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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\tham{\tilde{{\mathcal H}}}
\def\ug{\tilde{\gamma}}
\def\tg{T_{\gamma}}
\def\tgt{T_{\tilde{\gamma}}}
\def\t{\tau}
\def\d{\delta}
\def\e{\epsilon}
\def\g{\gamma}
%
\begin{document} 

\begin{center}
{\bf Solution to Exercise 6}
\end{center}

We want to examine the effect of hermitian conjugation on the operator 
expression 
\newline
$\tg {\rm exp} [ (-i) \int_{\g} \ham(\t) d \t ]$. The expression can be 
expanded as
\ben
\tg {\rm exp} \left[ (-i) \int_{\g} \ham(\t) d \t \right]
= \sum_n \frac{(-i)^n}{n!} \int_{z_1}^{z_N} d \t_1 \cdots \int_{z_1}^{z_N} 
d \t_n \tg \left[ \ham(\t_1) \cdots \ham(\t_n) \right].
\een
The time ordering
operator $\tg$ orders the operators along the directed path $\g$ such that, 
for example,
\[
\tg [ \ham(z_2) \ham (z_1) \ham (z_N) ] = \ham (z_N)  \ham(z_2) \ham (z_1) 
\]
if $z_N > z_2 > z_1$ along the path $\g$.

We will first note the effect of hermitian conjugation on the integration 
measure of each term on the right hand side of equation (1). We get, 
$-i \rightarrow i$, $d \t \rightarrow d \t^{\ast}$, and the limits of 
integration $(z_1, z_N) \rightarrow (z_1^{\ast}, z_N^{\ast})$. The difference 
of a negative sign can be absorbed by reversing the limits of integration. 
Thus
\ben
\left( -i \int_{z_1}^{z_N} d \t \right)^{\dagger} 
= \left( -i \int_{z_N^{\ast}}^{z_1^{\ast}} d \t^{\ast} \right).
\een

Next, we will examine the behaviour of the integrand, which is a product of 
time ordered operators, under hermitian conjugation. Let us look at 
$\tg [ \ham(\t_1) \ham(\t_2) \cdots \ham(\t_m) ]$ where, for definiteness, 
we assume $\t_1 < \t_2 < \cdots < \t_m$ as we go along some path $\g$. Then,
\bea
\left[\tg \left( \ham(\t_1) \ham(\t_2) \cdots \ham(\t_m) \right)\right]
^{\dagger}
&=& \left[ \ham(\t_m) \cdots \ham(\t_2) \ham(\t_1) \right]^{\dagger}
\nonumber \\
&=& \ham(\t_1)^{\dagger} \ham(\t_2)^{\dagger} \cdots \ham(\t_m)^{\dagger}
\nonumber \\
&=& \tham(\t_1^{\ast})\tham(\t_2^{\ast}) \cdots \tham(\t_m^{\ast})
\nonumber \\
&=& \tgt \left[ \tham(\t_1^{\ast})\tham(\t_2^{\ast}) \cdots 
\tham(\t_m^{\ast}) \right].
\eea
In the last but one line of the above equation we have used the definition
$\tham(z) = (\ham(z^{\ast}))^{\dagger}$. From the expression in this line we 
see that the effect of hermitian conjugation on a product of operators time
ordered along the contour $\g = (z_1, z_N)$, is to time order the operators
along the contour $\ug = (z_N^{\ast}, z_1^{\ast})$.

From equations (1), (2) and (3) we can therefore write,
\bea
\left(\tg {\rm exp} \left[ (-i) \int_{\g} \ham(\t) d \t \right]\right)
^{\dagger} &=& 
\sum_n \frac{(-i)^n}{n!} \int_{z_N^{\ast}}^{z_1^{\ast}} d \t_1^{\ast}
 \cdots  \int_{z_N^{\ast}}^{z_1^{\ast}} d \t_n^{\ast}
\tgt \left[ \ham(\t_1^{\ast}) \cdots \ham(\t_n^{\ast}) \right] \nonumber \\
&=& \left(\tgt {\rm exp} \left[ (-i) \int_{\ug} \tham(\t) d \t \right]\right)
^{\dagger}.
\eea

\end{document}