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\newtheorem{thm}{Теорема}

\begin{document}

\title[Лекция 14. Точные формфакторы]{Лекция 14.\\Точные формфакторы квазилокальных операторов}
\author{Михаил Лашкевич}
\date{}

\frame{\titlepage}

\begin{frame}
\frametitle{Алгебра Фаддеева"--~Замолодчикова}

Вспомним алгебру Фаддеева"--~Замолодчикова:
\vspace{-8pt}
\Multline$$
V^{\alpha_1}(\theta_1)V^{\alpha_2}(\theta_2)
-\sum_{\alpha'_1\alpha'_2}S(\theta_1-\theta_2)^{\alpha_1}_{\alpha'_1}{}^{\alpha_2}_{\alpha'_2}V^{\alpha'_2}(\theta_2)V^{\alpha'_1}(\theta_1)
=2\pi C^{\alpha_1\alpha_2}\delta(\theta_1-\theta_2-i\pi)
\\
-2\pi\sum_{\alpha'_1,\alpha'_2}C^{\alpha'_1\alpha'_2}S(-i\pi)^{\alpha_2\alpha_1}_{\alpha'_2\alpha'_1}\delta(\theta_1-\theta_2+i\pi),
\label{VV-unif-commut-rep}
$$
где
\eq$$
\theta_i\in\cCrr\equiv\R\cup(\R-i\pi).
\label{cC-twolines-rep}
$$
\pause
При этом $V^\alpha(\theta)$ с $\theta\in\R$ имеют смысл операторов уничтожения, а
\eq$$
V^+_\alpha(\theta)=\sum_\beta C_{\alpha\beta}V^\beta(\theta-i\pi)
\quad
(\theta\in\R)
\label{V+-V-rel-rep}
$$
"--- операторов рождения.
\pause

Определим нормальное упорядочение
\eq$$
\Gathered{
\lcolon X\rcolon=X,
\text{ если }
X=V^+_{\beta_1}(\vartheta_1)\cdots V^+_{\beta_L}(\vartheta_L)V^{\alpha_K}(\theta_K)\cdots V^{\alpha_1}(\theta_1),
\quad
\theta_i,\vartheta_j\in\R;
\\
\lcolon XV^{\alpha_1}(\theta_1)V^{\alpha_2}(\theta_2)Y\rcolon
=\sum_{\alpha'_1\alpha'_2}S^{\alpha_1\alpha_2}_{\alpha'_1\alpha'_2}(\theta_1-\theta_2)
\lcolon XV^{\alpha'_2}(\theta_2)V^{\alpha'_1}(\theta_1)Y\rcolon.
}\label{VV-normorder}
$$
\pause
\emph{Тривиальный пример:} если $S(\theta)=\pm1$ мы имеем операторы рождения"=уничтожения свободных бозонов или фермионов.

\end{frame}

\begin{frame}
\frametitle{Трансляционно"=инвариантные операторы}

Определим оператор
\vspace{-8pt}
\Multline$$
O(x)=\sum^\infty_{n=0}{1\over n!}\sum_{\{\alpha_i\}}\prod^n_{i=1}\int_\cCrr{d\theta_i\over2\pi}\,
e^{-ix\sum^n_{i=1}P_{\alpha_i}(\theta_i)}
F_O(\theta_1,\ldots,\theta_n)_{\alpha_1\ldots\alpha_n}
\\
\times\lcolon V^{\alpha_n}(\theta_n)\cdots V^{\alpha_1}(\theta_1)\rcolon,
\label{O-ffdecomp}
$$
заданный бесконечным набором мероморфных функций $F_O(\theta_1,\ldots,\theta_N)$ (\emph{формфакторов}) комплексных переменных~$\theta_i$.
\pause
Здесь
$$
P^0_\alpha(\theta)=m_\alpha\ch\theta,
\qquad
P^1_\alpha(\theta)=m_\alpha\sh\theta.
$$
\pause
Сходимость интегралов в мнимом времени требует, чтобы формфакторы росли медленнее, чем $e^{\tau e^\theta}$ с любым положительным~$\tau$.
\pause

\emph{Трансляционная инвариантность:}
\eq$$
i\d_\mu O(x)=[O(x),P_\mu],
\label{O-translinv}
$$
\pause
где $P_\mu$ "--- оператор импульса:
\eq$$
P_\mu=\sum_\alpha\int_\R{d\theta\over2\pi}\,P_{\alpha\mu}(\theta)V^+_\alpha(\theta)V^\alpha(\theta)
\pause
\quad\Rightarrow\quad
[P_\mu,V^\alpha(\theta)]=-P_{\alpha\mu}(\theta)V^\alpha(\theta).
\label{Pmu-def}
$$
\pause
Так как $P_0=H$ "--- гамильтониан, оператор $O(x)$ является \emph{решением уравнения Гайзенберга}.

\end{frame}

\begin{frame}
\frametitle{Лоренц"=инвариантные операторы}

Потребуем от операторов \emph{лоренц"=инвариантности} в виде:
\eq$$
F_O(\theta_1+\lambda,\ldots,\theta_n+\lambda)_{\alpha_1\ldots\alpha_n}
=e^{s_O\lambda}F_O(\theta_1,\ldots,\theta_n)_{\alpha_1\ldots\alpha_n}.
\label{FO-spin}
$$
Число $s_O$ называется \emph{лоренцевым спином} оператора.
\pause
Отсюда следует, что
\eq$$
i(x^\mu\d^\nu-x^\nu\d^\mu)O(x)=\epsilon^{\mu\nu}\left(is_OO(x)+[O(x),L]\right),
\label{O-Lorentzinv}
$$
\pause
где оператор момента $L$ определяется как
\eq$$
L=i\int_\R{d\theta\over2\pi}\,V^+_\alpha(\theta){d\over d\theta}V^\alpha(\theta)
\pause
\quad\Rightarrow\quad
[L,V^\alpha(\theta)]=-i{d\over d\theta}V^\alpha(\theta).
\label{L-def}
$$
\pause
При преобразовании Лоренца, характеризуемом быстротой $\lambda$, корреляционные функции операторов будут умножаться на постоянный множитель:
$$
\left\langle\prod^n_{i=1}O(z_i,\bz_i)\right\rangle=e^{\lambda\sum s_{O_i}}\left\langle\prod^n_{i=1}O(z_ie^\lambda,\bz_ie^{-\lambda})\right\rangle.
$$

\end{frame}

\begin{frame}
\frametitle{Матричные элементы операторов}

На вещественной оси функции $F_O$ являются матричными элементами:
\eq$$
\langle\vac|O(x)|\theta_1,\ldots,\theta_n\rangle_{\alpha_1\ldots\alpha_n}
=e^{-ix\sum^n_{i=1}P_{\alpha_i}(\theta_i)}F_O(\theta_1,\ldots,\theta_n)_{\alpha_1\ldots\alpha_n}
\label{O-vac-matel}
$$
при $\theta_1>\cdots>\theta_n$.
\pause
Более общо,
\Align$$
&{}_{\beta_1\ldots\beta_{n'}}\langle\vartheta_1,\ldots,\vartheta_{n'}|O(x)|\theta_1,\ldots,\theta_n\rangle_{\alpha_1\ldots\alpha_n}
\notag
\\
&\quad
=\sum_{\{\beta'_i\}}e^{-ix\sum^n_{i=1}P_{\alpha_i}(\theta_i)+ix\sum^{n'}_{i=1}P_{\beta_i}(\vartheta_i)}
\prod^{n'}_{i=1}C_{\beta_i\beta'_i}
\notag
\\
&\qquad
\times
F_O(\theta^-_1,\ldots,\theta^-_n,\vartheta^+_{n'}-i\pi,\ldots,\vartheta^+_1-i\pi)_{\alpha_1\ldots\alpha_n\beta'_{n'}\ldots\beta'_1},
\label{O-matel}
$$
если $\theta_1>\ldots>\theta_n$, $\vartheta_1>\ldots>\vartheta_{n'}$, $\theta^\pm=\theta\pm i0$.
\pause

Графически это равенство можно изобразить так:
$$
\tikz[baseline=-2pt,scale=.5]{
% lower legs
  \coordinate (t) at (-2.5,-2.5); \coordinate (tar) at ($(t)!.15!(0,0)$);
  \draw[->] (t) node[below] {$\theta_1$} -- (tar) node[left=-2pt] {$\alpha_1\,$}; \draw[-] (tar) -- (0,0);
  \coordinate (t) at (-1,-2.5); \coordinate (tar) at ($(t)!.15!(0,0)$);
  \draw[->] (t) node[below] {$\theta_2$} -- (tar) node[left=-2pt] {$\alpha_2$}; \draw[-] (tar) -- (0,0);
  \coordinate (t) at (0,-2.5); \coordinate (tar) at ($(t)!.15!(0,0)$);
  \draw[->] (t) node[below] {$\>\theta_3$} -- (tar) node[left=-2pt] {$\alpha_3$}; \draw[-] (tar) -- (0,0);
  \node at (1,-2.5) {$\cdots$};
  \coordinate (t) at (2,-2.5); \coordinate (tar) at ($(t)!.15!(0,0)$);
  \draw[->] (t) node[below] {$\>\>\theta_n$} -- (tar) node[left=-2pt] {$\alpha_n$}; \draw[-] (tar) -- (0,0);
% upper legs
  \coordinate (t) at (1.5,2.5); \coordinate (tar) at ($(0,0)!.8!(t)$);
  \draw[->] (0,0) -- (tar) node[right=-2pt] {$\beta_1$}; \draw[-] (tar) -- (t) node[above] {$\>\vartheta_1$};
  \coordinate (t) at (.5,2.5); \coordinate (tar) at ($(0,0)!.8!(t)$);
  \draw[->] (0,0) -- (tar) node[left=-2pt] {$\beta_2$}; \draw[-] (tar) -- (t) node[above] {$\vartheta_2$};
  \node at (-.5,2.5) {$\cdots$};
  \coordinate (t) at (-1.6,2.5); \coordinate (tar) at ($(0,0)!.8!(t)$);
  \draw[->] (0,0) -- (tar) node[left=-2pt] {$\>\beta_{n'}$}; \draw[-] (tar) -- (t) node[above] {$\vartheta_{n'}$};
% circle
  \draw[thick,fill=lightgray!30] (0,0) node {$O(x)$} circle (1cm);
}
\quad=\quad
\tikz[baseline=-2pt,scale=.5]{
% lower legs
  \coordinate (t) at (-2.5,-2.5); \coordinate (tar) at ($(t)!.3!(0,0)$);
  \draw[->] (t) node[below] {$\theta_1$} -- (tar) node[left=-2pt] {$\alpha_1\,$}; \draw[-] (tar) -- (0,0);
  \coordinate (t) at (-1,-2.5); \coordinate (tar) at ($(t)!.15!(0,0)$);
  \draw[->] (t) node[below] {$\theta_2$} -- (tar) node[left=-2pt] {$\alpha_2$}; \draw[-] (tar) -- (0,0);
  \coordinate (t) at (0,-2.5); \coordinate (tar) at ($(t)!.15!(0,0)$);
  \draw[->] (t) node[below] {$\>\theta_3$} -- (tar) node[left=-2pt] {$\alpha_3$}; \draw[-] (tar) -- (0,0);
  \node at (1,-2.5) {$\cdots$};
  \coordinate (t) at (2,-2.5); \coordinate (tar) at ($(t)!.15!(0,0)$);
  \draw[->] (t) node[below] {$\>\>\theta_n$} -- (tar) node[left] {$\alpha_n$}; \draw[-] (tar) -- (0,0);
% upper legs
  \coordinate (t) at (1.5,2.5); \coordinate (tar) at ($(0,0)!.8!(t)$);
  \draw[-] (0,0) -- (tar) node[right=-2pt] {$\bar\beta_1$}; \draw[<-] (tar) -- (t) node[above right] {$\vartheta_1-i\pi$};
  \coordinate (t) at (.5,2.5); \coordinate (tar) at ($(0,0)!.8!(t)$);
  \draw[-] (0,0) -- (tar) node[left=-2pt] {$\bar\beta_2$}; \draw[<-] (tar) -- (t) node[above] {$\vartheta_2-i\pi$};
  \node at (-.5,2.5) {$\cdots$};
  \coordinate (t) at (-1.6,2.5); \coordinate (tar) at ($(0,0)!.8!(t)$);
  \draw[-] (0,0) -- (tar) node[left=-2pt] {$\>\bar\beta_{n'}$}; \draw[<-] (tar) -- (t) node[above left] {$\vartheta_{n'}-i\pi$};
% circle
  \draw[thick,fill=lightgray!30] (0,0) node {$O(x)$} circle (1cm);
}
$$

\end{frame}

\begin{frame}
\frametitle{Корреляционные функции}

Корреляционные функции могут быть записаны в виде \emph{спектральных разложений}. Например, двухточечная функция:
\Multline$$
\langle O_1(x)O_2(0)\rangle
=\sum^\infty_{n=0}{1\over n!}\sum_{\{\alpha_i\},\{\alpha'_i\}}\int_\R{d^n\theta\over(2\pi)^n}\,
e^{-ix\sum^n_{i=1}P_{\alpha_i}(\theta_i)}\prod^n_{i=1}C^{\alpha_i\alpha'_i}
\\
\times
F_{O_1}(\theta_1,\ldots,\theta_n)_{\alpha_1\ldots\alpha_n}F_{O_2}(\theta_n-i\pi,\ldots,\theta_1-i\pi)_{\alpha'_n\ldots\alpha'_1}.
\label{2point-corrfun-spectral}
$$
\pause
Эта формула позволяет эффективно вычислять корреляционные функции, только когда ряд убывает.
\pause
Это верно для \emph{больших расстояний} между операторами.
\pause
Действительно, рассмотрим операторы, разделенные мнимым временем~$\tau$. Тогда множитель
$$
\left|e^{-ix\sum^n_{i=1}P_{\alpha_i}(\theta_i)}\right|=e^{-\tau\sum_im_{\alpha_i}\ch\theta_i}\le e^{-\tau\sum_im_{\alpha_i}}
$$
экспоненциально быстро убывает с числом частиц, причем тем быстрее, чем больше~$\tau$.
\pause

\emph{Условие эрмитовости:}
\vspace{-4pt}
\Multline$$
O^+(x)=O(x)
\quad\Leftrightarrow
\\
\left(F_O(\theta_1,\ldots,\theta_n)_{\alpha_1\ldots\alpha_n}\right)^*
=\smashop{\sum_{\{\alpha'_i\}}\prod^{n'}_{i=1}}C_{\alpha_i\alpha'_i}\cdot F_O(\theta_n-i\pi,\ldots,\theta_1-i\pi)_{\alpha'_n\ldots\alpha'_1}
\\
(\forall n\in\Z_{\ge0},\ \theta_i\in\R).
\label{O-hermitian}
$$

\end{frame}

\begin{frame}
\frametitle{Циклическое свойство}

Постулируем \emph{циклическое свойство}:
\eq$$
F_O(\theta_1,\theta_2,\ldots,\theta_n)_{\alpha_1\alpha_2\ldots\alpha_n}
=\sum_{\alpha'_1}(\Omega^{-1}(O))^{\alpha'_1}_{\alpha_1}F_O(\theta_2,\ldots,\theta_n,\theta_1-2\pi i)_{\alpha_2\ldots\alpha_n\alpha'_1}.
\label{FO-cyclic}
$$
Здесь $\Omega(O)$ "--- некоторая постоянная матрица.
\pause
Графически
$$
\tikz[baseline=-2pt]{
% lower legs
  \coordinate (t) at (-2.5,-2.5); \coordinate (tar) at ($(t)!.3!(0,0)$);
  \draw[->] (t) node[below] {$\theta_1$} -- (tar) node[left] {$\alpha_1\,$}; \draw[-] (tar) -- (0,0);
  \coordinate (t) at (-1,-2.5); \coordinate (tar) at ($(t)!.3!(0,0)$);
  \draw[->] (t) node[below] {$\theta_2$} -- (tar) node[left] {$\alpha_2$}; \draw[-] (tar) -- (0,0);
  \node at (-.2,-2.5) {$\cdots$};
  \coordinate (t) at (.6,-2.5); \coordinate (tar) at ($(t)!.3!(0,0)$);
  \draw[->] (t) node[below] {$\ \theta_{n-1}$} -- (tar) node[left] {$\alpha_{n-1}\!\!$}; \draw[-] (tar) -- (0,0);
  \coordinate (t) at (1.6,-2.5); \coordinate (tar) at ($(t)!.3!(0,0)$);
  \draw[->] (t) node[below] {$\ \theta_n$} -- (tar) node[right] {$\alpha_n$}; \draw[-] (tar) -- (0,0);
% circle
  \draw[thick,fill=lightgray!30] (0,0) node {$O(x)$} circle (.5cm);
}
\quad=\quad
\tikz[baseline=-2pt]{
% lower legs
  \coordinate (t) at (3,-2.5); \coordinate (tar) at ($(t)!.3!(0,0)$);
    \coordinate (Omega) at ($(t)!.45!(0,0)$); \coordinate (t1ar) at ($(t)!.7!(0,0)$);
  \draw[->] (t) node[below] {$\theta_1-2\pi i$} -- (tar) node[right] {$\>\alpha_1$};
    \draw[->] (tar) -- (Omega) -- node[pos=0,cross] {} node [pos=0,above right] {$\Omega^{-1}$} (t1ar); \draw[-] (t1ar) -- (0,0);
  \coordinate (t) at (-1,-2.5); \coordinate (tar) at ($(t)!.3!(0,0)$);
  \draw[->] (t) node[below] {$\theta_2$} -- (tar) node[left] {$\alpha_2$}; \draw[-] (tar) -- (0,0);
  \node at (-.2,-2.5) {$\cdots$};
  \coordinate (t) at (.6,-2.5); \coordinate (tar) at ($(t)!.3!(0,0)$);
  \draw[->] (t) node[below] {$\ \theta_{n-1}$} -- (tar) node[left] {$\alpha_{n-1}\!\!$}; \draw[-] (tar) -- (0,0);
  \coordinate (t) at (1.6,-2.5); \coordinate (tar) at ($(t)!.3!(0,0)$);
  \draw[->] (t) node[below] {$\theta_n$} -- (tar) node[right] {$\alpha_n$}; \draw[-] (tar) -- (0,0);
% circle
  \draw[thick,fill=lightgray!30] (0,0) node {$O(x)$} circle (.5cm);
}
$$
\pause
Нетрудно проверить, что
\eq$$
S_{12}(\theta)\Omega_1(O)\Omega_2(O)=\Omega_1(O)\Omega_2(O)S_{12}(\theta).
\label{SOmegaOmega-commut}
$$

\end{frame}

\begin{frame}
\frametitle{Перестановочное свойство}

Постулируем \emph{перестановочное свойство}:
\Multline$$
F_O(\ldots,\theta_i,\theta_{i+1},\ldots)_{\ldots\alpha_i\alpha_{i+1}\ldots}
\\
=\sum_{\alpha'_i\alpha'_{i+1}}S(\theta_i-\theta_{i+1})^{\alpha'_i\alpha'_{i+1}}_{\alpha_i\alpha_{i+1}}
F_O(\ldots,\theta_{i+1},\theta_i,\ldots)_{\ldots\alpha'_{i+1}\alpha'_i\ldots}.
\label{FO-trasnposition}
$$
\pause

$$
\tikz[baseline=-2pt]{
% lower legs
  \coordinate (t) at (-1.6,-2.5); \coordinate (tar) at ($(t)!.3!(0,0)$);
  \draw[->] (t) node[below] {$\theta_1$} -- (tar) node[left] {$\alpha_1\,$}; \draw[-] (tar) -- (0,0);
  \node at (-.9,-2.5) {$\cdots$};
  \coordinate (t) at (-.2,-2.5); \coordinate (tar) at ($(t)!.3!(0,0)$);
  \draw[->] (t) node[below] {$\theta_i$} -- (tar) node[left] {$\alpha_i$}; \draw[-] (tar) -- (0,0);
  \coordinate (t) at (.6,-2.5); \coordinate (tar) at ($(t)!.3!(0,0)$);
  \draw[->] (t) node[below] {$\ \ \theta_{i+1}$} -- (tar) node[right] {$\alpha_{i+1}$}; \draw[-] (tar) -- (0,0);
  \node at (1.6,-2.5) {$\cdots$};
  \coordinate (t) at (2.6,-2.5); \coordinate (tar) at ($(t)!.3!(0,0)$);
  \draw[->] (t) node[below] {$\ \theta_n$} -- (tar) node[right] {$\alpha_n$}; \draw[-] (tar) -- (0,0);
% circle
  \draw[thick,fill=lightgray!30] (0,0) node {$O(x)$} circle (.5cm);
}
\quad=\quad
\tikz[baseline=-2pt]{
% lower legs
  \coordinate (t) at (-1.6,-2.5); \coordinate (tar) at ($(t)!.3!(0,0)$);
  \draw[->] (t) node[below] {$\theta_1$} -- (tar) node[left] {$\alpha_1\,$}; \draw[-] (tar) -- (0,0);
  \node at (-.9,-2.5) {$\cdots$};
  \coordinate (t1) at (-.2,-2.5); \coordinate (t2ar) at ($(t1)!.6!(0,0)$);
  \coordinate (t2) at (.6,-2.5); \coordinate (t1ar) at ($(t2)!.6!(0,0)$);
  \draw[->] (t1) node[below] {$\theta_i$} node[above left] {$\alpha_i\!\!$} .. controls ($(t1)!.3!(0,0)$) and ($(t2)!.3!(0,0)$) .. (t1ar);
    \draw[-] (t1ar) -- (0,0);
  \draw[->] (t2) node[below] {$\ \ \theta_{i+1}$} node[above right] {$\!\!\alpha_{i+1}$} .. controls ($(t2)!.3!(0,0)$) and ($(t1)!.3!(0,0)$) .. (t2ar);
    \draw[-] (t2ar) -- (0,0);
  \node at (1.6,-2.5) {$\cdots$};
  \coordinate (t) at (2.6,-2.5); \coordinate (tar) at ($(t)!.3!(0,0)$);
  \draw[->] (t) node[below] {$\ \theta_n$} -- (tar) node[right] {$\alpha_n$}; \draw[-] (tar) -- (0,0);
% circle
  \draw[thick,fill=lightgray!30] (0,0) node {$O(x)$} circle (.5cm);
}
$$

\end{frame}

\begin{frame}
\frametitle{Кинематический полюс}

Пока что не было никаких условий, которые связывают формфакторы с различным количеством частиц. Постулируем следующее условие \emph{кинематического полюса}.
\pause

Формфакторы имеют полюсы на границе физического листа $0\le\Im(\theta_i-\theta_{i+1})\le\pi$ в точках $\theta_i-\theta_{i+1}=i\pi$.
\pause
Вычет в полюсе для $(n+2)$"=частичного формфактора выражается через $n$"=частичный формфактор:
\vspace{-4pt}
\Multline$$
\Res_{\vartheta'=\vartheta}F_O(\vartheta'+i\pi,\vartheta,\theta_1,\ldots,\theta_n)_{\beta'\beta\alpha_1\ldots\alpha_n}=
\\*
=i\sum_{\beta'',\{\alpha'_i\}}C_{\beta'\beta''}\left(\delta^{\beta''}_\beta\delta^{\alpha'}_\alpha
  -\sum_{\{\gamma_i\}}\Omega^{\beta''}_{\gamma_n}(O)\delta^{\gamma_0}_\beta
  \prod^n_{i=1}S(\vartheta-\theta_i)^{\gamma_i}_{\gamma_{i-1}}{}^{\alpha'_i}_{\alpha_i}\right)
\\
\times F_O(\theta_1,\ldots,\theta_n)_{\alpha'_1\ldots\alpha'_n}.
\label{FO-kinpole}
$$
\pause
\vspace{-8pt}
$$
\tikz[baseline=-2pt,scale=.6]{
% lower legs
  \coordinate (t) at (-2.5,-4); \coordinate (tar) at ($(t)!.2!(0,0)$);
    \coordinate (t2ar) at ($(t)!1.2!(0,0)$); \coordinate (tfin) at ($(t)!1.25!(0,0)$);
  \draw[->] (t) node[below] {$\vartheta$} -- (tar) node[left=-2pt] {$\beta\,$};
    \draw[->] (tar) -- (t2ar); \draw[-] (t2ar) node[right] {$\bar\beta'$} -- (tfin) node[above] {$\vartheta'$};
  \coordinate (t) at (-.2,-4); \coordinate (tar) at ($(t)!.2!(0,0)$);
  \draw[->] (t) node[below] {$\theta_1$} -- (tar) node[left] {$\alpha_1$}; \draw[-] (tar) -- (0,0);
  \coordinate (t) at (.6,-4); \coordinate (tar) at ($(t)!.2!(0,0)$);
  \draw[->] (t) node[below] {$\>\theta_2$} -- (tar) node[right] {$\alpha_2$}; \draw[-] (tar) -- (0,0);
  \node at (1.4,-4) {$\cdots$};
  \coordinate (t) at (2.2,-4); \coordinate (tar) at ($(t)!.2!(0,0)$);
  \draw[->] (t) node[below] {$\>\>\theta_n$} -- (tar) node[right] {$\alpha_n$}; \draw[-] (tar) -- (0,0);
% circle
  \draw[thick,fill=lightgray!30] (0,0) node {$O(x)$} circle (.7cm); \node[left] at (-.7,0) {$-i\Res\limits_{\vartheta'=\vartheta}$};
}
=
\tikz[baseline=-2pt,scale=.6]{
% lower legs
  \coordinate (t) at (-3.5,-4); \coordinate (tar) at ($(t)!.2!(-1,0)$); \coordinate (t1ar) at ($(t)!.63!(-1,0)$);
    \coordinate (t2ar) at ($(t)!1.2!(-1,0)$); \coordinate (tfin) at ($(t)!1.25!(-1,0)$);
  \draw[-] (t) node[below] {$\vartheta$} -- (tar) node[left=-2pt] {$\beta\,$}; \draw[->] (tar) -- (t1ar);
    \draw[-] (t1ar) -- (t2ar); \draw[-] (t2ar) node[left] {$\bar\beta'$} -- (tfin) node[above] {$\vartheta'$};
  \coordinate (t) at (-.2,-4); \coordinate (tar) at ($(t)!.2!(0,0)$);
  \draw[->] (t) node[below] {$\theta_1$} -- (tar) node[left] {$\alpha_1$}; \draw[-] (tar) -- (0,0);
  \coordinate (t) at (.6,-4); \coordinate (tar) at ($(t)!.2!(0,0)$);
  \draw[->] (t) node[below] {$\>\theta_2$} -- (tar) node[right] {$\alpha_2$}; \draw[-] (tar) -- (0,0);
  \node at (1.4,-4) {$\cdots$};
  \coordinate (t) at (2.2,-4); \coordinate (tar) at ($(t)!.2!(0,0)$);
  \draw[->] (t) node[below] {$\>\>\theta_n$} -- (tar) node[right] {$\alpha_n$}; \draw[-] (tar) -- (0,0);
% circle
  \draw[thick,fill=lightgray!30] (0,0) node {$O(x)$} circle (.7cm);
}
\kern -1em-\quad
\tikz[baseline=-2pt,scale=.6]{
% lower legs
  \coordinate (t) at (-1.5,-4); \coordinate (tar) at ($(t)!.2!(1,0)$); \coordinate (Omega) at($(t)!1.05!(1,0)$);
    \coordinate (t2ar) at ($(t)!1.2!(1,0)$); \coordinate (tfin) at ($(t)!1.25!(1,0)$);
  \draw[->] (t) node[below] {$\vartheta$} -- (tar) node[left] {$\beta\,$};
    \draw[->] (tar) -- (Omega) -- node[pos=0,cross] {} node[pos=0,right] {$\Omega$} (t2ar);
    \draw[-] (t2ar) node[left] {$\bar\beta'$} -- (tfin) node[above] {$\vartheta'$};
  \coordinate (t) at (-.2,-4); \coordinate (tar) at ($(t)!.2!(0,0)$);
  \draw[->] (t) node[below] {$\theta_1$} -- (tar) node[left=-2pt] {$\alpha_1$}; \draw[-] (tar) -- (0,0);
  \coordinate (t) at (.6,-4); \coordinate (tar) at ($(t)!.2!(0,0)$);
  \draw[->] (t) node[below] {$\>\theta_2$} -- (tar) node[right] {$\alpha_2$}; \draw[-] (tar) -- (0,0);
  \node at (1.4,-4) {$\cdots$};
  \coordinate (t) at (2.2,-4); \coordinate (tar) at ($(t)!.2!(0,0)$);
  \draw[->] (t) node[below] {$\>\>\theta_n$} -- (tar) node[right] {$\alpha_n$}; \draw[-] (tar) -- (0,0);
% circle
  \draw[thick,fill=lightgray!30] (0,0) node {$O(x)$} circle (.7cm);
}
$$

\end{frame}

\begin{frame}
\frametitle{Согласованность с циклическим свойством}

Проверим согласованность с циклическим условием. С одной стороны
$$
\begin{tikzpicture}[baseline=-.2pt]
\draw[->] (-120:1.5) node[below] {$\vartheta'+i\pi$} -- (-120:1) node[left] {$\beta'$}; \draw[-] (-120:1) -- (0,0);
\draw[->] (-90:1.5) node[below] {$\vartheta$} -- (-90:1) node[left] {$\beta$}; \draw[-] (-90:1) -- (0,0);
\draw[double distance=3pt,thick] (-60:1.5) -- (0,0);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (-60:1.5) node[below] {$\{\theta\}$} -- (-60:1) node[left] {$\alpha$};
% circle
\draw[thick,fill=lightgray!30] (0,0) node {$O$} circle (.3cm) node[left=.3cm] {$-i\Res\limits_{\vartheta'=\vartheta}$};
\end{tikzpicture}
=
\begin{tikzpicture}[baseline=-.2pt]
\begin{scope}[xshift=-.5cm]
  \draw[->] (-120:1.5) node[below] {$\vartheta+i\pi$} -- (-120:1) node[left] {$\beta'$}; \draw[-] (-120:1) -- (0,0);
  \draw[->] (-90:1.5) node[below] {$\vartheta$} -- (-90:1) node[left] {$\beta$}; \draw[-] (-90:1) -- (0,0);
  \draw[fill] (0,0) node[above] {$C$} circle (1pt);
\end{scope}
\draw[double distance=3pt,thick] (-60:1.5) -- (0,0);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (-60:1.5) node[below] {$\{\theta\}$} -- (-60:1) node[left] {$\alpha$};
% circle
\draw[thick,fill=lightgray!30] (0,0) node {$O$} circle (.3cm);
\end{tikzpicture}
-\quad
\begin{tikzpicture}[baseline=-.2pt]
\begin{scope}[xshift=.7cm]
  \draw[->] (-90:1.5) node[below] {$\vartheta$} -- (-90:1) node[below left] {$\beta$};
    \draw[-] (-90:1) -- node[pos=.7,cross] {} node[pos=.8,left] {$\Omega$} (0,0);
  \draw[->] (-45:1.5) node[below] {$\>\>\vartheta+i\pi$} -- (-45:1) node[right] {$\beta'$}; \draw[-] (-45:1) -- (0,0);
  \draw[fill] (0,0) node[above] {$C$} circle (1pt);
\end{scope}
\draw[double distance=3pt,thick] (-45:1.5) -- (0,0);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (-45:1.5) -- (-45:1) node[left] {$\alpha$};
% circle
\draw[thick,fill=lightgray!30] (0,0) node {$O$} circle (.3cm);
\uncover<5->{\draw[-,thick,color=red,overlay] ([xshift=.7cm]-45:.25) circle (.6cm);}
\end{tikzpicture}
$$
\pause
С другой стороны, тот же вычет можно записать в виде
$$
\begin{tikzpicture}[baseline=-.2pt]
\draw[->] (-60:1.5) node[below] {$\vartheta'-i\pi$} -- (-60:1) node[right] {$\beta'$};
  \draw[-] (-60:1) -- node[pos=.4,cross] {} node[pos=.4,right] {$\Omega^{-1}$} (0,0);
\draw[->] (-120:1.5) node[below] {$\vartheta$} -- (-120:1) node[left] {$\beta$}; \draw[-] (-120:1) -- (0,0);
\draw[double distance=3pt,thick] (-90:1.5) -- (0,0);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (-90:1.5) -- (-90:1) node[left] {$\alpha$};
% circle
\draw[thick,fill=lightgray!30] (0,0) node {$O$} circle (.3cm) node[left=.3cm] {$-i\Res\limits_{\vartheta'=\vartheta}$};
\end{tikzpicture}
=
\begin{tikzpicture}[baseline=-.2pt]
\draw[->] (-60:1.5) node[below] {$\vartheta-i\pi$} -- (-60:1.3) node[right] {$\beta'$};
  \draw[-] (-60:1.3) -- node[pos=1,cross] {} node[pos=1,above right] {$\Omega^{-1}$} (-60:1.1) to[out=120,in=-105] (-105:.5) -- (-105:.3);
\draw[->] (-120:1.5) node[below] {$\vartheta'$} -- (-120:1) node[left] {$\beta$}; \draw[-] (-120:1) -- (0,0);
\draw[double distance=3pt,thick] (-90:1.5) -- (0,0);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (-90:1.5) -- (-90:1) node[left] {$\alpha$};
% circle
\draw[thick,fill=lightgray!30] (0,0) node {$O$} circle (.3cm) node[left=.3cm] {$i\Res\limits_{\vartheta'=\vartheta}$};
\end{tikzpicture}
\pause
=-
\begin{tikzpicture}[baseline=-.2pt]
  \draw[->] (-60:1.5) node[below] {$\vartheta-i\pi$} -- (-60:1.3) node[right] {$\beta'$};
    \draw[-] (-60:1.3) -- node[pos=1,cross] {} node[pos=1,above right] {$\Omega^{-1}$} (-60:1.1) to[out=120,in=-60] ([xshift=-.5cm]-60:.5) -- (-.5,0);
\begin{scope}[xshift=-.5cm]
  \draw[->] (-120:1.5) node[below] {$\vartheta$} -- (-120:1) node[left] {$\beta$}; \draw[-] (-120:1) -- (0,0);
  \draw[fill] (0,0) node[above] {$C$} circle (1pt);
\end{scope}
\draw[double distance=3pt,thick] (-90:1.5) -- (0,0);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (-90:1.5) -- (-90:1) node[left] {$\alpha$};
% circle
\draw[thick,fill=lightgray!30] (0,0) node {$O$} circle (.3cm);
\end{tikzpicture}
+
\begin{tikzpicture}[baseline=-.2pt]
\begin{scope}[xshift=.7cm]
  \draw[->] (-90:1.5) node[below] {$\vartheta-i\pi$} -- (-90:1.3) node[right] {$\beta$};
    \draw[-] (-90:1.3) -- node[pos=1,cross] {} node[pos=1,left=-2pt] {$\Omega^{-1}$} (-90:1.1) to[out=90,in=-90] (-1,-.8) to[out=90,in=-90] (-90:.5)
      -- node[pos=0,cross] {} node[pos=0,above left] {$\Omega$} (0,0);
  \draw[->] (-60:1.5) node[below] {$\>\>\vartheta$} -- (-60:1) node[right] {$\beta'$}; \draw[-] (-60:1) -- (0,0);
  \draw[fill] (0,0) node[above] {$C$} circle (1pt);
\end{scope}
\draw[double distance=3pt,thick] (-90:1.5) -- (0,0);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (-90:1.5) -- (-90:1) node[below left] {$\alpha$};
% circle
\draw[thick,fill=lightgray!30] (0,0) node {$O$} circle (.3cm);
\end{tikzpicture}
$$
\pause
$$
=-\quad
\begin{tikzpicture}[baseline=-.2pt]
\begin{scope}[xshift=.7cm]
  \draw[->] (-90:1.5) node[below] {$\vartheta$} -- (-90:1) node[below left] {$\beta$}; \draw[-] (-90:1) -- (0,0);
  \draw[->] (-45:1.5) node[below] {$\>\>\vartheta-i\pi$} -- (-45:1) node[right] {$\beta'$};
    \draw[-] (-45:1) -- node[cross] {} node[above right] {$\Omega^{-1}$} (0,0);
  \draw[fill] (0,0) node[above] {$C$} circle (1pt);
\end{scope}
\draw[double distance=3pt,thick] (-45:1.5) -- (0,0);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (-45:1.5) -- (-45:1) node[left] {$\alpha$};
% circle
\draw[thick,fill=lightgray!30] (0,0) node {$O$} circle (.3cm);
\uncover<5->{\draw[-,thick,color=red,overlay] ([xshift=.7cm]-45:.25) circle (.6cm);}
\end{tikzpicture}
+
\begin{tikzpicture}[baseline=-.2pt]
\begin{scope}[xshift=.7cm]
  \draw[->] (-90:1.5) node[below] {$\vartheta-i\pi$} -- (-90:1) node[below left] {$\beta$}; \draw[-] (-90:1) -- (0,0);
  \draw[->] (-60:1.5) node[below] {$\>\>\vartheta$} -- (-60:1) node[right] {$\beta'$}; \draw[-] (-60:1) -- (0,0);
  \draw[fill] (0,0) node[above] {$C$} circle (1pt);
\end{scope}
\draw[double distance=3pt,thick] (-90:1.5) -- (0,0);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (-90:1.5) -- (-90:1) node[left] {$\alpha$};
% circle
\draw[thick,fill=lightgray!30] (0,0) node {$O$} circle (.3cm);
\end{tikzpicture}
\pause
{\red\Rightarrow}
\vtop{\hsize=3cm\vspace{-2.5\medskipamount}
$$
\Gathered{
C\Omega=(\Omega^{-1})^tC
\\
\uncover<6->{\text{or}}
\\
\uncover<6->{\sum_{\alpha,\beta}C_{\alpha\beta}\Omega^\alpha_{\alpha'}(O)\Omega^\beta_{\beta'}(O)=C_{\alpha'\beta'}.}
}
$$
}
$$


\end{frame}

\begin{frame}
\frametitle{Согласованность с циклическим свойством}

Итак,
\eq$$
\sum_{\alpha,\beta}C_{\alpha\beta}\Omega^\alpha_{\alpha'}(O)\Omega^\beta_{\beta'}(O)=C_{\alpha'\beta'}.
\label{COmega-rel}
$$
\pause
Всегда можно выбрать базис из нейтральных и заряженных частиц. В нем матрица $C$ будет иметь блочно"=диагональный вид:
\begin{enumerate}
\item для нейтральной частицы $\alpha$ блок $1\times1$ вида $C_\alpha=1$;
\pause
\item для пары заряженных частиц $\alpha,\bar\alpha$ блоки $2\times2$ вида $C_\alpha=\sigma^2$.
\end{enumerate}
\pause
Дополнительной заменой базиса матрицу $\Omega$ можно диагонализовать: $\Omega=\diag(\Omega_\alpha)$.
\pause
Тогда
$$
\Omega_\alpha\Omega_{\bar\alpha}=1.
$$
\pause
В частности, для нейтральных частиц $\Omega_\alpha=\pm1$.
\pause
Кроме того, в $C$"=симметричной модели $\Omega_{\bar\alpha}=\Omega_\alpha^*$, так что $|\Omega_\alpha|=1$.

\end{frame}

\begin{frame}
\frametitle{Динамические полюсы}

Второй тип полюсов происходит от связанных состояниях и называется \emph{динамическими полюсами}.
\pause
Пусть имеется набор связанных состояний $\gamma$ состояний $\beta'$ и $\beta$.
\pause
Тогда формфакторы имеют полюс при $\theta_i-\theta_j=iu^\gamma_{\beta'\beta}$ и
\vspace{-8pt}
\Multline$$
\Res_{\vartheta'=\vartheta}F_O(\vartheta'+i\bar u^{\bar\beta}_{\bar\gamma\beta'},\vartheta-i\bar u^{\bar\beta'}_{\bar\beta\gamma},
  \theta_1,\ldots,\theta_n)_{\beta'\beta\alpha_1\ldots\alpha_n}
\\
=i\sum_\gamma\Gamma^\gamma_{\beta'\beta}F_O(\vartheta,\theta_1,\ldots,\theta_n)_{\gamma\alpha_1\ldots\alpha_n}.
\label{FO-dynpole}
$$
\pause
Графически
$$
\tikz[baseline=-2pt,scale=.8]{
% lower legs
  \coordinate (t) at (-3,-3); \coordinate (tar) at ($(t)!.3!(0,0)$);
  \draw[->] (t) -- node[pos=0,above,sloped] {$\vartheta'+i\bar u^{\bar\beta}_{\bar\gamma\beta'}\ $} (tar) node[above left=-4pt] {$\beta'$};
    \draw[-] (tar) -- (0,0);
  \coordinate (t) at (-2,-3); \coordinate (tar) at ($(t)!.3!(0,0)$);
  \draw[->] (t) -- node[pos=0,below,sloped] {$\vartheta-i\bar u^{\bar\beta'}_{\beta\bar\gamma}\ $} (tar) node[below right=-2pt] {$\beta$};
    \draw[-] (tar) -- (0,0);
  \coordinate (t) at (-.2,-3); \coordinate (tar) at ($(t)!.2!(0,0)$);
  \draw[->] (t) node[below] {$\theta_1$} -- (tar) node[left] {$\alpha_1$}; \draw[-] (tar) -- (0,0);
  \coordinate (t) at (.6,-3); \coordinate (tar) at ($(t)!.2!(0,0)$);
  \draw[->] (t) node[below] {$\>\theta_2$} -- (tar) node[left] {$\alpha_2$}; \draw[-] (tar) -- (0,0);
  \node at (1.4,-3) {$\cdots$};
  \coordinate (t) at (2.2,-3); \coordinate (tar) at ($(t)!.2!(0,0)$);
  \draw[->] (t) node[below] {$\>\>\theta_n$} -- (tar) node[right] {$\alpha_n$}; \draw[-] (tar) -- (0,0);
% circle
  \draw[thick,fill=lightgray!30] (0,0) node {$O(x)$} circle (.5cm); \node at (-1.2,0) {$-i\Res\limits_{\vartheta'=\vartheta}$};
}
=
\tikz[baseline=-2pt,scale=.8]{
% lower legs
  \coordinate (t1) at (-3,-3); \coordinate (t1Gamma) at ($(t1)!.6!(0,0)$);
  \coordinate (t2) at (-2,-3); \coordinate (t2Gamma) at ($(t2)!.6!(0,0)$); \coordinate (Gamma) at ($(t1Gamma)!.5!(t2Gamma)$);
  \coordinate (t1ar) at ($(t1)!.5!(Gamma)$); \coordinate (t2ar) at ($(t2)!.5!(Gamma)$);
  \draw[->] (t1) -- node[pos=0,above,sloped] {$\vartheta'+i\bar u^{\bar\beta}_{\bar\gamma\beta'}\ $} (t1ar) node[above left=-4pt] {$\beta'$};
    \draw[-] (t1ar) -- (Gamma);
  \draw[->] (t2) -- node[pos=0,below,sloped] {$\vartheta-i\bar u^{\bar\beta'}_{\beta\bar\gamma}\ $} (t2ar) node[below right=-2pt] {$\beta$};
    \draw[-] (t2ar) -- (Gamma);
  \coordinate (tar) at ($(Gamma)!.4!(0,0)$);
  \draw[->] (Gamma) node[above left=-2pt] {$\Gamma$} -- (tar);
    \draw[-] (tar) -- (0,0);
  \coordinate (t) at (-.2,-3); \coordinate (tar) at ($(t)!.2!(0,0)$);
  \draw[->] (t) node[below] {$\theta_1$} -- (tar) node[left] {$\alpha_1$}; \draw[-] (tar) -- (0,0);
  \coordinate (t) at (.6,-3); \coordinate (tar) at ($(t)!.2!(0,0)$);
  \draw[->] (t) node[below] {$\>\theta_2$} -- (tar) node[left] {$\alpha_2$}; \draw[-] (tar) -- (0,0);
  \node at (1.4,-3) {$\cdots$};
  \coordinate (t) at (2.2,-3); \coordinate (tar) at ($(t)!.2!(0,0)$);
  \draw[->] (t) node[below] {$\>\>\theta_n$} -- (tar) node[right] {$\alpha_n$}; \draw[-] (tar) -- (0,0);
% circle
  \draw[thick,fill=lightgray!30] (0,0) node {$O(x)$} circle (.5cm);
}
$$

\end{frame}

\begin{frame}
\frametitle{Произведение операторов}

Рассмотрим произведение $O_1(x)O_2(y)$ двух операторов при условии
\eq$$
x^0=y^0+0,
\qquad
x^1>y^1.
\label{xy-order}
$$
\pause
Пусть $\xi=\{\xi_i\}^k_{i=1}$. Обозначим $\overrightarrow\xi=(\xi_1,\ldots,\xi_k)$, $\overleftarrow\xi=(\xi_k,\ldots,\xi_1)$, $\overrightarrow{V^\alpha(\xi)}=V^{\alpha_1}(\xi_1)\cdots V^{\alpha_k}(\xi_k)$, $P_\alpha(\xi)=\sum P_{\alpha_i}(\xi_i)$ и т.~п.
\pause

Тогда
\Align$$
O_1(x)O_2(y)
&=\sum^\infty_{m,n,k=0}{1\over m!n!k!}\sum_{\{\alpha_i\},\ldots,\{\gamma'_i\}}
\int_\R{d^k\xi\over(2\pi)^k}\int_\cCrr{d^m\theta\over(2\pi)^m}\int_\cCrr{d^n\vartheta\over(2\pi)^n}\,
\notag
\\*
&\quad\times e^{-iP_\gamma(\xi)(x-y)-iP_\alpha(\theta)x-iP_\beta(\vartheta)y}
\notag
\\*
&\quad\times
C^{\gamma\gamma'}
F_{O_1}(\overrightarrow{\xi^-},\overrightarrow{\theta})_{\overrightarrow{\gamma}\overrightarrow{\alpha}}
F_{O_2}(\overrightarrow{\vartheta},\overleftarrow{\xi^+-i\pi})_{\overrightarrow{\beta}\overleftarrow{\gamma'}}
\lcolon\overleftarrow{V^\alpha(\theta)}\overleftarrow{V^\beta(\vartheta)}\rcolon.
\label{O1O2-prod}
$$
\pause
Графически
$$
O_1(x)O_2(y)=\quad
\begin{tikzpicture}[darrow/.style={double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}},baseline=-2pt]
\draw[darrow] (-1,-1.5) node[below] {$V(\vartheta)$} -- node[left] {$\alpha$} (-1,-.5);
\draw[darrow] (-.5,0) -- node[above] {$\gamma$} (.5,0);
\draw[darrow] (1,-1.5) node[below] {$V(\theta)$} -- node[right] {$\beta$} (1,-.5);
\draw[thick,fill=lightgray!30] (-1,0) node {$O_2(x)$} circle (.5cm);
\draw[thick,fill=lightgray!30] (1,0) node {$O_1(x)$} circle (.5cm);
\draw[-] (-1.5,-1.5) -- (1.5,-1.5);
\end{tikzpicture}
$$

\end{frame}

\begin{frame}
\frametitle{Коммутирование операторов: сдвиг контуров}

Чтобы прокоммутировать операторы, сдвинем контур интегрирования по $\xi_i$ вверх на $i\pi$:
$$
\tikz{
\draw[->] (-4,0) -- (4,0);
\draw[-] (-4,1) -- (4,1);
\draw[fill] (-1.9,1) node[above left] {$\theta_j+i\pi$} circle (1pt);
\draw[fill] (-1.9,.3) node[left=2pt] {$\theta_j+iu$} circle (1pt);
\draw[fill] (2.3,0) node[below right] {$\vartheta_k$} circle (1pt);
\draw[fill] (2.3,.7) node[right=2pt] {$\vartheta_k+i\bar u$} circle (1pt);
\draw[->] (0,-.2) -- (0,0) node[below right] {$0$} -- (0,1) node[above right] {$i\pi$} -- (0,1.4);
\only<2>{\draw[-,very thick,color=red] (-4,0) -- (2.2,0) arc (180:0:.1cm) -- (4,0);}
\only<3,4>{\draw[-,very thick,color=red] (-4,1) -- (-2,1) arc (180:0:.1cm) -- (4,1);}
\only<5>{
  \draw[-,very thick,color=red] (-1.9,1) circle (.1cm);
  \draw[-,very thick,color=red] (-1.9,.3) circle (.1cm);
  \draw[-,very thick,color=red] (2.3,.7) circle (.1cm);
}
}
$$
\pause\pause
\Align*$$
&O_1(x)O_2(x)=
\\
&\sum^\infty_{m,n,k=0}{1\over m!n!k!}\sum_{\{\alpha_i\},\ldots,\{\gamma'_i\}}
\int_\R{d^k\xi\over(2\pi)^k}\int_\cCrr{d^m\theta\over(2\pi)^m}\int_\cCrr{d^n\vartheta\over(2\pi)^n}\,
\notag
\\*
&\times e^{iP_\gamma(\xi)(x-y)-iP_\alpha(\theta)x-iP_\beta(\vartheta)y}
\notag
\\*
&\times
C^{\gamma\gamma'}\only<4->{{\only<4>\red\Omega^{\gamma''}_{\gamma'}}}
F_{O_1}({\only<3>{\blue\overrightarrow{\xi^++i\pi},\overrightarrow{\theta}}\only<4->{\only<4>\blue\overrightarrow{\theta},\overrightarrow{\xi^+-i\pi}}})
_{\only<3>{{\blue\overrightarrow{\gamma}\overrightarrow{\alpha}}}\only<4->{{\only<4>\blue\overrightarrow{\alpha}\overrightarrow\gamma}}}
F_{O_2}(\overrightarrow{\vartheta},\overleftarrow{\xi^-})_{\overrightarrow{\beta}\overleftarrow{\only<4>\red\gamma''}}
\\*
&\times
\only<3>{{\blue\lcolon\overleftarrow{V^\alpha(\theta)}\overleftarrow{V^\beta(\vartheta)}\rcolon}}
\only<4->{{\only<4>\blue\lcolon\overleftarrow{V^{\beta'}(\vartheta)}\overleftarrow{V^{\alpha'}(\theta)}\rcolon}
{\only<4>\red\smash{\prod_{i,j}S(\theta_i-\vartheta_j)^{\alpha_i\beta_j}_{\alpha'_i\beta'_j}}}}
\only<5->{{\red{}\quad+\quad\textbf{????}}}
$$

\end{frame}

\begin{frame}
\frametitle{Коммутирование операторов: кинематические полюсы}

Рассмотрим вклад кинематического полюса от $\theta_1$ в интеграл по~$\xi_k$:
\Multline*$$
-e^{-iP_{(\gamma)_k}((\xi)_k)(x-y)-iP_{(\alpha)_1}((\theta)_1)x-i(P_\beta(\vartheta)+P_{\alpha_1}(\theta_1))y}
\\
\times\smashoperator{\sum_{\{\alpha\},\ldots,\{\delta'\}}}
C^{(\gamma)_1(\gamma')_1}
F_{O_1}\left((\overrightarrow{\xi^-})_k,(\overrightarrow\theta)_1\right)_{(\overrightarrow{\gamma''})_k(\overrightarrow{\alpha'})_1}
F_{O_2}\left(\overrightarrow\vartheta,\theta_1,(\overleftarrow{\xi^+-i\pi})_k\right)_{\overrightarrow\beta,\alpha'_1,
  (\overleftarrow\gamma)_1}
\\
\times\left(\delta^{\alpha'}_\alpha\delta^{\gamma''}_\gamma
  -\delta^{\delta_1}_{\alpha_1}\delta^{\alpha'_1}_{\delta'_k}\Omega^{\delta'_1}_{\delta_m}(O_1)
  \prod^{k-1}_{i=1}S(\theta_1-\xi_i+2\pi i)^{\delta'_{i+1}}_{\delta'_i}{}^{\gamma''_i}_{\gamma_i}
  \prod^m_{i=2}S(\theta_1-\theta_i)^{\delta_i}_{\delta_{i-1}}{}^{\alpha'_i}_{\alpha_i}\right)
\\
\times\lcolon\overleftarrow{V^\alpha(\theta)}\overleftarrow{V^\beta(\vartheta)}\rcolon,
$$
где $(\xi)_j=\{\xi_i\}_{i\ne j}$.
\pause

Графически
$$
i\Res_{\xi_k=\theta_1}
\tikz[baseline=-2pt,scale=.7]{
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-2) node[left] {$\beta$} node[below] {$\vartheta$} -- (0,-1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,2.5) node[left] {$(\alpha)_1$} node[above] {$\theta$} -- (0,1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-.5) -- node[left] {$(\gamma)_k$} (0,.5);
%
  \coordinate (a) at ($(0,-1)+(30:.5)$); \coordinate (b) at ($(0,1)+(-30:.5)$); \coordinate (c) at($(0,1)+(30:.5)$);
  \begin{scope}[decoration={markings,mark=at position 0.5 with {\arrow{>}}}]
    \draw[-,postaction={decorate}] (a) to[out=30,in=-30] node[pos=.5,right] {$\gamma_k$} (b);
    \draw[-,postaction={decorate}] (.8,2.7) to[out=-90,in=30] node[pos=.5,right] {$\alpha_1$} (c);
  \end{scope}
%
  \draw[thick,fill=lightgray!30] (0,1) node {$O_1$} circle (.5cm);
  \draw[thick,fill=lightgray!30] (0,-1) node {$O_2$} circle (.5cm);
}
\pause
=-
\tikz[baseline=-2pt,scale=.7]{
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-2) node[left] {$\beta$} node[below] {$\vartheta$} -- (0,-1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,2.5) node[left] {$(\alpha)_1$} node[above] {$\theta$} -- (0,1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-.5) -- node[left] {$(\gamma)_k$} (0,.5);
%
  \coordinate (a) at ($(0,-1)+(30:.5)$); \coordinate (b) at ($(0,1)+(-30:.5)$); \coordinate (c) at($(0,1)+(30:.5)$);
  \begin{scope}[decoration={markings,mark=at position 0.5 with {\arrow{>}}}]
    \draw[-,postaction={decorate}] (.8,2.7) to[out=-90,in=30] node[pos=.5,right] {$\alpha_1$} (a);
  \end{scope}
%
  \draw[thick,fill=lightgray!30] (0,1) node {$O_1$} circle (.5cm);
  \draw[thick,fill=lightgray!30] (0,-1) node {$O_2$} circle (.5cm);
}
+
\tikz[baseline=-2pt,scale=.7]{
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-2) node[left] {$\beta$} node[below] {$\vartheta$} -- (0,-1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,2.5) node[left] {$(\alpha)_1$} node[above] {$\theta$} -- (0,1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-.5) -- node[left] {$(\gamma)_k$} (0,.5);
%
  \coordinate (a) at ($(0,-1)+(30:.5)$); \coordinate (b) at ($(0,1)+(-30:.5)$); \coordinate (c) at($(0,1)+(30:.5)$);
  \begin{scope}[decoration={markings,mark=at position 0.15 with {\arrow{>}}}]
    \draw[-,postaction={decorate}] (.8,2.7) node[below right] {$\alpha_1$} to[out=-90,in=90] node[pos=1,cross] {} node[pos=1,left] {$\Omega_1$} (-.7,1)
      to[out=-90,in=90] (.5,-.5) to[out=-90,in=30] (a);
  \end{scope}
%
  \draw[thick,fill=lightgray!30] (0,1) node {$O_1$} circle (.5cm);
  \draw[thick,fill=lightgray!30] (0,-1) node {$O_2$} circle (.5cm);
}
\pause
\hspace{-8pt}=-
\tikz[baseline=-.5cm-2pt,scale=.7]{
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-2.5) node[left] {$\beta$} node[below] {$\vartheta$} -- (0,-1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,2) node[left] {$(\alpha)_1$} node[above] {$\theta$} -- (0,1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-.5) -- node[left] {$(\gamma)_k$} (0,.5);
%
  \coordinate (a) at ($(0,-1)+(-150:.5)$);
  \begin{scope}[decoration={markings,mark=at position 0.5 with {\arrow{>}}}]
    \draw[-,postaction={decorate},overlay] (-.8,-2.7) to[out=90,in=-150] node[pos=.5,left] {$\alpha_1$} (a);
  \end{scope}
%
  \draw[thick,fill=lightgray!30] (0,1) node {$O_1$} circle (.5cm);
  \draw[thick,fill=lightgray!30] (0,-1) node {$O_2$} circle (.5cm);
}
+
\tikz[baseline=-.5cm-2pt,scale=.7]{
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-2.5) node[left] {$\beta$} node[below] {$\vartheta$} -- (0,-1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,2) node[left] {$(\alpha)_1$} node[above] {$\theta$} -- (0,1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-.5) -- node[left] {$(\gamma)_k$} (0,.5);
%
  \coordinate (a) at ($(0,-1)+(-150:.5)$);
  \begin{scope}[decoration={markings,mark=at position 0.3 with {\arrow{>}}}]
    \draw[-,postaction={decorate}] (-.8,-2.7) to[out=90,in=-150] node[pos=.3,left] {$\alpha_1$} node[pos=.6,cross] {} node[pos=.6,left] {$\Omega_1$} (a);
  \end{scope}
%
  \draw[thick,fill=lightgray!30] (0,1) node {$O_1$} circle (.5cm);
  \draw[thick,fill=lightgray!30] (0,-1) node {$O_2$} circle (.5cm);
}
$$

\end{frame}

\begin{frame}
\frametitle{Коммутирование операторов: кинематические полюсы}

Перепишем это равенство по"=другому:
\vspace{-4pt}
$$
\tikz[baseline=-2pt,scale=.6]{
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-2.5) node[below] {$\vartheta$} -- node[below left] {$\beta$} (0,-1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,2) node[above] {$\theta$} -- node[above left] {$\alpha$} (0,1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-.5) -- node[below left] {$\gamma$} (0,.5);
%
  \draw[thick,fill=lightgray!30] (0,1) node {$O_1$} circle (.5cm);
  \draw[thick,fill=lightgray!30] (0,-1) node {$O_2$} circle (.5cm);
}
+i\Res_{\xi_{k+1}=\vartheta_1}
\tikz[baseline=-2pt,scale=.6]{
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-2.5) node[below] {$(\vartheta)_1$} -- node[below right] {$(\beta)_1$} (0,-1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,2) node[above] {$\theta$} -- node[above left] {$\alpha$} (0,1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-.5) -- node[below left] {$\gamma$} (0,.5);
%
  \coordinate (a) at ($(0,-1)+(30:.5)$); \coordinate (b) at ($(0,1)+(-30:.5)$); \coordinate (c) at($(0,1)+(-150:.5)$);
  \begin{scope}[decoration={markings,mark=at position 0.5 with {\arrow{>}}}]
    \draw[-,postaction={decorate}] (a) to[out=30,in=-30] node[pos=.5,right] {$\gamma_{k+1}$} (b);
    \draw[-,postaction={decorate}] (-.8,-2.5) node[below] {$\vartheta_1$} to[out=90,in=-150] node[pos=.3,left] {$\beta_1$} (c);
  \end{scope}
%
  \draw[thick,fill=lightgray!30] (0,1) node {$O_1$} circle (.5cm);
  \draw[thick,fill=lightgray!30] (0,-1) node {$O_2$} circle (.5cm);
}
\quad=
\tikz[baseline=-2pt,scale=.6]{
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-2.5) node[below] {$(\vartheta)_1$} -- node[below right] {$(\beta)_1$} (0,-1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,2) node[above] {$\theta$} -- node[above left] {$\alpha$} (0,1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-.5) -- node[below left] {$\gamma$} (0,.5);
%
  \coordinate (a) at ($(0,-1)+(-150:.5)$);
  \begin{scope}[decoration={markings,mark=at position 0.3 with {\arrow{>}}}]
    \draw[-,postaction={decorate}] (-.8,-2.5) node[below] {$\vartheta_1$}
      to[out=90,in=-150] node[pos=.3,left] {$\beta_1$} node[pos=.6,cross] {} node[pos=.6,left] {$\Omega_1$} (a);
  \end{scope}
%
  \draw[thick,fill=lightgray!30] (0,1) node {$O_1$} circle (.5cm);
  \draw[thick,fill=lightgray!30] (0,-1) node {$O_2$} circle (.5cm);
}
$$
\pause
Многократно применяя эту процедуру (и аккуратно учитывая комбинаторику) получаем правило:
\vspace{-4pt}
$$
\tikz[baseline=-2pt,scale=.6]{
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-2.5) node[below] {$\vartheta$} -- node[below left] {$\beta$} (0,-1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,2) node[above] {$\theta$} -- node[above left] {$\alpha$} (0,1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-.5) -- node[below left] {$\gamma$} (0,.5);
%
  \draw[thick,fill=lightgray!30] (0,1) node {$O_1$} circle (.5cm);
  \draw[thick,fill=lightgray!30] (0,-1) node {$O_2$} circle (.5cm);
}
\quad\longrightarrow\quad
\tikz[baseline=-2pt,scale=.6]{
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-2.5) node[below] {$\vartheta$}
    -- node[pos=.3,cross=4pt] {} node[pos=.3,right=2pt] {$\Omega_1^{\otimes n}$} node[below left=2pt] {$\beta$} (0,-1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,2) node[above] {$\theta$} -- node[above left] {$\alpha$} (0,1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-.5) -- node[below left] {$\gamma$} (0,.5);
%
  \draw[thick,fill=lightgray!30] (0,1) node {$O_1$} circle (.5cm);
  \draw[thick,fill=lightgray!30] (0,-1) node {$O_2$} circle (.5cm);
}
\pause
\quad
\vcenter{\hsize=6cm
При этом $V(\vartheta)$ оказываются\\
слева от $V(\theta)$: $\lcolon\overleftarrow{V^\beta(\vartheta)}\overleftarrow{V^\alpha(\theta)}\rcolon$.\\
$\Rightarrow$ разложение для $O_2(y)O_1(x)$.
}
$$

\end{frame}

\begin{frame}
\frametitle{Коммутирование операторов: динамические полюсы}

Но мы не учли \emph{динамические полюсы}.
\pause
Динамические полюсы встречаются всегда парами: один от $O_1$ и один от $O_2$.
$$
-i\!\!\!\!\Res_{\substack{\xi_k=\theta_1+\\\quad+iu^\delta_{\gamma_k\alpha_1}}}
\tikz[baseline=-2pt,scale=.6]{
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-2) node[left] {$(\beta)_n$} node[below] {$(\vartheta)_n$} -- (0,-1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,2) node[left] {$(\alpha)_1$} node[above] {$\theta$} -- (0,1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-.5) -- node[left] {$(\gamma)_k$} (0,.5);
%
  \draw[thick,fill=lightgray!30] (0,1) node {$O_1$} circle (.5cm);
  \draw[thick,fill=lightgray!30] (0,-1) node {$O_2$} circle (.5cm);
}
=
\tikz[baseline=-2pt,scale=.6]{
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-2) node[left] {$(\beta)_n$} node[below] {$(\vartheta)_n$} -- (0,-1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,2) node[left] {$(\alpha)_1$} node[above] {$(\theta)_1$} -- (0,1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-.5) -- node[left] {$(\gamma)_k$} (0,.5);
%
  \coordinate (a) at ($(0,-1)+(30:.5)$); \coordinate (b) at ($(0,1)+(-30:.5)$);
  \begin{scope}[decoration={markings,mark=at position 0.25 with {\arrow{>}},mark=at position 0.8 with {\arrow{>}}}]
    \draw[-,postaction={decorate}] (a) to[out=30,in=-30] node[pos=.5] {} coordinate (c)
      node[pos=.25,right] {$\gamma$} node[pos=.75,above right] {$\delta$} (b);
  \end{scope}
  \draw[->] (1.5,0) node[right] {$\theta_1$} -- node[pos=.2,above] {$\alpha_1$} (1,0); \draw[-] (1,0) -- (c);
%
  \draw[thick,fill=lightgray!30] (0,1) node {$O_1$} circle (.5cm);
  \draw[thick,fill=lightgray!30] (0,-1) node {$O_2$} circle (.5cm);
}
\pause
,\quad
i\!\!\!\!\Res_{\substack{\xi_k=\vartheta_n+\\\quad+i\bar u^{\delta'}_{\beta_n\gamma_k}}}
\tikz[baseline=-2pt,scale=.6]{
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-2) node[left] {$(\beta)_n$} node[below] {$(\vartheta)_n$} -- (0,-1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,2) node[left] {$(\alpha)_1$} node[above] {$\theta$} -- (0,1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-.5) -- node[left] {$(\gamma)_k$} (0,.5);
%
  \draw[thick,fill=lightgray!30] (0,1) node {$O_1$} circle (.5cm);
  \draw[thick,fill=lightgray!30] (0,-1) node {$O_2$} circle (.5cm);
}
=
\tikz[baseline=-2pt,scale=.6]{
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-2) node[left] {$(\beta)n$} node[below] {$(\vartheta)_n$} -- (0,-1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,2) node[left] {$(\alpha)_1$} node[above] {$(\theta)_1$} -- (0,1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-.5) -- node[left] {$(\gamma)_k$} (0,.5);
%
  \coordinate (a) at ($(0,-1)+(30:.5)$); \coordinate (b) at ($(0,1)+(-30:.5)$);
  \begin{scope}[decoration={markings,mark=at position 0.25 with {\arrow{<}},mark=at position 0.8 with {\arrow{<}}}]
    \draw[-,postaction={decorate}] (a) to[out=30,in=-30] node[pos=.5] {} coordinate (c)
          node[pos=.25,right] {$\delta'$} node[pos=.75,above right] {$\gamma'$} (b);
  \end{scope}
  \draw[->] (1.5,0) node[right] {$\vartheta_n$} -- node[pos=.2,above] {$\beta_n$} (1,0); \draw[-] (1,0) -- (c);
%
  \draw[thick,fill=lightgray!30] (0,1) node {$O_1$} circle (.5cm);
  \draw[thick,fill=lightgray!30] (0,-1) node {$O_2$} circle (.5cm);
}
$$
\pause
В итоге суммы вкладов сокращается:
$$
\tikz[baseline=-2pt,scale=.6]{
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-2) node[left] {$(\beta)_n$} node[below] {$(\vartheta)_n$} -- (0,-1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,2) node[left] {$(\alpha)_1$} node[above] {$(\theta)_1$} -- (0,1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-.5) -- node[left] {$(\gamma)_k$} (0,.5);
%
  \coordinate (a) at ($(0,-1)+(30:.5)$); \coordinate (b) at ($(0,1)+(-30:.5)$);
  \begin{scope}[decoration={markings,mark=at position 0.25 with {\arrow{>}},mark=at position 0.8 with {\arrow{>}}}]
    \draw[-,postaction={decorate}] (a) to[out=30,in=-30] node[pos=.5] {} coordinate (c)
      node[pos=.25,right] {$\gamma$} node[pos=.75,above right] {$\delta$} (b);
  \end{scope}
  \draw[->] (1.5,0) node[right] {$\theta_1$} -- node[pos=.2,above] {$\alpha_1$} (1,0); \draw[-] (1,0) -- (c);
%
  \draw[thick,fill=lightgray!30] (0,1) node {$O_1$} circle (.5cm);
  \draw[thick,fill=lightgray!30] (0,-1) node {$O_2$} circle (.5cm);
}
\quad-\quad
\tikz[baseline=-2pt,scale=.6]{
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-2) node[left] {$(\beta)n$} node[below] {$(\vartheta)_n$} -- (0,-1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,2) node[left] {$(\alpha)_1$} node[above] {$(\theta)_1$} -- (0,1.5);
  \draw[double distance=3pt,thick,-{Stealth[length=3mm,width=3mm]}] (0,-.5) -- node[left] {$(\gamma)_k$} (0,.5);
%
  \coordinate (a) at ($(0,-1)+(30:.5)$); \coordinate (b) at ($(0,1)+(-30:.5)$);
  \begin{scope}[decoration={markings,mark=at position 0.25 with {\arrow{<}},mark=at position 0.8 with {\arrow{<}}}]
    \draw[-,postaction={decorate}] (a) to[out=30,in=-30] node[pos=.5] {} coordinate (c)
          node[pos=.25,right] {$\delta'$} node[pos=.75,above right] {$\gamma'$} (b);
  \end{scope}
  \draw[->] (1.5,0) node[right] {$\vartheta_n$} -- node[pos=.2,above] {$\beta_n$} (1,0); \draw[-] (1,0) -- (c);
%
  \draw[thick,fill=lightgray!30] (0,1) node {$O_1$} circle (.5cm);
  \draw[thick,fill=lightgray!30] (0,-1) node {$O_2$} circle (.5cm);
}
\quad\mathrel{\mathop=\limits_{\substack{\vartheta_n=\theta_1\\\beta_n=\alpha_1\\\gamma'=\bar\delta\\\delta'=\bar\gamma}}}0.
$$

\end{frame}

\begin{frame}
\frametitle{Взаимная локальность}

Пусть
\eq$$
\smashop{\sum_{\{\alpha'_i\}}\prod^n_{i=1}}\Omega^{\alpha'_i}_{\alpha_i}(O_I)F_{O_J}(\theta_1,\ldots,\theta_n)_{\alpha'_1\ldots\alpha'_n}
=C(O_I,O_J)F_{O_J}(\theta_1,\ldots,\theta_n)_{\alpha_1\ldots\alpha_n},
\label{O1O2-quasilocal}
$$
для любого набора частиц.
\pause
Этому условию нетрудно удовлетворить, поскольку в подходящем базисе левая часть равна $\prod_\alpha\Omega_\alpha$, причем вклады частицы и античастицы сокращаются.
\pause
Тогда мы будем говорить, что операторы $O_1$ и $O_2$ \emph{взаимно"=квазилокальны} и
\eq$$
O_1(x)O_2(y)=\Cases{C(O_1,O_2)O_2(y)O_1(x),&x^0=y^0,\ x^1>y^1;\\C^{-1}(O_2,O_1)O_2(y)O_1(x),&x^0=y^0,\ x^1<y^1.}
\label{O1O2-commut}
$$
\pause
Произведение
\eq$$
e^{2\pi i\omega(O_1,O_2)}=\Omega(O_1,O_2)=C(O_1,O_2)C(O_2,O_1)
\label{omega(O1,O2)}
$$
называют \emph{индексом взаимной локальности}, а величину $\omega$ "--- \emph{показателем} взаимной локальности.
\pause
В эвклидовой плоскости имеем
$$
O_1(ze^{2\pi i},\bz e^{-2\pi i})O_2(0,0)=\Omega(O_1,O_2)O_1(z,\bz)O_2(0,0).
$$
\pause
Пусть
$$
A^\alpha(x)=\int_\cCrr{d\theta\over2\pi}\,e^{-iP_\alpha(\theta)x}V^\alpha(\theta)+\cdots
$$
"--- некоторый локальный оператор, уничтожающий частицу сорта $\alpha$ (и рождающий ее античастицу).\uncover<6>{{\red\vrule width 2pt height 8pt depth 4pt}}
\pause
Тогда
\eq$$
\Omega_\alpha(O)=C(O,A^\alpha)=\Omega(O,A^\alpha)
\label{Omega(O)-mutualloc}
$$

\end{frame}

\begin{frame}
\frametitle{Теорема и гипотеза Смирнова}

Тем самым мы получили

\begin{block}{Теорему Смирнова}

Если наборы формфакторов двух операторов удовлетворяют формфакторным аксиомам, т.е.\ ограничению на рост, условиям лоренцева спина, цикличности, перестановки, кинематического и динамического полюсов, эти операторы взаимно"=квазилокальны с индексом квазилокальности, определенным выше.

\end{block}
\pause

Имеется обратная к этой теореме

\begin{block}{Гипотеза Смирнова}

Для любого набора взаимно"=квазилокальных операторов наборы формфакторов этих операторов удовлетворяют формфакторным аксиомам.

\end{block}

\end{frame}

\begin{frame}
\frametitle{Модель Изинга}

Рассмотрим случай $S(\theta)=-1$, что соответствует свободному фермиону или модели Изинга:
$$
V(\theta_1)V(\theta_2)+V(\theta_2)V(\theta_1)=2\pi\delta(\theta_1-\theta_2-i\pi)+2\pi\delta(\theta_1-\theta_2+i\pi).
$$
\pause
Поле свободного фермиона:
\eq$$
\psi_\pm(x)=\pm\sqrt{m\over2}\int_\cCrr{d\theta\over2\pi}\,e^{-iP(\theta)x\pm{1\over2}(\theta+{i\pi\over2})}V(\theta),
\quad
s_{\psi_\pm}=\pm{1\over2},
\
\Omega_1(\psi_\pm)=-1.
\label{IsingT-psi}
$$
\pause
Формфакторы свободного фермиона:
$$
F_{\psi_\pm}(\theta_1,\ldots,\theta_n)\only<3>{\rlap{\red\vrule width 2pt height 8pt depth 4pt}}
\uncover<4->{=\pm\delta_{n1}\sqrt{m\over2}e^{\pm{1\over2}(\theta_1+{i\pi\over2})}.}
$$
\pause\pause
Для тензора энергии"=импульса
\eq$$
T_{zz}=-{i\over2}\lcolon\psi_+\,\d\psi_+\rcolon,
\qquad
T_{\bz\bz}={i\over2}\lcolon\psi_-\,\bd\psi_-\rcolon,
\qquad
T_{z\bz}={im\over2}\lcolon\psi_+\psi_-\rcolon,
\label{IsingT-T}
$$
\uncover<5>{{\red\hrule}}
\pause
имеем ненулевые формфакторы только при $n=2$:
\eq$$
\Aligned{
F_{T_{zz}}(\theta_1,\theta_2)
&={im^2\over4}e^{\theta_1+\theta_2}\sh{\theta_1-\theta_2\over2},
&\quad&s_{T_{zz}}=2,
&\quad&\Omega_1(T_{zz})=1,
\\
F_{T_{\bz\bz}}(\theta_1,\theta_2)
&={im^2\over4}e^{-\theta_1-\theta_2}\sh{\theta_1-\theta_2\over2},
&&s_{T_{\bz\bz}}=-2,
&&\Omega_1(T_{\bz\bz})=1,
\\
F_{T_{z\bz}}(\theta_1,\theta_2)
&={im^2\over4}\sh{\theta_1-\theta_2\over2},
&\quad&s_{T_{z\bz}}=0,
&\quad&\Omega_1(T_{z\bz})=1.
}
\label{IsingT-FT}
$$

\end{frame}

\begin{frame}
\frametitle{Модель Изинга}

Найдем \emph{минимальный формфактор}, то есть функцию $R(\theta)$, такую, что
\eq$$
R(\theta)=R(2\pi i-\theta),
\qquad
R(\theta)=S(\theta)R(-\theta),
\label{R-relations}
$$
не имеющую полюсов на физическом листе $0\le\Im\theta\le\pi$ кроме точки $\theta=i\pi$.
\pause
В случае модели Изинга минимальный формфактор равен
$$
R(\theta)=\only<2>{\rlap{\red\vrule width 2pt height 8pt depth 4pt}}\uncover<3->{\th{\theta\over2}}.
$$
\pause\pause
Можно ввести два поля $\sigma(x)$ и $\mu(x)$ с формфакторами
\Align$$
F_\sigma(\theta_1,\ldots,\theta_{2n})
&=i^nG_\sigma m^{1/8}\prod^{2n}_{i<j}\th{\theta_i-\theta_j\over2},
&&s_\sigma=0,\ \Omega_1(\sigma)=-1,
\label{IsingT-sigma}
\\
F_\mu(\theta_1,\ldots,\theta_{2n+1})
&=i^{n-1}G_\mu m^{1/8}\prod^{2n+1}_{i<j}\th{\theta_i-\theta_j\over2},
&&s_\mu=0,\ \Omega_1(\mu)=1.
\label{IsingT-mu}
$$
\pause
По свойствам взаимной локальности имеется четыре класса операторов:
\begin{itemize}
\item[$\{\ve\}$:] Бозонные четные операторы с $\Omega_1(O)=1$. Это семейство включает в себя, например, $1$ и~$T_{\mu\nu}$.
\pause
\item[$\{\mu\}$:] Бозонные нечетные операторы с $\Omega_1(O)=1$. Семейство включает~$\mu$.
\pause
\item[$\{\sigma\}$:] Бозонные четные операторы с $\Omega_1(O)=-1$. Семейство включает~$\sigma$.
\pause
\item[$\{\psi\}$:] Фермионные нечетные операторы с $\Omega_1(O)=-1$. Семейство включает~$\psi_\pm$.
\end{itemize}

\end{frame}

\begin{frame}
\frametitle{Модель Изинга: взаимная локальность}

Для представителей этих классов имеем ($x^0=y^0$)
\eq$$
\Gathered{
\ve(x)\ve(y)=\ve(y)\ve(x),
\\
\ve(x)\mu(y)=\mu(y)\ve(x),
\quad
\ve(x)\sigma(y)=\sigma(y)\ve(x),
\quad
\ve(x)\psi(y)=\psi(y)\ve(x),
\\
\uncover<2->{\mu(x)\mu(y)=\mu(y)\mu(x),}
\\
\uncover<2->{\mu(x)\sigma(y)=\sign(x^1-y^1)\sigma(y)\mu(x),
\quad
\mu(x)\psi(y)=\sign(x^1-y^1)\psi(y)\mu(x),}
\\
\uncover<3->{\sigma(x)\sigma(y)=\sigma(y)\sigma(x),
\quad
\sigma(x)\psi(y)=-\sign(x^1-y^1)\psi(y)\sigma(x),}
\\
\uncover<4->{\psi(x)\psi(y)=-\psi(y)\psi(x).}
}\label{ve-mu-sigma-psi-commut}
$$
\pause\pause\pause\pause
Имеется три сектора взаимно"=локальных операторов:
\begin{itemize}
\item«бозонный» сектор $\{\ve,\mu\}$;
\pause
\item«фермионный» сектор $\{\ve,\psi\}$;
\pause
\item«двойственный бозонный» сектор $\{\ve,\sigma\}$.
\end{itemize}
\pause
Бозонный и двойственный бозонный сектор отвечают параметру беспорядка и параметру порядка при $T<T_c$. По двойственности Крамерса"--~Ваннье они меняются местами при $T>T_c$.

\end{frame}

\begin{frame}
\frametitle{Модель sh"=Гордона}

Рассмотрим более общую $S$"=матрицу. Уравнения на минимальный формфактор
\eq$$
R(\theta)=R(2\pi i-\theta),
\quad
R(\theta)=S(\theta)R(-\theta),
\tag{\ref{R-relations}}
$$
\pause
в случае одной частицы можно по правилу
\eq$$
S(\theta)=\exp i\int^\infty_0{dt\over t}\,f(t)\sin\theta t
\pause
\ \Rightarrow\
R(\theta)=R_0\exp\int^\infty_0{dt\over t}\,{f(t)(\cos(\theta-i\pi)t-1)\over2\sh\pi t},
\label{R-S-rel}
$$
\pause
В модели sh"=Гордона имеем
\eq$$
R(\theta)=\exp\left(4\int^\infty_0{dt\over t}\,
{\sh{\pi t\over2}\sh{\pi pt\over2}\sh{\pi(p+1)t\over2}\over\sh^2\pi t}\cos(\theta-i\pi)t\right).
\label{ShG-R}
$$
\pause
Можно проверить, что
\eq$$
R(\theta)R(\theta+i\pi)={\sh\theta\over\sh\theta-i\sin\pi p}=1/f(e^{-\theta}),
\qquad
f(z)=1+{2i\sin\pi p\over z-z^{-1}}.
\label{ShG-RR-id}
$$
\pause
Функция $f(z)$ обладает свойством
\eq$$
{f(e^\theta)\over f(e^{-\theta})}=S(\theta).
\label{ShG-f-S-rel}
$$
\pause
Кроме того, положим
\eq$$
\rho=(-R(i\pi)\sin\pi p)^{-1/2}.
\label{ShG-rhoconst}
$$

\end{frame}

\begin{frame}
\frametitle{Модель sh"=Гордона}

Тогда
\eq$$
F_O(\theta_1,\ldots,\theta_n)
=\rho^n\prod^n_{i<j}R(\theta_i-\theta_j)\cdot J_O(e^\theta_1,\ldots,e^\theta_n),
\label{FO-JO-rel}
$$
\pause
где $J_O(x_1,\ldots,x_n)$ "--- рациональные симметричные функции с полюсами в точка $x_i=-x_j$:
\eq$$
\Res_{z'=-z}J_O(z',z,x_1,\ldots,x_n)=-iz\sin\pi p\cdot 
\left(\prod^n_{i=1}f\left(z\over x_i\right)-\prod^n_{i=1}f\left(x_i\over z\right)\right)J(x_1,\ldots,x_n).
\label{JO-kinpole}
$$
\pause
Функции
\eq$$
J_a(x_1,\ldots,x_n)
=\sum_{I_n=I_-\sqcup I_+}e^{i\pi a(\#I_--\#I_+)}\prod_{i\in I_-\atop j\in I_+}f\left(x_i\over x_j\right),
\label{Ja-def}
$$
удовлетворяют (\ref{JO-kinpole}) и дают формфакторы операторов $e^{\alpha\varphi(x)}/\langle e^{\alpha\varphi(0)}\rangle$, где
$$
a={1\over2}-\alpha\sqrt{(-p)(1+p)\over2}.
$$




\end{frame}

\end{document}
