Well, it becomes a bit clearer when we see the final formulas of Ref. 1:

$$\delta \langle a_f , t_f |a_i , t_i \rangle
~=~ \frac{i}{\hbar} \int_{t_i}^{t_f} \! dt
\langle a_f , t_f | \delta L(t) |a_i , t_i \rangle \tag{7.126}
$$

$$ \delta^{\prime} \delta \langle a_f , t_f |a_i , t_i \rangle ~=~\frac{1}{2}\left(\frac{i}{\hbar}\right)^2
\int_{t_i}^{t_f} \! dt \int_{t_i}^{t_f} \!dt^{\prime} $$
$$\times \langle a_f , t_f |T[ \delta L(t)~\delta^{\prime}L(t^{\prime}) ] |a_i , t_i \rangle. \tag{7.131}$$

Recall that the Schwinger action principle can be described via a time integral $\int_{t_i}^{t_f}\!dt~ \delta L(t)$ of an operator $\delta L(t)$. Imagine that the time interval

$$[t_i,t_f]~=~\cup_{n=1}^N I_n, \qquad I_n~:=~[t_{n-1},t_n],$$

is divided into a sufficiently fine discretization $t_i=t_0<t_1, \ldots t_{N-1} < t_N=t_f $, where the integer $N$ is sufficiently large.

By inserting many completeness identities $\sum_b |b , t_n \rangle\langle b , t_n |={\bf 1}$, we can split a total variation (7.126) into many small contribution labelled by the time intervals $I_n$, $n\in\{1,2, \ldots, N\}$.

Similarly, when performing a double variation (7.131), we will get $N$ diagonal and $N(N-1)$ off-diagonal contributions labelled by two time intervals $I_n$ and $I_m$, where $n,m\in\{1,2, \ldots, N\}$. (A diagonal contribution $n=m$ refers to the same time interval $I_n=I_m$.) If in the limit $N\to\infty$, the $N(N-1)$ off-diagonal contributions dominate over the $N$ diagonal contributions, the two variations becomes *effectively* independent.

However in hindsight, it seems that Ref. 1 is assuming that the two variation $\delta$ and $\delta^{\prime}$ are *manifestly* independent and not just *effectively* independent. Manifest independence here means that the $\delta^{\prime}$ variation simply doesn't act on the $\delta L(t)$ operator, and vice-versa.

References:

- D.J. Toms,
*The Schwinger Action Principle and Effective Action,* 1997, Section 7.6.

This post imported from StackExchange Physics at 2014-04-20 15:03 (UCT), posted by SE-user Qmechanic