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\begin{document}

\subsection*{Defenitions}

\begin{eqnarray*}
I_{+}(r) & = & 2\pi \ln r\\
I_{-}(r) & = & \pi \cos 2\theta _{r}\\
F_{\nu }(r) & = & K_{\nu }\ln r+d_{\nu }\cos 2\theta _{r}\\
A\tilde{f}(\tau ) & =Af\left(\frac{u_{\sigma }\tau }{\alpha }\right) & =\frac{1}{\sqrt{2\pi \left\langle \left(u(\tau )-u(0)\right)^{2}\right\rangle }}\\
R_{\nu }^{(0)} & = & e^{-F_{\nu }(r)}\\
D & = & \frac{2D_{\xi }\alpha }{\pi u_{\sigma ^{2}}}\left(\frac{u_{\sigma }}{u_{\rho }}\right)^{K_{\rho }}A\\
y & = & \frac{g_{1\bot }}{\pi u_{\sigma }}
\end{eqnarray*}



\subsection*{Perturbation}

\begin{eqnarray*}
\frac{\delta R_{\sigma }(r)}{R_{\sigma }^{(0)}} & = & 1+4\pi K_{\sigma }^{2}\left(\frac{g_{1\bot }}{(2\pi )^{2}u_{\sigma }}\right)^{2}I_{+}(r)\int drr^{3-4K_{\sigma }}+\\
 &  & +\frac{2}{u_{\sigma ^{2}}}\left(\frac{u_{\sigma }}{u_{\rho }}\right)^{K_{\rho }}\left(\frac{D_{\xi }\alpha }{4\pi ^{2}}\right)K_{\sigma }^{2}\left(I_{+}(r)-I_{-}(r)\right)A\int drr^{2-K_{\sigma }-K_{\rho }}f(r)-\\
 &  & -\frac{dl}{2\pi }K_{\sigma }^{2}yDI_{+}(r)Af(1)\int drr^{3-4K_{\sigma }}-\frac{dl}{4\pi }K_{\sigma }^{2}yD\left(I_{+}(r)-I_{-}(r)\right)A\int drr^{2-K_{\sigma }-K_{\rho }}f(r)=\\
 & = & 1+\frac{K_{\sigma }^{2}}{2}\left[\left(y^{2}-2yDf(1)dl\right)\int drr^{3-4K_{\sigma }}+\left(D-yDdl\right)\int drr^{2-K_{\sigma }-K_{\rho }}f(r)\right]\ln r+\\
 &  & -\frac{K_{\sigma }^{2}}{4}\left(D-yDdl\right)\int drr^{2-K_{\sigma }-K_{\rho }}f(r)\cos 2\theta _{r}=\\
 & = & e^{-\left(F_{\sigma }^{eff}(r)-F_{\sigma }(r)\right)}\\
\frac{\delta R_{\rho }(r)}{R_{\rho }^{(0)}} & = & 1+\frac{2}{u_{\sigma ^{2}}}\left(\frac{u_{\sigma }}{u_{\rho }}\right)^{K_{\rho }}\left(\frac{D_{\xi }\alpha }{4\pi ^{2}}\right)\frac{u_{\rho }}{u_{\sigma }}K_{\rho }^{2}\left(I_{+}(r)-I_{-}(r)\right)A\int drr^{2-K_{\sigma }-K_{\rho }}f(r)-\\
 &  & -\frac{dl}{4\pi }\frac{u_{\rho }}{u_{\sigma }}K_{\rho }^{2}yD\left(I_{+}(r)-I_{-}(r)\right)\int drr^{2-K_{\sigma }-K_{\rho }}f(r)=\\
 & = & 1+\frac{1}{2}\frac{u_{\rho }}{u_{\sigma }}K_{\rho }^{2}\left(D-yDdl\right)\int drr^{2-K_{\sigma }-K_{\rho }}f(r)\ln r-\\
 &  & -\frac{1}{4}\frac{u_{\rho }}{u_{\sigma }}K_{\rho }^{2}\left(D-yDdl\right)\int drr^{2-K_{\sigma }-K_{\rho }}f(r)\cos 2\theta _{r}=\\
 & = & e^{-\left(F_{\rho }^{eff}(r)-F_{\rho }(r)\right)}
\end{eqnarray*}


\begin{eqnarray*}
K_{\sigma }^{eff} & = & K_{\sigma }-\frac{1}{2}K_{\sigma }^{2}\left(y^{2}-2yDf(1)dl\right)\int drr^{3-4K_{\sigma }}-\frac{1}{2}K_{\sigma }^{2}\left(D-yDdl\right)\int drr^{2-K_{\sigma }-K_{\rho }}f(r)\\
d_{\sigma }^{eff} & = & d_{\sigma }+\frac{1}{4}K_{\sigma }^{2}\left(D-yDdl\right)\int drr^{2-K_{\sigma }-K_{\rho }}f(r)\\
K_{\rho }^{eff} & = & K_{\rho }-\frac{1}{2}\frac{u_{\rho }}{u_{\sigma }}K_{\rho }^{2}\left(D-yDdl\right)\int drr^{2-K_{\sigma }-K_{\rho }}f(r)\\
d_{\rho }^{eff} & = & d_{\rho }+\frac{1}{4}\frac{u_{\rho }}{u_{\sigma }}K_{\rho }^{2}\left(D-yDdl\right)\int drr^{2-K_{\sigma }-K_{\rho }}f(r)
\end{eqnarray*}



\subsection*{Elimination of degrees of freedom}

\begin{eqnarray*}
\int _{1}^{\infty }drr^{3-4K_{\sigma }} & = & \int _{1}^{e^{l}}drr^{3-4K_{\sigma }}+\int _{e^{l}}^{\infty }drr^{3-4K_{\sigma }}=\int _{1}^{e^{l}}drr^{3-4K_{\sigma }}+e^{(4-4K_{\sigma })l}\int _{1}^{\infty }drr^{3-4K_{\sigma }}=\\
 & = & dl+\left[1+\left(4-4K_{\sigma }\right)dl\right]\int _{1}^{\infty }drr^{3-4K_{\sigma }}\\
\int _{1}^{\infty }drr^{2-K_{\sigma }-K_{\rho }}f(r) & = & f(1)dl+\left[1+\left(3-K_{\sigma }-K_{\rho }\right)dl\right]\int _{1}^{\infty }drr^{2-K_{\sigma }-K_{\rho }}f(e^{l}r)
\end{eqnarray*}


\begin{eqnarray*}
K_{\sigma }^{eff} & = & K_{\sigma }-\frac{1}{2}K_{\sigma }^{2}\left(y^{2}+Df(1)\right)dl-\frac{1}{2}K_{\sigma }^{2}\left[y^{2}+\left(\left(4-4K_{\sigma }\right)y^{2}-2yDf(1)\right)dl\right]\int drr^{3-4K_{\sigma }}-\\
 &  & -\frac{1}{2}K_{\sigma }^{2}\left[D+\left(3-K_{\sigma }-K_{\rho }-y\right)Ddl\right]\int drr^{2-K_{\sigma }-K_{\rho }}f(r)=\\
 & = & K_{\sigma }+dK_{\sigma }-\frac{1}{4}K_{\sigma }^{2}\left[y^{2}+dy^{2}\right]\int drr^{3-4K_{\sigma }}-\frac{1}{4}K_{\sigma }^{2}\left[D+dD\right]\int drr^{2-K_{\sigma }-K_{\rho }}f(r)
\end{eqnarray*}



\subsection*{RG equations}

\begin{eqnarray*}
\dot{K}_{\sigma } & = & -\frac{1}{2}K_{\sigma }^{2}\left(y^{2}+Df(1)\right)\\
\dot{K}_{\rho } & = & -\frac{1}{2}\frac{u_{\rho }}{u_{\sigma }}K_{\rho }^{2}Df(1)\\
\dot{y} & = & \left(2-2K_{\sigma }\right)y-Df(1)\\
\dot{D} & = & \left(3-K_{\sigma }-K_{\rho }-y\right)D\\
\dot{d}_{\sigma } & = & \frac{1}{4}K_{\sigma }^{2}Df(1)\\
\dot{d}_{\rho } & = & \frac{1}{4}\frac{u_{\rho }}{u_{\sigma }}K_{\rho }^{2}Df(1)\\
\dot{\eta } & = & \eta 
\end{eqnarray*}



\subsection*{Spin rotation invariance}

the full hamilotinan is described by 6 independent parameters: $v_{F},g_{1\Vert },g_{1\bot },g_{2\Vert },g_{2\bot },D_{\xi }$
or $u_{\sigma },K_{\sigma },u_{\rho },K_{\rho },g_{1\bot },D_{\xi }$.
In the case of spin rotation invariance there are only 4 independed
paramteters. The constrains are in fermion representaion:

\begin{eqnarray*}
g_{1\Vert } & = & g_{1\bot }\\
g_{2\Vert } & = & g_{2\bot }
\end{eqnarray*}
and in boson representaion:

\begin{eqnarray*}
u_{\sigma } & = & v_{F}+o\left(g_{1\bot }^{2}\right)\\
K_{\sigma } & = & 1+\frac{y}{2}+o\left(y^{2}\right)
\end{eqnarray*}


\begin{flushleft}for the rg equations to be consistent we should have
$\dot{K}_{\sigma }=\frac{1}{2}\dot{y}+o\left(y^{2}\right)$ \end{flushleft}

\begin{eqnarray*}
\dot{y} & = & \left(2-2K_{\sigma }\right)y-Df(1)=-y^{2}-Df(1)\\
\dot{K}_{\sigma } & = & -\frac{1}{2}K_{\sigma }^{2}\left(y^{2}+Df(1)\right)=-\frac{1}{2}\left(1+\frac{y}{2}\right)^{2}\left(y^{2}+Df(1)\right)=-\frac{1}{2}\left(y^{2}+Df(1)\right)+o\left(y^{2}\right)\quad \surd 
\end{eqnarray*}



\subsection*{Non interaction point}

seperating inealstic term from the backscattering

\begin{eqnarray*}
S & = & -D_{\xi }\int dxd\tau d\tau '\psi _{R}^{\dagger }\psi _{L}(x,\tau )\psi _{L}^{\dagger }\psi _{R}(x,\tau ')A\tilde{f}(\tau -\tau ')=-D_{\xi }\oint dxd\tau d\tau '\psi _{R}^{\dagger }\psi _{L}(x,\tau )\psi _{L}^{\dagger }\psi _{R}(x,\tau ')A\tilde{f}(\tau -\tau ')\\
 &  & -D_{\xi }A\int dxdT\int _{-\alpha /u_{\sigma }}^{\alpha /u_{\sigma }}d\tau \psi _{R}^{\dagger }\psi _{L}(x,T+\frac{\tau }{2})\psi _{L}^{\dagger }\psi _{R}(x,T-\frac{\tau }{2})\tilde{f}(\tau )\simeq \\
 & \simeq  & -D_{\xi }\oint dxd\tau d\tau '(...)-D_{\xi }A\int _{-\alpha /u_{\sigma }}^{\alpha /u_{\sigma }}d\tau \tilde{f}(\tau )\int dxd\tau \psi _{R}^{\dagger }\psi _{L}(x,\tau )\psi _{L}^{\dagger }\psi _{R}(x,\tau )
\end{eqnarray*}


\begin{flushleft}Define\end{flushleft}

\[
F(l)=\int _{0}^{1}drf(r)=\frac{u_{\sigma }}{\alpha }\int _{0}^{\alpha /u_{\sigma }}d\tau \tilde{f}(\tau )\]


\begin{flushleft}Then the $g_{1}$ process is modifed by\end{flushleft}

\[
g_{1}=g_{1}^{in}-\frac{2D_{\xi }\alpha }{u_{\sigma }}F\]


\begin{flushleft}at the noninteracting point $u_{\sigma }=v_{F}\quad D=\frac{2D_{\xi }\alpha A}{\pi v_{F}^{2}}\quad \dot{y}=-Df(1)$,
therefore\end{flushleft}

\[
\frac{1}{\pi v_{F}}\frac{dg_{1\perp }^{in}}{dl}=\frac{dy}{dl}+\frac{d}{dl}(DF)=-Df(1)+\frac{d}{dl}(DF)\]


\begin{flushleft}The string flauctuaions is of the form $f(r)=g(\eta r)$,
therefore\end{flushleft}

\begin{eqnarray*}
\frac{d\eta }{dl} & = & \eta \\
\frac{df}{d\eta } & = & \frac{r}{\eta }\frac{df}{dr}
\end{eqnarray*}


\begin{flushleft}suppose also that r$f(r)\rightarrow 0$ as $r\rightarrow 0$
then\end{flushleft}

\begin{eqnarray*}
\frac{dF}{dl} & = & \int _{0}^{1}\frac{df}{d\eta }\frac{d\eta }{dl}dr=\int _{0}^{1}r\frac{df}{dr}dr=f(1)-F\\
\frac{d}{dl}(DF) & = & \frac{dD}{dl}F+D\frac{dF}{dl}=D(F+\frac{dF}{dl})=Df(1)
\end{eqnarray*}


\begin{flushleft}thus we get\end{flushleft}

\[
\frac{dg_{1\perp }^{in}}{dl}=0\]



\subsection*{String fluctuations}

The string dynamics is modeled by

\[
S=\frac{1}{2}\int dxd\tau \left[\rho \left(\partial _{\tau }u\right)^{2}+a^{2(n-1)}\sigma \left(\partial _{x}^{n}u\right)^{2}+\rho \omega _{0}^{2}u^{2}\right]\]


\begin{flushleft}The fluctuation is given by\end{flushleft}

\begin{eqnarray*}
G(r,\tau ,0,0) & = & \left\langle T_{\tau }u(r,\tau )u(0,0)\right\rangle =\int \frac{dkd\omega }{\left(2\pi \right)^{2}}\frac{e^{ikx-i\omega \tau }}{a^{2(n-1)}\sigma k^{2n}+\rho \omega ^{2}+\rho \omega _{0}^{2}}\\
 & = & \frac{v}{a^{\frac{n-1}{n}}\sigma }\int \frac{dkd\omega }{\left(2\pi \right)^{2}}\frac{e^{ika^{\frac{1-n}{n}}x-i\omega v\tau }}{k^{2n}+\omega ^{2}+\left(\frac{\omega _{0}}{v}\right)^{2}}
\end{eqnarray*}


\begin{flushleft}integration over $\omega $ yields\end{flushleft}

\begin{eqnarray*}
G(\tau ) & = & \frac{v\pi }{a^{\frac{n-1}{n}}\sigma }\int _{-\infty }^{\infty }\frac{dk}{\left(2\pi \right)^{2}}\frac{e^{-\sqrt{k^{2n}+\left(\frac{\omega _{0}}{v}\right)^{2}}v\left|\tau +a\right|}}{\sqrt{k^{2n}+\left(\frac{\omega _{0}}{v}\right)^{2}}}=\frac{v}{2\pi \sigma a^{\frac{n-1}{n}}}\int _{2\pi a^{\frac{n-1}{n}}/L}^{\infty }dk\frac{e^{-\sqrt{k^{2n}+\left(\frac{\omega _{0}}{v}\right)^{2}}v\left|\tau +a\right|}}{\sqrt{k^{2n}+\left(\frac{\omega _{0}}{v}\right)^{2}}}\simeq \\
 & \simeq  & \frac{v}{2\pi a^{\frac{n-1}{n}}\sigma }\int _{2\pi a^{\frac{n-1}{n}}/L}^{\left|v\tau +a\right|^{-\frac{1}{n}}}\frac{dk}{\sqrt{k^{2n}+\left(\frac{\omega _{0}}{v}\right)^{2}}}
\end{eqnarray*}


\begin{enumerate}
\item rigid string: $\sigma =0$\\
frist integrate over $k$ to give the length of the string, $L$.
define $m=\rho L$\\
\begin{eqnarray*}
G(\tau ) & = & \frac{e^{-\omega _{0}\left|\tau \right|}}{2m\omega _{0}}\\
\left\langle T_{\tau }\left(u(\tau )-u(\tau ')\right)^{2}\right\rangle  & = & \frac{1-e^{-\omega _{0}\left|\tau \right|}}{m\omega _{0}}\\
D & = & \frac{2D_{\xi }\alpha }{\pi u_{\sigma ^{2}}}\left(\frac{u_{\sigma }}{u_{\rho }}\right)^{K_{\rho }}\left(\frac{m\omega _{0}}{2\pi }\right)^{\frac{1}{2}}\\
\eta  & = & \frac{\alpha \omega _{0}}{u_{\sigma }}\\
f(r) & = & \frac{1}{\sqrt{1-e^{-\eta r}}}
\end{eqnarray*}

\item fluctuating string: $\omega _{0}=0\quad n=1\quad v=\sqrt{\frac{\sigma }{\rho }}$
\\
\begin{eqnarray*}
G(x,\tau ) & = & \int \frac{dkd\omega }{\left(2\pi \right)^{2}}\frac{e^{ikx-i\omega (\tau +a)}}{\sigma k^{2}+\rho \omega ^{2}}\simeq \frac{v}{2\pi \sigma }\int _{(2\pi /L)^{-1}}^{\left(x^{2}+(v\tau +a)^{2}\right)^{-1}}\frac{dk}{k}=\frac{v}{2\pi \sigma }\ln \frac{L}{2\pi \sqrt{x^{2}+(v\tau +a)^{2}}}\\
\left\langle T_{\tau }\left(u(\tau )-u(\tau ')\right)^{2}\right\rangle  & = & \frac{v}{2\pi \sigma }\ln \left(1+\frac{v\left|\tau \right|}{a}\right)\simeq \frac{v}{2\pi \sigma }\ln \left(\frac{v\left|\tau \right|}{a}\right)\\
D & = & \frac{2D_{\xi }\alpha }{\pi u_{\sigma ^{2}}}\left(\frac{u_{\sigma }}{u_{\rho }}\right)^{K_{\rho }}\left(\frac{\sigma }{2v}\right)^{\frac{1}{2}}\\
\eta  & = & \frac{\alpha v}{au_{\sigma }}\\
f(r) & = & \frac{1}{\sqrt{\ln \left(1+\eta r\right)}}
\end{eqnarray*}

\item mad hatter string: $\omega _{0}=0\quad n>1$\\
\begin{eqnarray*}
G(\tau ) & = & \frac{v}{2\pi \sigma (n-1)}\left[\left(\frac{L}{2\pi a}\right)^{n-1}-\left(1+\frac{v\left|\tau \right|}{a}\right)^{\frac{n-1}{n}}\right]\\
\left\langle T_{\tau }\left(u(\tau )-u(\tau ')\right)^{2}\right\rangle  & = & \frac{v}{\pi \sigma (n-1)}\left(\frac{v\left|\tau \right|}{a}\right)^{\frac{n-1}{n}}\\
D & = & \frac{2D_{\xi }\alpha }{\pi u_{\sigma ^{2}}}\left(\frac{u_{\sigma }}{u_{\rho }}\right)^{K_{\rho }}\left(\frac{(n-1)\sigma }{2v}\right)^{\frac{1}{2}}\eta ^{\frac{1-n}{2n}}\\
\eta  & = & \frac{\alpha v}{au_{\sigma }}\\
f(r) & = & r^{\frac{1-n}{2n}}
\end{eqnarray*}
\\
in that case \\
\[
\dot{D}=\left(\frac{5}{2}+\frac{1}{2n}-K_{\sigma }-K_{\rho }-y\right)D\]

\end{enumerate}

\subsection*{RG equations for spin symmetric hamilotinan}


\subsubsection*{stationary string}

\begin{eqnarray*}
\dot{K}_{\rho } & = & -\frac{1}{2}\frac{u_{\rho }}{v_{F}}K_{\rho }^{2}D\\
\dot{y} & = & -y^{2}-D\\
\dot{D} & = & (2-K_{\rho }-\frac{3}{2}y)D
\end{eqnarray*}


\begin{center}%
\begin{figure}

\caption{\label{fig:statFlow}stationary string flow}

\includegraphics[  width=10cm,bb = 0 0 200 100, draft, type=eps]{statFlow.eps not found!}
\end{figure}
\end{center}


\subsubsection*{rigid vibrating strnig}

\begin{eqnarray*}
\dot{K}_{\rho } & = & -\frac{1}{2}\frac{u_{\rho }}{v_{F}}\frac{K_{\rho }^{2}D}{\sqrt{1-e^{-\eta }}}\\
\dot{y} & = & -y^{2}-\frac{D}{\sqrt{1-e^{-\eta }}}\\
\dot{D} & = & (2-K_{\rho }-\frac{3}{2}y)D\\
\dot{\eta } & = & \eta 
\end{eqnarray*}


\begin{center}%
\begin{figure}

\caption{\label{fig:rigidFlow}rigid string flow}

\includegraphics[  width=10cm,bb = 0 0 200 100, draft, type=eps]{rigidFlow.eps not found!}
\end{figure}
\end{center}


\subsubsection*{fluctuating string}

\begin{eqnarray*}
\dot{K}_{\rho } & = & -\frac{1}{2}\frac{u_{\rho }}{v_{F}}\frac{K_{\rho }^{2}D}{\sqrt{\ln \left(1+\eta \right)}}\\
\dot{y} & = & -y^{2}-\frac{D}{\sqrt{\ln \left(1+\eta \right)}}\\
\dot{D} & = & (2-K_{\rho }-\frac{3}{2}y)D\\
\dot{\eta } & = & \eta 
\end{eqnarray*}


\begin{center}%
\begin{figure}

\caption{\label{fig: flucFlow}fluctuatin string flow}

\includegraphics[  width=10cm,bb = 0 0 200 100, draft, type=eps]{fluctFlow.eps not found!}
\end{figure}
\end{center}


\subsubsection*{mad hatter string with $n\rightarrow \infty $}

\begin{eqnarray*}
\dot{K}_{\rho } & = & -\frac{1}{2}\frac{u_{\rho }}{v_{F}}K_{\rho }^{2}D\\
\dot{y} & = & -y^{2}-D\\
\dot{D} & = & \frac{3}{2}(1-\frac{2}{3}K_{\rho }-y)D
\end{eqnarray*}



\subsubsection*{change of seperatrix}

%
\begin{figure}

\caption{\label{fig: rigidsep}seperatrices}

\includegraphics[  width=10cm,bb = 0 0 200 100, draft, type=eps]{rigidSep.eps not found!}

\includegraphics[  width=10cm,bb = 0 0 200 100, draft, type=eps]{fluctSep.eps not found!}
\end{figure}

\end{document}
