Stara Lesna 2002

On the equivalence
between Sine-Gordon Model and Thirring Model
in the chirally broken phase

Andrei Ivanov, Manfried Faber


Sine-Gordon model

Thirring model

Sine-Gordon Thirring model equivalence

Coleman's proof of equivalence

ThM \bgroup\color{red}$ \equiv$\egroup Nambu-Jona-Lasinio
in 1+1 dimensions

Thirring model describes attractive $ \delta$ -interaction

$\displaystyle \textcolor{red}{{\cal L}(x)}$ $\displaystyle \textcolor{red}{=}$ $\displaystyle \textcolor{red}{\bar{\psi}(x)(i\gamma^{\mu}\partial_{\mu} - m)\psi(x)}$  
  $\displaystyle \textcolor{red}{-}$ $\displaystyle \textcolor{red}{\frac{1}{2} \; g \; \bar{\psi}(x)\gamma^{\mu}\psi(x) \; \bar{\psi}(x) \gamma_{\mu}\psi(x)},$  

\bgroup\color{red}$\displaystyle \psi = \left( \begin{array}{c} \psi_1 \\ \psi_2\end{array} \right)
$\egroup

Fiertz transformation

$\displaystyle \bar{\psi}\gamma^{\mu}\psi \; \bar{\psi} \gamma_{\mu}\psi = 4 \ps...
...si_1 \psi_2^\dagger \psi_2 = - (\bar{\psi}\psi)^2 - (\bar{\psi}i\gamma^5\psi)^2$      


$\displaystyle \textcolor{red}{{\cal L}(x)}$ $\displaystyle \textcolor{red}{=}$ $\displaystyle \textcolor{red}{\bar{\psi}(x)(i\gamma^{\mu}\partial_{\mu} - m)\psi(x)}$  
  $\displaystyle \textcolor{red}{+}$ $\displaystyle \textcolor{red}{\frac{1}{2} \; g \,\{[\bar{\psi}(x)\psi(x)]^2 +
[\bar{\psi}(x)i\gamma^5\psi(x)]^2\}},$  

Path integral approach to massless ThM

    $\displaystyle Z_{\rm Th} = \int {\cal D}\psi{\cal D}\bar{\psi}\,\exp \,i\int
d^2x \Big\{\bar{\psi}(x)i\gamma^{\mu}\partial_{\mu}\psi(x)$  
    $\displaystyle + \frac{1}{2}\,g\,[(\bar{\psi}(x)\psi(x))^2 +
(\bar{\psi}(x)i\gamma^5\psi(x))^2]\Big\}.$  

introduce collective fields
    $\displaystyle Z_{\rm Th} = \int {\cal D}\psi{\cal D}
\bar{\psi}{\cal D}\sigma {...
...rphi \exp i\!\!\int
d^2x\,\Big\{\bar{\psi}(x)i\gamma^{\mu}\partial_{\mu}\psi(x)$  
    $\displaystyle - \bar{\psi}(x)(\sigma(x) + i\gamma^5\varphi(x))\psi(x) -
\frac{1}{2g}\,[\sigma^2(x) + \varphi^2(x)]\Big\}$  

integrate out fermions
    $\displaystyle {\rm Det}(i\gamma^{\mu}\partial_{\mu} - \sigma - i\gamma^5\varphi) =$  
    $\displaystyle =\exp {\rm Tr}\,{\ell n}(i\gamma^{\mu}\partial_{\mu} - \sigma -
i\gamma^5\varphi) =$  
    $\displaystyle =\exp \,i\int d^2x\,(-i)\,{\rm tr}\,
\langle x\vert{\ell n}(i\gamma^{\mu}\partial_{\mu}
- \sigma - i\gamma^5\varphi)\vert x\rangle$  

evaluate path integral

$\displaystyle Z_{\rm Th} = \int {\cal D}\sigma {\cal D}\varphi \,
exp \,i \int d^2x\,{\cal L}_{\rm eff}(x)
$


$\displaystyle {\cal L}_{\rm eff}(x)$ $\displaystyle =$ $\displaystyle (-i)\,{\rm tr}\,
\langle x\vert{\ell n}(i\gamma^{\mu}\partial_{\mu}
- \sigma - i\gamma^5\varphi)\vert x\rangle$  
    $\displaystyle - \frac{1}{2g}\,[\sigma^2(x) + \varphi^2(x)]$  

neglect \bgroup\color{red}$ \partial_{\mu}\sigma$\egroup and \bgroup\color{red}$ \partial_{\mu}\varphi$\egroup      \bgroup\color{red}$ \rightarrow$\egroup     effective potential
    $\displaystyle {\rm Det}(i\gamma^{\mu}\partial_{\mu} - \sigma - i\gamma^5\varphi) =$  
    $\displaystyle \hspace{1cm}= {\rm Det} \left( \begin{array}{cc}
-\sigma - i\varp...
...rtial_t + i \partial_x & -\sigma + i\varphi
\end{array} \right)
= \Box + \rho^2$  

polar coordinates \bgroup\color{red}$ \rho$\egroup and \bgroup\color{red}$ \vartheta$\egroup

\bgroup\color{red}$\displaystyle \sigma(x) = \rho(x)\,\cos \vartheta(x),\qquad
\varphi(x) = \rho(x)\,\sin \vartheta(x)
$\egroup

integrals of type

\bgroup\color{red}$\displaystyle \int^{\Lambda}_{\mu}\frac{dk^2_{\rm E}}{(k^2_{\rm E})^m}
$\egroup

with cut-offs \bgroup\color{red}$ \Lambda$\egroup and \bgroup\color{red}$ \mu$\egroup

Effective potential

    $\displaystyle V[\sigma(x), \varphi(x)] =
- {\cal L}_{\rm eff}(x) \Big\vert _{\partial_{\mu}\sigma = \partial_{\mu}\varphi = 0} =$  
    $\displaystyle = \frac{\Lambda^2}{4\pi}\Bigg[ \frac{2\pi}{g}\,\frac{\rho^2(x)}{\Lambda^2}
+ \frac{\rho^2(x)}{\Lambda^2}{\ell n}\,\frac{\rho^2(x)}{\Lambda^2}$  
    $\displaystyle - \Big( 1 + \frac{\rho^2(x)}{\Lambda^2}\Big){\ell
n}\Big(1+ \frac{\rho^2(x)}{\Lambda^2}\Big) \Bigg]$  

for \bgroup\color{red}$ \mu \rightarrow 0$\egroup and neglecting \bgroup\color{red}$ (\Lambda^2 - \Lambda^2\,{\ell n}\,\Lambda^2)/4\pi$\egroup
(-70,60)k=1 (-150,110)k=2 (-200,140)k=3 (-300,200) $ \frac{2 \pi}{g}={\ell n}2^k$
oscillator part and centrifugal part of potential

Non-trivial solution \bgroup\color{red}$ g > 0 \rightarrow$\egroup minimum for non-trivial solution

\bgroup\color{red}$\displaystyle M = \rho_{min}(x) = \frac{\Lambda}{\displaystyle \sqrt{e^{\textstyle
2\pi/g} - 1}}.
$\egroup

mexican hat potential

\bgroup\color{red}$\displaystyle \sigma(x) = \rho(x)\,\cos \vartheta(x),\qquad
\varphi(x) = \rho(x)\,\sin \vartheta(x)
$\egroup


(-400,150) $ \frac{2 \pi}{g}={\ell n}2^2$ (-15,65)$ \sigma$ (-55,80)(1,0)40 (-195,170)$ \varphi$ (-180,140)(1,4)5

equations of motion

\bgroup\color{red}$\displaystyle \bar{\psi}(x)\psi(x) = -\,\frac{\sigma(x)}{g},\qquad
\bar{\psi}(x)i\gamma^5\psi(x) = -\,\frac{\varphi(x)}{g},
$\egroup

kinetic term
$ \Lambda \rightarrow \infty$ \bgroup\color{red}$ \;
\Longrightarrow \; \rho \rightarrow M \;$\egroup


\bgroup\color{red}$ \Longrightarrow \; \sigma(x) + i \gamma_5 \, \varphi(x) =M\,e^{\textstyle i \, \gamma_5 \, \vartheta(x)}$\egroup

path integral, up to constant

$\displaystyle Z_{\rm Th} =\int {\cal D}\vartheta\,{\rm Det}\Big(i\gamma^{\mu}\partial_{\mu} -
M\,e^{\textstyle i\,\gamma^5\,\vartheta}\,\Big) =$      
$\displaystyle \int {\cal D}\vartheta\,\exp\,i\,\int d^2x\,{\cal L}_{\rm eff}(x)$      

transform functional determinant
    $\displaystyle {\rm Det}\Big(i\gamma^{\mu}\partial_{\mu} - M\,e^{\textstyle
i\,\gamma^5\,\vartheta}\,\Big) =$  
    $\displaystyle = {\rm Det}\Big(e^{\textstyle
i\,\gamma^5\,\vartheta/2}\,(i\gamma...
...u} +
\gamma^{\mu}A_{\mu} - M)\,e^{\textstyle
i\,\gamma^5\,\vartheta/2}\,\Big) =$  
    $\displaystyle =J[\vartheta]\,{\rm
Det}(i\gamma^{\mu}\partial_{\mu} + \gamma^{\mu}A_{\mu} - M),$  


with \bgroup\color{red}$ A_{\mu}(x) =
\frac{1}{2}\,\varepsilon_{\mu\nu}\,\partial^{\nu}\vartheta(x)$\egroup

appropriate regularisation scheme

\bgroup\color{red}$\displaystyle J[\vartheta] =1.
$\egroup

1-loop = leading contribution for \bgroup\color{red}$ \Lambda \rightarrow \infty$\egroup

$\displaystyle {\cal L}_{\rm eff}(x) = \frac{1}{16\pi}\,\Big(1 - e^{\textstyle
-2\pi/g}\,\Big)\,\partial_{\mu}\vartheta(x)\,\partial^{\mu}\vartheta(x)
$

Bosonization rules

\bgroup\color{red}$\displaystyle \bar{\psi}(x)\psi(x) = -\,\frac{\sigma(x)}{g},\qquad
\bar{\psi}(x)i\gamma^5\psi(x) = -\,\frac{\varphi(x)}{g}
$\egroup

\bgroup\color{red}$\displaystyle \sigma(x) = \rho(x)\,\cos \vartheta(x),\qquad
\varphi(x) = \rho(x)\,\sin \vartheta(x)
$\egroup



\bgroup\color{red}$\displaystyle \hspace{-10cm} \langle \bar{\psi}\psi\rangle = - M/g
$\egroup


ThM:\bgroup\color{red}$ {\quad \cal L}_{\rm eff}(x) = \frac{1}{16\pi}\,\Big(1 - e^{\...
...
-2\pi/g}\,\Big)\,\partial_{\mu}\vartheta(x)\,\partial^{\mu}\vartheta(x)$\egroup


SG:\bgroup\color{red}$ {\qquad \cal L}_{\rm eff}(x) = \frac{1}{2 \beta^2}\,\partial_{\mu}(\beta \vartheta)\,\partial^{\mu}(\beta \vartheta)$\egroup

bosonisation rules

$\displaystyle Z\,\bar{\psi}(x)(1\mp \gamma^5)\psi(x) = \langle
\bar{\psi}\psi\rangle \,e^{\textstyle \pm i \beta \vartheta(x)}
$

1-loop: \bgroup\color{red}$ Z=1$\egroup

\bgroup\color{red}$\displaystyle \textcolor{red}{\frac{8\pi}{\beta^2} = 1 - e^{\textstyle -2\pi/g}}
$\egroup

Coleman in massive Thirring model

$\displaystyle \frac{4\pi}{\beta^2} = 1 + \frac{g}{\pi}
$

Massive Thirring model



    $\displaystyle Z_{\rm Th} = \int {\cal D}\psi{\cal D}\bar{\psi}\,\exp \,i\int
d^2x \Big\{\bar{\psi}(x) ( i\gamma^{\mu}\partial_{\mu} - m ) \psi(x)$  
    $\displaystyle + \frac{1}{2}\,g\,[(\bar{\psi}(x)\psi(x))^2 +
(\bar{\psi}(x)i\gamma^5\psi(x))^2]\Big\}.$  


    $\displaystyle V[\rho(x),\vartheta(x)] = \frac{1}{4\pi}\Bigg[ \rho^2(x){\ell
n}\,\frac{\rho^2(x)}{\Lambda^2} -$  
    $\displaystyle \qquad - (\Lambda^2 + \rho^2(x)){\ell
n}\Bigg(1 + \frac{\rho^2(x)}{\Lambda^2}\Bigg)
+\frac{2\pi}{g}\,\rho^2(x)$  
    $\displaystyle \qquad - \frac{4\pi m}{g}\,\rho(x) -
\frac{4\pi m}{g}\,\rho(x) \,(\cos\vartheta(x) - 1)\Bigg]$  

(-400,150) $ \frac{2 \pi}{g}={\ell n}2^2$ (-400,30) $ \frac{4 \pi m}{g}=0.2$ (-15,65)$ \sigma$ (-55,80)(1,0)40 (-195,170)$ \varphi$ (-180,140)(1,4)5
Bosonization of massive Thirring model
$ \Lambda \rightarrow \infty$ bosonized massive Thirring model

$\displaystyle {\cal L}_{\rm eff}(x) = \frac{1}{16\pi}\,\Big(1 -
e^{\textstyle -2\pi/g}\,\Big)
\,\partial_{\mu}\vartheta(x)\,\partial^{\mu}\vartheta(x) +$      
$\displaystyle + \frac{m M}{g}\,(\cos \vartheta(x) - 1).$      

Sine-Gordon model
$\displaystyle {\cal L}_{\rm eff}(x) =
\frac{1}{2 \beta^2}\,\partial_{\mu}(\beta...
...al^{\mu}(\beta\vartheta) +
\frac{\alpha}{\beta^2}\,(\cos \beta\vartheta(x) - 1)$      

equations of motion:

\bgroup\color{red}$\displaystyle \bar{\psi}(x)\psi(x) = -\frac{\sigma(x) - m}{g}, \qquad
\bar{\psi}(x)i\gamma^5\psi(x) = -\frac{\varphi(x)}{g}
$\egroup

generalized bosonization rules:

$\displaystyle m\,\bar{\psi}(x)(1\mp \gamma^5)\psi(x) =
-\frac{\alpha}{\beta^2}\,e^{\textstyle \pm i \beta \vartheta(x)} +
\frac{m^2}{g}.
$

\bgroup\color{red}$\displaystyle \textcolor{red}{\frac{8\pi}{\beta^2} = 1 - e^{\textstyle -2\pi/g} \le 1 }
$\egroup

Operator formalism normal ordering \bgroup\color{red}$ j^\mu = \; :\bar{\psi}(x) \gamma_{\mu}\psi(x):$\egroup

\bgroup\color{red}$ {\cal L}(x,t) =:\bar{\psi}(x,t)i\gamma^{\mu}\partial_{\mu}\psi(x,t): -$\egroup

\bgroup\color{red}$ -\frac{1}{2}\,g :\bar{\psi}(x,t)\gamma_{\mu}\psi(x,t)
\bar{\psi}(x,t)\gamma^{\mu}\psi(x,t):$\egroup

two-body term

\bgroup\color{red}$ :\bar{\psi}(x,t)\gamma_{\mu}\psi(x,t)
\bar{\psi}(x,t)\gamma^{\mu}\psi(x,t): \to j_{\mu}(x,t)j^{\mu}(x,t)$\egroup



\bgroup\color{red}$ j_{\mu} j^{\mu} = :\bar{\psi}(x,t)\gamma_{\mu}\psi(x,t):
:\bar{\psi}(x,t)\gamma^{\mu}\psi(x,t): =$\egroup

    $\displaystyle = :\bar{\psi}(x,t)\gamma_{\mu}\psi(x,t) \;
\bar{\psi}(x,t)\gamma^{\mu}\psi(x,t):$  
    $\displaystyle + 2:\bar{\psi}(x,t)\gamma_{\mu}\langle
0\vert\psi(x,t)\bar{\psi}(x,t)\vert\rangle \gamma_{\mu}\psi(x,t):$  
    $\displaystyle - {\rm tr}\,\{\gamma_{\mu}\langle
0\vert\psi(x,t)\bar{\psi}(x,t)\vert\rangle \gamma^{\mu}\langle
0\vert\psi(x,t)\bar{\psi}(x,t)\vert\rangle\}$  

Trivial vacuum:

\bgroup\color{red}$ {\cal L}(x) = :\bar{\psi}(x,t)i\gamma^{\mu}\partial_{\mu}\psi(x,t):
- \frac{1}{2}\,g\,j_\mu \; j^\mu$\egroup

Non-trivial vacuum ?

normal ordering for which M ?

Self consistency !

Mass \bgroup\color{red}$ M$\egroup given by 1-body term

1-loop: \bgroup\color{red}$ M - g\,\frac{M}{2\pi}\,{\ell n}\Big(1 + \frac{\Lambda^2}{M^2}\Big) = 0$\egroup gap-equation

Solutions of gap equation



\bgroup\color{red}$ {\cal L}(x,t) =:\bar{\psi}(x,t)(i\gamma^{\mu}\partial_{\mu} -
M)\psi(x,t): -$\egroup

\bgroup\color{red}$\displaystyle -\frac{1}{2}\,g:\bar{\psi}(x,t)\gamma_{\mu}\psi(x,t)\bar{\psi}(x,t)
\gamma^{\mu}\psi(x,t): + \frac{M^2}{2g}
$\egroup

non-trivial vacuum \bgroup\color{red}$ {\cal E}(M) = \langle 0\vert{\cal H}(x,t)\vert\rangle = \textcolor{red}{- \frac{M^2}{2g}}$\egroup

BCS formalism Hamilton density

$\displaystyle {\cal H}(x,t)$ $\displaystyle =$ $\displaystyle - :\psi^{\dagger}_1(x,t)i\frac{\partial }{\partial
x}\psi_1(x,t):$  
    $\displaystyle + :\psi^{\dagger}_2(x,t)i\frac{\partial }{\partial
x}\psi_2(x,t):$  
    $\displaystyle + 2\,g:\psi^{\dagger}_1(x,t)\psi_1(x,t)
\psi^{\dagger}_2(x,t)\psi_2(x,t):$  

trial wave function (massive fermions)
$\displaystyle \vert\Omega\rangle$ $\displaystyle =$ $\displaystyle \prod_{\textstyle p^1}[u_{\textstyle p^1} +
v_{\textstyle p^1}\, a^{\dagger}(p^1)b^{\dagger}(-p^1)]\vert\rangle$  
$\displaystyle u_{\textstyle p^1}$ $\displaystyle =$ $\displaystyle \sqrt{\frac{1}{2}\Bigg( 1 +
\frac{\vert p^1\vert}{\sqrt{(p^1)^2 + M^2}}\Bigg)}$  
$\displaystyle v_{\textstyle p^1}$ $\displaystyle =$ $\displaystyle \varepsilon(p^1)\,\sqrt{\frac{1}{2}\Bigg( 1 -
\frac{\vert p^1\vert}{\sqrt{(p^1)^2 + M^2}}\Bigg)}$  

Energy density
    $\displaystyle {\cal E}(M) =$  
    $\displaystyle = \frac{M^2}{4\pi}\,\Bigg[\,{\ell n}\frac{M^2}{\Lambda^2} - (1 + ...
...ambda^2})\,{\ell n}\Bigg(1 + \frac{M^2}{\Lambda^2}\Bigg) +
\frac{2\pi}{g}\Bigg]$  

gap equation
$\displaystyle M$ $\displaystyle =$ $\displaystyle M\,\frac{g}{2\pi}\,{\ell n}\Bigg(1 + \frac{\Lambda^2}{M^2}\Bigg)$  

non-trivial solution
$\displaystyle M = \frac{\Lambda}{\displaystyle e^{\textstyle 2\pi/g} - 1}$      

Equations of motion for massless Thirring model

$\displaystyle i\gamma^{\mu}\partial_{\mu}\psi(x,t)$ $\displaystyle =$ $\displaystyle g\,j^{\mu}(x,t)\gamma_{\mu}\psi(x,t) ,$  
$\displaystyle -i\partial_{\mu}\bar{\psi}(x,t)\gamma^{\mu}$ $\displaystyle =$ $\displaystyle g\,\bar{\psi}(x,t)\gamma_{\mu}j^{\mu}(x,t).$  

equivalent to
$\displaystyle i\partial_{\mu}\psi(x,t)$ $\displaystyle =$ $\displaystyle a\,j_{\mu}(x,t)\psi(x,t) +
b\,\varepsilon_{\mu\nu}j^{\nu}(x,t)\gamma^5\psi(x,t)$  
$\displaystyle -i\partial_{\mu}\bar{\psi}(x,t)$ $\displaystyle =$ $\displaystyle a\,\bar{\psi}(x,t)j_{\mu}(x,t) +
b\,\bar{\psi}(x,t)\gamma^5j^{\nu}(x,t)\varepsilon_{\nu\mu}$  

with \bgroup\color{red}$ a+b = g$\egroup

derive constant of motion

\bgroup\color{red}$\displaystyle \partial_{\alpha}([\bar{\psi}(x,t)\psi(x,t)]^2 +
[\bar{\psi}(x,t)i\gamma^5\psi(x,t)]^2) = 0
$\egroup

in chirally broken phase

\bgroup\color{red}$\displaystyle [\bar{\psi}(x,t)\psi(x,t)]^2 + [\bar{\psi}(x,t)i\gamma^5\psi(x,t)]^2
= \frac{M^2}{g^2}
$\egroup

special solution

\bgroup\color{red}$\displaystyle \psi(x,t) = \sqrt{-\frac{M}{2g}}{\displaystyle
...
...\textstyle - \omega/2}\,e^{\textstyle
+ i\eta(x,t)}
\end{array}\right)}
$\egroup

helical wave

\bgroup\color{red}$ \beta\vartheta(x,t) = \xi(x,t) +\eta(x,t)
=\xi_0 + \eta_0 - 2M(x + \frac{1}{gc}\,t)$\egroup Schwinger term

Canonical anti-commutation relation

\bgroup\color{red}$ \{\psi(x,t),\psi^{\dagger}(y,t)\} = \delta(x - y)$\egroup

equal time commutation relations

\bgroup\color{red}$ [j_0(x,t), j_0(y,t)] = [j_1(x,t), j_1(y,t)] = 0$\egroup

\bgroup\color{red}$ [j_0(x,t), j_1(y,t)] = 0$\egroup \bgroup\color{red}$ \Rightarrow$\egroup Schwinger, 1959

\bgroup\color{red}$ \to [j_0(x,t), j_1(y,t)] = i\,c\,\frac{\partial}{\partial x}\delta(x-y)$\egroup

ThM: Schwinger term \bgroup\color{red}$ c = \frac{1}{\pi}$\egroup \bgroup\color{red}$ \Rightarrow$\egroup Sommerfield, 1963

$ \langle 0\vert[j_0(x,t), j_1(y,t)]\vert\rangle =
i\,\frac{1}{\pi}\,\frac{\Lambda^2}{M^2 + \Lambda^2}\,\frac{\partial}{\partial x}\delta(x-y)$

symmetric phase: \bgroup\color{red}$ c = \frac{1}{\pi}$\egroup

broken phase: $ c = \frac{1}{\pi}\,\Big(1 - e^{\textstyle
-2\pi/g}\Big)$ Fermion number as topological charge


local gauge transformation of Thirring fermions

\bgroup\color{red}$\displaystyle \psi(x,t) \to e^{\textstyle i\alpha_{\rm V}(x,t)}\psi(x,t)
$\egroup

\bgroup\color{red}$\displaystyle {\rm Det}[i\gamma^{\mu}\partial_{\mu} + \gamma^...
...al_{\mu} + \gamma^{\mu} ( A_{\mu} + \partial_\mu \alpha_{\rm V} ) - M ]
$\egroup


\bgroup\color{red}$\displaystyle {\delta \cal L}_{\rm eff}(x) = i\,\Big\langle x...
...mma^{\mu}\Big\}\Big\vert x\Big\rangle\,
\partial_{\mu}\alpha_{\rm V}(x)
$\egroup

Noether current

\bgroup\color{red}$\displaystyle \hspace{-80mm} J^{\mu}(x) = - \frac{\delta {\ca...
...{\rm eff}[\alpha_{\rm
V}(x)]}{\delta \partial_{\mu}\alpha_{\rm V}(x)} =
$\egroup

\bgroup\color{red}$\displaystyle = - i\,\Big\langle
x\Big\vert{\rm tr}\,\Big\{\f...
...eta}\,\textcolor{red}{\varepsilon^{\mu\nu}\,\partial_{\nu}\vartheta(x)}
$\egroup

topological current

\bgroup\color{red}$\displaystyle {\cal J}^{\mu}(x,t) =
\frac{\beta}{2\pi}\,\textcolor{red}{\varepsilon^{\mu\nu}\,\partial_{\nu}\vartheta(x,t)}.
$\egroup

integer topological charge

\bgroup\color{red}$\displaystyle q = \int\limits^{\infty}_{-\infty}dx\,{\cal J}^0(x,t)
= \frac{\beta}{2\pi}\,[\vartheta(\infty) -
\vartheta(-\infty)]
$\egroup

Solitons versus Goldstone bosons

bosonized Thirring model

\bgroup\color{red}$\displaystyle S = \int d^2x\,\Big[\,\frac{1}{2 \beta^2}\,\par...
...artheta) + \frac{\alpha }{\beta^2} \,(\cos \beta
\vartheta(x) - 1)\Big]
$\egroup

\bgroup\color{red}$ \beta\,\vartheta(x) \to \vartheta(x)$\egroup

\bgroup\color{red}$\displaystyle S = \frac{1}{\beta^2} \int
d^2x\,\Big[\,\frac{1...
...ta(x)
\partial^{\mu}\vartheta(x)
+ \alpha \,(\cos\vartheta(x) - 1)\Big]
$\egroup

interpret $ \beta^2$ as ``$ \hbar$ '' in Sine-Gordon model
$\displaystyle M^2_{\vartheta}$ $\displaystyle =$ $\displaystyle \alpha = \frac{m M}{g} =
-\,m\,\beta^2\,\langle\bar{\psi}\psi\rangle + \frac{m^2}{g}\,\beta^2$  
$\displaystyle M^2_{\rm sol}$ $\displaystyle =$ $\displaystyle \frac{64\alpha}{\beta^4}
= - \frac{64}{\beta^2}\,m\,\langle \bar{\psi}\psi\rangle + O(m^2)$  

from equivalence SG - ThM

\bgroup\color{red}$ \frac{8\pi}{\beta^2} = 1 - e^{\textstyle -2\pi/g} \le 1 \qquad \Rightarrow \qquad \textcolor{red}{\beta^2 > 8\pi}$\egroup

Sine-Gordon model in quantum world

mass ratio of particles in Thirring model

\bgroup\color{red}$ \frac{M_{\rm sol}}{M_{\vartheta}} = \frac{8}{\beta^2} < 1 \qquad \Rightarrow \qquad \textcolor{siesta}{M_{\rm sol} < M_{\vartheta}}$\egroup

\bgroup\color{red}$ \beta^2 >> 8\pi \qquad \Rightarrow \qquad$\egroup 1+1-quantum world

mainly populated by solitonsConclusion

Speculation
Does 3+1-dim nature consist of solitons ?

pro:

contra:

A Soliton model in 3+1 dimensions






























   soliton field\bgroup\color{red}$\displaystyle \quad\vec{n}, \quad \mid \vec{n} \mid^2=1
$\egroup

   Dirac-monopole, Wu-Yang-monopole

   singular in origin

   long-range field\bgroup\color{red}$\displaystyle =$\egroup   Coulomb field

   two physical degrees of freedom\bgroup\color{red}$\displaystyle \hat{=}$\egroup   electromagnetic field

A regular Soliton

   dissolve origin

\bgroup\color{red}$\displaystyle U = \cos \alpha(r) + i \vec{\sigma} \vec{n} \, \sin \alpha(r), \quad \alpha(r) \; = \;$\egroup   atan\bgroup\color{red}$\displaystyle \frac{r}{r_0}
$\egroup

dual vector potential = connection

\bgroup\color{red}$\displaystyle \vec{C}_\mu = - \frac{e_0}{4 \pi \varepsilon_0} \vec{\Gamma}_\mu , \quad \Gamma_{\mu k} =
\frac{i}{2} \,$\egroup   Tr\bgroup\color{red}$\displaystyle \, ( \partial_\mu U \, U^\dagger \sigma_k ) .
$\egroup

field strength = curvature

\bgroup\color{red}$\displaystyle \hspace{0.8mm}{^*}\hspace{-0.8mm}\vec{F}_{\mu \...
...nu}, \quad \vec{R}_{\mu \nu} = \vec{\Gamma}_\mu \times \vec{\Gamma}_\nu
$\egroup


$\displaystyle \vec{R}_{\mu \nu}$ $\displaystyle =$ $\displaystyle \vec{\Gamma}_\mu \times \vec{\Gamma}_\nu = \frac{1}{2} ( \partial_\mu \vec{\Gamma}_\nu - \partial_\nu \vec{\Gamma}_\mu ) =$  
  $\displaystyle =$ $\displaystyle \partial_\mu \vec{\Gamma}_\nu - \partial_\nu \vec{\Gamma}_\mu - \vec{\Gamma}_\mu \times \vec{\Gamma}_\nu$  

dual ``QCD''-Lagrangian

\bgroup\color{red}$\displaystyle {\cal L}_{dual} = - \frac{\alpha_f \hbar c}{4 \...
...mm}\vec{F}_{\mu \nu} \hspace{0.8mm}{^*}\hspace{-0.8mm}\vec{F}^{\mu \nu}
$\egroup

dissolves soliton

compressing ``Higgs''-potential

\bgroup\color{red}$\displaystyle {\cal L}_{p} = - {\cal H}_{p} \; = \; \frac{\alpha_f \hbar c}{4 \pi r_0^4} \left( \frac{\text{Tr} U(x)}{2} \right)^{2m}
$\egroup

A stable Soliton

\bgroup\color{red}$\displaystyle {\cal L} = - \frac{\alpha_f \hbar c}{4 \pi}\lef...
... \vec{R}_{\mu \nu} \vec{R}^{\mu \nu} + \frac{1}{r_0^4} u_0^{2m} \right)
$\egroup

minimizing hedge-hog solution

\bgroup\color{red}$ U = \cos \alpha(r) + i \vec{\sigma} \vec{n} \, \sin \alpha(r), \quad \vec{n}=\frac{\vec{r}}{r}$\egroup

non-linear differential equation for \bgroup\color{red}$ \alpha(\rho), \rho=\frac{r}{r_0}$\egroup

\bgroup\color{red}$\displaystyle \partial_\rho^2 \cos \alpha +\frac{\left(1-\cos^2 \alpha \right) \cos\alpha}{\rho^2}
- m \rho^2 \cos^{2m-1} \alpha = 0
$\egroup

analytical solution for \bgroup\color{red}$ m=3$\egroup

\bgroup\color{red}$\displaystyle \alpha(\rho)=\arctan(\rho)
$\egroup

Conclusion




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