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%% Indranil's LaTeX macros
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% Equation environments
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def\ben{\begin{equation}}
\def\een{\end{equation}}
\def\benu{\begin{enumerate}}
\def\enu{\end{enumerate}}
%
% Special letters 
\def\ham{{\mathcal H}}
\def\mu{{\mathcal U}}
\def\ut{\tilde{U}}
\def\tt{\tilde{T}}
\def\t{\tau}
\def\d{\delta}
\def\e{\epsilon}
%

\begin{document} 

\begin{center}
{\bf Solution to Exercise 7}
\end{center}

We want to find the differential equation satisfied by the operator 
$\ut(t, t_0)$, which is defined as 
\[
\ut(t,t_0) = \tt {\rm exp}\left[ -i \int_{t_0}^{t} \ham(\t) d \t \right].
\]
Here $\tt$ is the antichronological ordering operator defined as 
\[
\tt \left[ A(t_1) B(t_2) \right] = \left \{ 
\begin{array}{cc}
A(t_1) B(t_2) & (t_1 < t_2) \\
B(t_2) A(t_1) & (t_1 > t_2).
\end{array} \right.
\]

We examine the time evolution of $\ut(t,0)$ by comparing it with $ \ut(t +
\d t, 0)$. We have,
\bea
\ut(t +\d t, t_0) &=& \tt {\rm exp} \left[ -i \int_{t_0}^{t} \ham(\t) d \t
-i \d t \ham(t+) \right] \nonumber \\
&=& \tt \sum_n \frac{(-i)^n}{n!}  \left[ \int_{t_0}^{t} \ham(\t) d \t
+ \d t \ham(t+) \right]^n \nonumber \\
&=& \ut(t, t_0) + (-i) \tt \sum_{n=1}^{\infty} \frac{(-i)^{n-1}}{(n-1)!}
\d t \ham(t+) \left[ \int_{t_0}^{t} \ham(\t) d \t \right]^{n-1} 
+ {\mathcal O}(\d t)^2 \nonumber \\
&=& \ut(t, t_0) -i \ut(t, t_0) \d t \ham(t) + {\mathcal O}(\d t)^2.
\eea
Here $t+ = t + \e$, with $\e \rightarrow 0$. Due to anti time ordering, in 
the last line of the above expression, $\ham(t+)$ moves to the right of all 
other terms. Thus, $\ut(t, t_0)$ obeys the differential equation
\ben
\frac{d}{dt} \ut(t, t_0) = \left( -i \right) \ut(t, t_0) \ham(t).
\een
\end{document}