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\begin{document}

\centerline{\bf Lecture~12}
\centerline{\bf Factorized $S$ matrix}
\bigskip

What is integrability? In classical mechanics there is an exhaustive criterion: a conservative nondegenerate Hamiltonian system with $n$ degrees of freedom $(q_i,p_i)$ is integrable, if it satisfies the hypotheses of the Liouville theorem, i.e.\ it possesses exactly $n$ independent first integrals of motion $I_i$, which are in involution, and the Jacobian $\d(I)/\d(p)\ne0$. In classical field theory there is no universal criterion, since the Liouville theorem cannot be generalized to infinite number of degrees of freedom. There are, nevertheless, partial criteria, like existence of an $LA$ pair, bi-Hamiltonian structure etc. In quantum mechanics we expect that the system is integrable, if the underlying classical system is integrable, and the integrals of motion preserve in the quantum case. Nevertheless, there are many quantum systems that do not have classical counterpart. Probably, the most general integrable systems are those that can be described by Skyanin's separation of variables procedure. Nevertheless, there are also limitations for this construction. In the quantum field theory there is no conventional criterion. A partial (necessary, but not sufficient) criterion is the existence of an infinite number of integrals of motion.

Now we will see that an infinite set of integrals of motion impose serious restriction on a relativistic field theory, which seem to be sufficient for integrability.

Any relativistic system in two dimensions, either integrable or non-integrable, possesses at least two integrals of motion: energy and momentum. In two dimensions it is more convenient to consider the currents $I_{\pm1}=E\pm P$, which possess Lorentz spin $\pm1$. Consider an in- or out-state $|\alpha_1\theta_1,\ldots,\alpha_N\theta_N\rangle_{\text{in/out}}$, which consists of $N$ stable particles with internal states $\alpha_i$ and rapidities $\theta_i$. Then it is evident that
\eq$$
I_{\pm1}|\alpha_1\theta_1,\ldots,\alpha_N\theta_N\rangle_{\text{in/out}}
=\sum^N_{i=1}m_{\alpha_i}\e^{\pm\theta_i}|\alpha_1\theta_1,\ldots,\alpha_N\theta_N\rangle_{\text{in/out}}.
\label{Ipm1-eigenvalues}
$$

For simplicity consider a model, which contains particles of the same mass and, which forms a multiplet of the symmetry group of the theory. For example, sine-Gordon model with $\beta^2\ge1/2$ satisfies this condition. Consider a scattering process. The eigenvalues of integrals of motion on the allowed out- and in-state must coincide. Hence,
\eq$$
\sum^{N'}_{i=1}\e^{\pm\theta'_i}=\sum^N_{i=1}\e^{\pm\theta_i},
\label{Ipm1-kineq}
$$
where variables with primes correspond to the out-state. Let $N=N'=2$ (the $2\to2$ scattering). Consider the equality (\ref{Ipm1-kineq}) as an equation on $\theta_1,\theta_2$ with given $\theta'_1,\theta'_2$. This equation has just two solutions: $\theta'_1=\theta_1,\theta'_2=\theta_2$ and $\theta'_1=\theta_2,\theta'_2=\theta_1$. We may assume the first solution, since the second one can be reduce to the first one by exchanging $\alpha'_1\leftrightarrow\alpha'_2$.

Suppose that there are more integrals of motion $I_s$, where $s$ is the spin. In the case of sine-Gordon/Thirring model there are no integrals of motion of even values of spin, and there is just one integral of motion for each odd value of spin. Integrals of motion are supposed to be \emph{local}, i.e.\ have the form
\eq$$
I_s=\int dx^1 j_s^0=\int dx^1(j_s^\bz-j_s^z).
\label{Is-local}
$$
Note that the spin of $j_s^\bz$ is $s+1$, while of $j_s^z$ is $s-1$. In the in-state particles are placed far from each other and the eigenvalue of $I_s$ splits into a sum of one-particle eigenvalues. But the one-particle eigenvalue must be proportional to~$\e^{s\theta}$. We have
\eq$$
I_s|\alpha_1\theta_1,\ldots,\alpha_N\theta_N\rangle_{\text{in/out}}
=I_s^{(0)}\sum^N_{i=1}\e^{s\theta_i}|\alpha_1\theta_1,\ldots,\alpha_N\theta_N\rangle_{\text{in/out}}.
\label{Is-eigen}
$$
Then we have for the $N\to N'$ process
\eq$$
\sum^{N'}_{i=1}\e^{s\theta'_i}=\sum^N_{i=1}\e^{s\theta_i}.
\label{Is-kineq}
$$
Again, consider this as a set of equations on $\{\theta_i\}$ for given $\{\theta'_i\}$. If $N'\ge N$ an infinite set of such equations has solutions for an $N$-dimensional region of values of $\theta'_i$, if $N=N'$. The solutions are evident:
\eq$$
\theta_i=\theta'_{\sigma_i},
\qquad
\sigma\in S^N.
\label{theta-solutions}
$$

Surely, for more general models we need more accurate consideration, but in general we may assume that a model is integrable, if it has an infinite number of commuting integrals of motion such that they do not allow arbitrary multiple particle production and uniquely define the momenta of ingoing particles by outgoing and vice versa. For relativistic systems it usually means that the numbers of particles of each mass conserve after scattering, and the momenta of particles remain unchanged.

{\itshape{\bfseries Assumption of factorized scattering.} The amplitude of scattering of $N$ particles into $N$ particles factorizes in a product of pairwise scatterings of particles in any order with summing over intermediate states.}

Graphically we may represent it as follows:
\eq$$
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%
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\ellipticalarc axes ratio  0.900:0.900  360 degrees 
	from  2.805 24.765 center at  1.905 24.765
\plot  0.572 23.241  1.333 24.098 /
\plot  3.524 26.099  2.667 25.241 /
\plot  1.333 23.241  1.715 23.908 /
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\plot  2.000 23.241  1.810 23.908 /
\plot  1.238 26.099  1.429 25.527 /
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\quad=\quad
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}
\label{factscat}
$$

In principle, the factorized scattering assumption can be checked order-by-order in perturbation theory. But we substantiate it in other way. Suppose that the interaction of particles is short-range in the following meaning. The stationary wave function of $N$ particles (we will consider bosons) becomes nearly equal to a combination of products of plane wave if all particles are distant enough from each other: $|x_i-x_j|\gg R$ ($\forall i,j$) with some interaction radius~$R$. This wave function reads:
\Multline$$
\psi_{\beta_1p_1,\ldots,\beta_np_n}(\alpha_1x_1,\ldots,\alpha_nx_n)
=\sum_{\tau\in S_n}
A^{\alpha_{\sigma_1}\ldots\alpha_{\sigma_n}}
_{\beta_{1\,\hphantom\sigma}\ldots\beta_n}[\tau]
e^{i\sum^n_{i=1}p_{\tau_i}x_{\sigma_i}},
\\
\text{if}\quad
x_{\sigma_1}<x_{\sigma_2}<\cdots<x_{\sigma_n},
\quad
|x_i-x_j|\gg R.
\label{psifunction}
$$
It is easy to check that the function $\psi$ is antisymmetric with respect to permutations of pairs $(\alpha_i,x_i)$. The parameters $\beta_i$ numerate somehow internal states of particles.

Transposition of two particles corresponds to their scattering:
\eq$$
A^{\alpha_1\ldots\alpha_{i+1}\alpha_i\ldots\alpha_n}
_{\beta_1\ldots\beta_i\,\beta_{i+1}\ldots\beta_n}[\tau s^i]
=\sum_{\alpha'_i\alpha'_{i+1}}
S(p_{\tau_i},p_{\tau_{i+1}})^{\alpha_i\alpha_{i+1}}_{\alpha'_i\alpha'_{i+1}}
A^{\alpha_1\ldots\alpha'_i\alpha'_{i+1}\ldots\alpha_n}
_{\beta_1\ldots\beta_i\,\beta_{i+1}\ldots\beta_n}[\tau].
\label{Arelation}
$$
Here $s^i$ is the permutation of two subsequent indexes and $S(p',p)^{\alpha_1\alpha_2}_{\alpha'_1\alpha'_2}$ is the two-particle scattering matrix. There are two equations the $S$ matrix, which follow from this definition.

First, apply the equation (\ref{Arelation}) twice:
$$
\Aligned{
A^{\alpha_1\ldots\alpha_{i+1}\alpha_i\ldots\alpha_n}_{\beta_1\ldots\beta_i\,\beta_{i+1}\ldots\beta_n}[\tau]
&=\sum_{\alpha'_i\alpha'_{i+1}}
S(p_{\tau_i},p_{\tau_{i+1}})^{\alpha_i\alpha_{i+1}}_{\alpha'_i\alpha'_{i+1}}
A^{\alpha_1\ldots\alpha'_i\alpha'_{i+1}\ldots\alpha_n}
_{\beta_1\ldots\beta_i\,\beta_{i+1}\ldots\beta_n}[\tau s^i]
\\
&=\sum_{\alpha'_i\alpha'_{i+1}\alpha''_i\alpha''_{i+1}}
S(p_{\tau_i},p_{\tau_{i+1}})^{\alpha_i\alpha_{i+1}}_{\alpha'_i\alpha'_{i+1}}
S(p_{\tau_{i+1}},p_{\tau_i})^{\alpha'_{i+1}\alpha'_i}_{\alpha''_{i+1}\alpha''_i}
A^{\alpha_1\ldots\alpha''_{i+1}\alpha'_i\ldots\alpha_n}_{\beta_1\ldots\beta_i\,\beta_{i+1}\ldots\beta_n}[\tau].
}
$$
Hence, the $S$ matrix must satisfy the \textbf{unitarity property}:
\eq$$
\sum_{\gamma_1\gamma_2}S(p_1,p_2)^{\beta_1\beta_2}_{\gamma_1\gamma_2}S(p_2,p_1)^{\gamma_2\gamma_1}_{\beta_2\beta_1}
=\delta^{\beta_1}_{\alpha_1}\delta^{\beta_2}_{\alpha_2}.
\label{Unitarity-ind}
$$
Note that the name `unitarity' is lame: it does not mean the unitarity in the physical sense. It is simply a consistency condition, which must be satisfied even for a nonunitary theory.

Let us rewrite equation (\ref{Unitarity-ind}) in an indexless form. To do it let us assume that the matrix $S_{ij}(p,p')$ acts on the tensor product $V_1\otimes V_2\otimes\cdots\otimes V_N$ of internal spaces of $N$ particles as follows. It acts as the unit operator on $V_k$, $k\ne i,j$ and as $S(p,p')$ on the tensor product $V_i\otimes V_j$. Note that $S_{ij}(p,p')$ is generally not the same as $S_{ji}(p,p')$. If we introduce the transposition operator $P$, we may say $S_{ij}(p,p')=P_{ij}S_{ji}(p,p')P_{ij}$. Besides, below we will need the charge conjugation operator $C$, which will be denoted as $C_i$, if acting on $V_i$.

With this notation the unitarity condition reads
\eq$$
S_{12}(p_1,p_2)S_{21}(p_2,p_1)=1.
\label{Unitarity}
$$
It also can be expressed graphically:
\eq$$
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\plot  0.546 26.317  0.286 26.289  0.529 26.191 /
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\endpicture
}
\quad=\quad
\vcenter{
\font\thinlinefont=cmr5
%
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\unitlength=1.04987cm
\linethickness=1pt
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\plot  0.349 26.035  0.286 26.289  0.222 26.035 /
\putrule from  1.333 24.098 to  1.333 26.289
\plot  1.397 26.035  1.334 26.289  1.270 26.035 /
\put{$\alpha_1$} [cB] at  0.386 26.484
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\linethickness=0pt
\putrectangle corners at  0.197 26.607 and  1.422 23.717
\endpicture
}
\label{Unitarity-graph}
$$

Another equation is obtained from the following argument. Suppose that we want to change the order of the particles, e.g.\ $123\to321$. We can do it in two ways:
$$
\Matrix{
            &        &132&\to&312&        &\\
            &\nearrow&            &   &            &\searrow&\\
123&        &            &   &            &        &321.\\
            &\searrow&            &   &            &\nearrow&\\
            &        &213&\to&231&        &
}
$$
The first way leads to the equation
\eq$$
A^{\alpha_3\alpha_2\alpha_1\ldots}_{\cdots}[321\ldots]
=\sum_{\gamma_1,\gamma_2,\gamma_3}
S^{\alpha_1\alpha_2}_{\gamma_1\gamma_2}(p_1,p_2)
S^{\gamma_1\alpha_3}_{\beta_1\gamma_3}(p_1,p_3)
S^{\gamma_2\gamma_3}_{\beta_2\beta_3}(p_2,p_3)
A^{\beta_1\beta_2\beta_3\ldots}_{\cdots}[123\ldots],
\label{rel1-ind}
$$
or, in the indexless form,
\eq$$
A_{321\ldots}
=S_{12}(p_1,p_2)S_{13}(p_1,p_3)S_{23}(p_2,p_3)
A_{123\ldots}.
\label{rel1}
$$
It can be presented in a graphic form:
$$
\sum_{\text{internal lines}}
\quad
\vcenter{
\font\thinlinefont=cmr5
%
\beginpicture
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\unitlength=1.04987cm
\linethickness=1pt
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\setlinear
\linethickness= 0.500pt
\setplotsymbol ({\thinlinefont .})
\plot  0.381 24.289  2.286 26.480 /
\plot  2.167 26.246  2.286 26.480  2.071 26.329 /
\plot  2.286 24.289  0.381 26.480 /
\plot  0.596 26.329  0.381 26.480  0.500 26.246 /
\putrule from  0.953 24.289 to  0.953 26.480
\plot  1.016 26.226  0.953 26.480  0.889 26.226 /
\put{$\alpha_3$} [lB] at  1.905 25.718
\put{$\alpha_2$} [lB] at  1.143 26.194
\put{$\alpha_1$} [lB] at  0.286 26.003
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\put{$p_2$} [lB] at  0.853 23.908
\put{$p_1$} [lB] at  2.286 23.908
\put{$\beta_1$} [lB] at  2.100 24.765
\put{$\beta_2$} [lB] at  1.048 24.384
\put{$\beta_3$} [lB] at  0.191 24.575
\put{$\gamma_1$} [lB] at  1.138 25.722
\put{$\gamma_2$} [lB] at  0.572 25.241
\put{$\gamma_3$} [lB] at  1.143 24.860
\linethickness=0pt
\putrectangle corners at  0.191 26.505 and  2.311 23.823
\endpicture
}
$$
The second way leads to
\eq$$
A^{\alpha_3\alpha_2\alpha_1\ldots}_{\cdots}[321\ldots]
=\sum_{\gamma_1,\gamma_2,\gamma_3}
S^{\alpha_2\alpha_3}_{\gamma_2\gamma_3}(p_2,p_3)
S^{\alpha_1\gamma_3}_{\gamma_1\beta_3}(p_1,p_3)
S^{\gamma_1\gamma_2}_{\beta_1\beta_2}(p_1,p_2)
A^{\beta_1\beta_2\beta_3\ldots}_{\cdots}[123\ldots],
\label{rel2-ind}
$$
or
\eq$$
A_{321\ldots}
=S_{23}(p_2,p_3)S_{13}(p_1,p_3)S_{12}(p_1,p_2)
A_{123\ldots},
\label{rel2}
$$
or graphically
$$
\sum_{\text{internal lines}}
\quad
\vcenter{
\font\thinlinefont=cmr5
%
\beginpicture
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\unitlength=1.04987cm
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\plot  2.286 24.289  0.381 26.480 /
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\put{$\alpha_3$} [lB] at  2.000 25.908
\put{$\beta_1$} [lB] at  2.195 24.575
\put{$\gamma_3$} [lB] at  1.334 24.860
\put{$\gamma_1$} [lB] at  1.238 25.718
\linethickness=0pt
\putrectangle corners at  0.191 26.505 and  2.311 23.823
\endpicture
}
$$
The condition that both way lead to the same result is called \textbf{Yang\--Baxter equation}, which reads
\eq$$
\sum_{\gamma_1,\gamma_2,\gamma_3}
S^{\alpha_1\alpha_2}_{\gamma_1\gamma_2}(p_1,p_2)
S^{\gamma_1\alpha_3}_{\beta_1\gamma_3}(p_1,p_3)
S^{\gamma_2\gamma_3}_{\beta_2\beta_3}(p_2,p_3)
=
\sum_{\gamma_1,\gamma_2,\gamma_3}
S^{\alpha_2\alpha_3}_{\gamma_2\gamma_3}(p_2,p_3)
S^{\alpha_1\gamma_3}_{\gamma_1\beta_3}(p_1,p_3)
S^{\gamma_1\gamma_2}_{\beta_1\beta_2}(p_1,p_2)
\label{YB-ind}
$$
or
\eq$$
S_{12}(p_1,p_2)S_{13}(p_1,p_3)S_{23}(p_2,p_3)
=S_{23}(p_2,p_3)S_{13}(p_1,p_3)S_{12}(p_1,p_2)
\label{YB}
$$
or
\eq$$
\vcenter{
\font\thinlinefont=cmr5
%
\beginpicture
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\unitlength=1.04987cm
\linethickness=1pt
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\setshadesymbol ({\thinlinefont .})
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\setplotsymbol ({\thinlinefont .})
\plot  0.381 24.289  2.286 26.480 /
\plot  2.167 26.246  2.286 26.480  2.071 26.329 /
\plot  2.286 24.289  0.381 26.480 /
\plot  0.596 26.329  0.381 26.480  0.500 26.246 /
\putrule from  0.953 24.289 to  0.953 26.480
\plot  1.016 26.226  0.953 26.480  0.889 26.226 /
\put{$\alpha_3$} [lB] at  1.905 25.718
\put{$\alpha_2$} [lB] at  1.093 26.194
\put{$\alpha_1$} [lB] at  0.286 26.003
\put{$p_3$} [lB] at  0.191 23.908
\put{$p_2$} [lB] at  0.853 23.908
\put{$p_1$} [lB] at  2.286 23.908
\put{$\beta_1$} [lB] at  2.100 24.765
\put{$\beta_2$} [lB] at  1.048 24.384
\put{$\beta_3$} [lB] at  0.191 24.575
\linethickness=0pt
\putrectangle corners at  0.191 26.505 and  2.311 23.823
\endpicture
}
\quad=\quad
\vcenter{
\font\thinlinefont=cmr5
%
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\linethickness=1pt
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\setshadesymbol ({\thinlinefont .})
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\setplotsymbol ({\thinlinefont .})
\plot  0.381 24.289  2.286 26.480 /
\plot  2.167 26.246  2.286 26.480  2.071 26.329 /
\plot  2.286 24.289  0.381 26.480 /
\plot  0.596 26.329  0.381 26.480  0.500 26.246 /
\putrule from  1.715 24.194 to  1.715 26.384
\plot  1.778 26.130  1.715 26.384  1.651 26.130 /
\put{$p_3$} [lB] at  0.191 23.908
\put{$p_2$} [lB] at  1.619 23.908
\put{$p_1$} [lB] at  2.286 23.908
\put{$\beta_2$} [lB] at  1.304 24.384
\put{$\alpha_2$} [lB] at  1.254 26.194
\put{$\beta_3$} [lB] at  0.381 24.765
\put{$\alpha_1$} [lB] at  0.476 25.813
\put{$\alpha_3$} [lB] at  2.000 25.908
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\linethickness=0pt
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\endpicture
}
\label{YB-graph}
$$

There is one more equation, which follows from relativistic invariance and is well-known in the perturbation theory, the \textbf{crossing symmetry}. It is better to write it from the very beginning in the graphical form:
\eq$$
\vcenter{
\font\thinlinefont=cmr5
%
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\linethickness=1pt
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\linethickness= 0.500pt
\setplotsymbol ({\thinlinefont .})
\plot  0.476 24.765  1.905 26.194 /
\plot  1.770 25.969  1.905 26.194  1.680 26.059 /
\plot  1.905 24.765  0.476 26.194 /
\plot  0.701 26.059  0.476 26.194  0.611 25.969 /
\put{$p_1$} [lB] at  2.000 24.479
\put{$p_2$} [lB] at  0.286 24.479
\put{$\alpha_1$} [lB] at  0.286 25.813
\put{$\alpha_2$} [lB] at  1.810 25.718
\put{$\beta_1$} [lB] at  1.810 24.955
\put{$\beta_2$} [lB] at  0.286 25.051
\put{$\phantom{-p_1}$} [lB] at  0.191 26.384
\linethickness=0pt
\putrectangle corners at  0.286 26.219 and  2.000 24.395
\endpicture
}
\quad=\quad
\vcenter{
\font\thinlinefont=cmr5
%
\beginpicture
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\unitlength=1.04987cm
\linethickness=1pt
\setplotsymbol ({\makebox(0,0)[l]{\tencirc\symbol{'160}}})
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\linethickness= 0.500pt
\setplotsymbol ({\thinlinefont .})
\plot  0.476 24.765  1.905 26.194 /
\plot  1.770 25.969  1.905 26.194  1.680 26.059 /
\plot  0.476 26.194  1.905 24.765 /
\plot  1.680 24.900  1.905 24.765  1.770 24.990 /
\plot  0.733 25.837  0.933 25.837 /
\plot  0.833 25.737  0.833 25.937 /
\plot  1.448 25.122  1.648 25.122 /
\plot  1.548 25.022  1.548 25.222 /
\put{$p_2$} [lB] at  0.286 24.479
\put{$\alpha_1$} [lB] at  0.286 25.813
\put{$\alpha_2$} [lB] at  1.810 25.718
\put{$\beta_1$} [lB] at  1.810 24.955
\put{$\beta_2$} [lB] at  0.286 25.051
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\linethickness=0pt
\putrectangle corners at  0.191 26.613 and  1.930 24.395
\endpicture
}
\label{Crossing-graph}
$$
where crosses mean the charge conjugation~$C$. Explicitly, we may write
\eq$$
S^{\alpha_1\alpha_2}_{\beta_1\beta_2}(p_1,p_2)
=\sum_{\gamma_1\delta_1}C_{\beta_1\delta_1}S^{\alpha_2\delta_1}_{\beta_2\gamma_1}(p_2,-p_1)C^{\gamma_1\alpha_1}
\label{Crossing-ind}
$$
or
\eq$$
S_{12}(p_1,p_2)=C_2^{-1}S_{21}(p_2,-p_1)C_2.
\label{Crossing}
$$
In the relativistic case it is convenient to use the rapidities $\theta_i$, so that $S(p_1,p_2)=S(\theta_1-\theta_2)$. Then these properties read:

\begin{enumerate}
\item\textbf{Yang\--Baxter equation}
\eq$$
S_{12}(\theta_1-\theta_2)S_{13}(\theta_1-\theta_3)S_{23}(\theta_2-\theta_3)
=S_{23}(\theta_2-\theta_3)S_{13}(\theta_1-\theta_3)S_{12}(\theta_1-\theta_2).
\label{YB-theta}
$$
\item\textbf{Unitarity}
\eq$$
S_{12}(\theta)S_{21}(-\theta)=1.
\label{Unitarity-theta}
$$
\item\textbf{Crossing symmetry}
\eq$$
S_{12}(\theta)
=C_2^{-1}S_{21}(i\pi-\theta)C_2.
\label{Crossing-theta}
$$
\end{enumerate}

The \emph{bootstrap equations} (\ref{YB-theta})\--(\ref{Crossing-theta}) are very restrictive. In many cases they make it possible to find the $S$ matrix, maybe up to some parameters. The variable $\theta=\theta_1-\theta_2$ is related to the Mandelstam variable $s$ according to
\eq$$
s=m_1^2+m_2^2+2m_1m_2\ch\theta.
\label{s-theta}
$$
It is easy to check that the physical sheet is
\eq$$
0\le\Im\theta<\pi.
\label{phys-sheet}
$$
The point $\theta=\i\pi$ corresponds to the branching point $s=(m_1-m_2)^2$, while the point $\theta=0$ corresponds to the branching point $s=(m_1+m_2)^2$. The real axis and the line $\Im\theta=\pi$ correspond to the right and left cuts in the $s$ plane correspondingly. On the imaginary axis $\theta=\i u$ the $S$ matrix is real, and a pole on the segment $u\in(0,\pi)$ corresponds to a bound state, if its residue in $u$ is positive.


In the case of sine-Gordon model the basic particles are soliton and antisoliton, which are charge and $U(1)$ symmetric. The only admissible form is
\eq$$
S(\theta)
=\begin{blockarray}{ccccc}
++&+-&-+&--&\\
\begin{block}{(cccc)c}
a(\theta)&&&&++\\&b(\theta)&c(\theta)&&+-\\&c(\theta)&b(\theta)&&-+\\&&&a(\theta)&--\\
\end{block}
\end{blockarray}
\label{S-U(1)}
$$
Here $a(\theta),b(\theta),c(\theta)$ are three analytic functions. By substituting this form into the Yang\--Baxter equation (\ref{YB-theta}) we obtain
\eq$$
b(\theta)={\sh{\theta\over p}\over\sh{\lambda-\theta\over\pi}}a(\theta),
\qquad
c(\theta)={\sh{\lambda\over p}\over\sh{\lambda-\theta\over\pi}}a(\theta).
\label{bc-a}
$$
Consistency with the crossing symmetry
$$
a(\theta)=b(\i\pi-\theta),
\qquad
c(\theta)=c(\i\pi-\theta)
$$
requires
\eq$$
\lambda=\i\pi
\label{lambda-ipi}
$$
and
\eq$$
{\sh{\theta\over p}\over\sh{\lambda-\theta\over\pi}}a(\theta)=a(\i\pi-\theta).
\label{a-crossing}
$$
The unitarity condition reduces to
\eq$$
a(\theta)a(-\theta)=1.
\label{a-unitarity}
$$
Solution to the equations (\ref{a-crossing}), (\ref{a-unitarity}) is not unique. Nevertheless, we may obtain a unique solution by imposing the following conditions
\begin{enumerate}
\item The solution is a meromorphic function of $\theta$ on the plane and of $p$ on the half-plane $\Re p>0$.
\item There are no poles of the $S$ matrix on the physical sheet except the segment $(0,\i\pi)$ of the imaginary axis, and there is no poles at all for $p\ge1$.
\item $a(0)=-1$, which reflects the fact known from the classical theory that two solitons cannot have the same momentum. In this case $S(0)=-P$ and it can be shown that the wave function~(\ref{psifunction}) vanishes.
\end{enumerate}

This only solution is
\eq$$
a(\theta)=-\exp\left(-\int{dt\over t}{\sh{\pi t\over2}\sh{\pi(p-1)t\over2}\over\sh\pi t\sh{\pi pt\over2}}\e^{-\i\theta t}\right)
=\exp\left(2\i\int{dt\over t}{\sh{\pi t\over2}\sh{\pi(p-1)t\over2}\over\sh\pi t\sh{\pi pt\over2}}\sin\theta t\right),
\label{a-solution}
$$
which differs from the function $S(\theta)$ of the Lecture~11 by the sign. It mean that the $S$ matrix defined by (\ref{S-U(1)})--(\ref{lambda-ipi}), (\ref{a-solution}) is the $S$ matrix of \emph{bosons} corresponding to the Thirring fermions. In one spacial dimension there is no physical difference between bosons and fermions. But it does not mean that the interaction between fermions and between bosons must be the same. For example, the free Majorana fermion can be described as a system of interacting bosons, generated by the order (or disorder) parameter. The $S$ matrices of bosons and fermions differ by the overall sign.

\bigbreak
\leftline{\bf Bibliography}
\medskip
[1]~A.~Zamolodchikov and Al.~Zamolodchikov, Factorized $S$ Matrices in Two-Dimensions as the Exact Solutions of Certain Relativistic Quantum Field Models, {\it Annals Phys.}\ {\bf 120} (1979) 253.









%{\raggedright
%\bibliographystyle{mybib}
%\bibliography{main}}


%\bigbreak
%\bigskip
%\centerline{\bf Problems}
%\medskip

%{\bf1.}



\end{document}
