

Non-trivial vacuum ?

normal ordering for which M ?

Self consistency !

Mass given by 1-body term

1-loop: gap-equation

Solutions of gap equation

- trivial solution: M=0
- non-trivial solution:

non-trivial vacuum

BCS formalism
Hamilton density

trial wave function (massive fermions)

Energy density

gap equation

non-trivial solution

Equations of motion
for massless Thirring model

equivalent to

with

derive constant of motion

in chirally broken phase

special solution

helical wave

Schwinger term

Canonical anti-commutation relation

equal time commutation relations

*Schwinger, 1959*

ThM: Schwinger term
*Sommerfield, 1963*

symmetric phase:

broken phase: Fermion number as topological charge

local gauge transformation of Thirring fermions

Noether current

topological current

integer topological charge

Solitons versus Goldstone bosons

bosonized Thirring model

interpret as `` '' in Sine-Gordon model

from equivalence SG - ThM

Sine-Gordon model in quantum world

mass ratio of particles in Thirring model

1+1-quantum world

mainly populated by solitonsConclusion

- massless Thirring model
ground state with broken chiral symmetry

- fermion condensate
- dynamical mass
- fermion-antifermion pairing

bosonization to massless free scalar field

- massive Thirring model
bosonization to Sine-Gordon model

(different to Coleman)

system populated by solitons mainly

- solitons have fermionic properties
Fermion number topological charge

Pauli's exclusion principle

Speculation

Does 3+1-dim nature consist of solitons ?

pro:

- mass as field energy (not a free parameter)

- mass finite at classical level (no divergencies)

- particles and antiparticles

- interaction a consequence of topology

- annihilation at classical level

- running of coupling at classical level

- only charges, no magnetic monopoles

- spin an intrinsic quantity

- velocity dependence of mass becomes obvious

- history

- ?

- ?

- ?

A Soliton model in 3+1 dimensions

soliton field

Dirac-monopole, Wu-Yang-monopole

singular in origin

long-range field Coulomb field

two physical degrees of freedom electromagnetic field

A regular Soliton

dissolve origin

atan

dual vector potential = connection

Tr

field strength = curvature

dual ``QCD''-Lagrangian

dissolves soliton

compressing ``Higgs''-potential

A stable Soliton

minimizing hedge-hog solution

non-linear differential equation for

analytical solution for

Conclusion

- Mass is field energy
- Electric charge is a topological quantum number,

only multiples of elementary charge - Interaction is a consequence of topolgy
- positive and negative charges:

``electrons'' and ``positrons'',

Coulomb field,

Maxwells electrodynamics,

- particle-antiparticle annihilation at the classical level
- Z(2) symmetry allows for two different two-particle combinations with only little difference in energy

and ? - many open questions

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