One of the most important tool in quantum mechanics is the Dyson series because it is the basis of the perturbative theory. There is a step in the derivation that I can't understand.
$\{H(t_i)\}$ are not commuting operators. The $T$ product is defined as follow:
T[H(t)H(t')] = \theta(t-t')H(t)H(t') + \theta(t'-t)H(t')H(t) where $\theta$ is the Heaviside function. You can extend it to $n$ factors, ordering them so that later times ($t$) stand to the left of earlier times.
I need to proof that: $ \int_{-\infty}^{t} dt_1 \int_{-\infty}^{t_1} dt_2 \ldots \int_{-\infty}^{t_{n-1}} dt_n H(t_1)H(t_2)\ldots H(t_n) $ is equal to $\frac{1}{n!}\int_{-\infty}^{t} dt_1 \int_{-\infty}^{t} dt_2 \ldots \int_{-\infty}^{t} dt_n T[H(t_1)H(t_2)\ldots H(t_n)] $
I tried to start with $n=2$, then I think it's easy to use induction, but I'm stuck:
$\int_{-\infty}^{t} dt_1 \int_{-\infty}^{t} dt_2T[H(t_1)H(t_2)] = \int_{-\infty}^{t} dt_1 \int_{-\infty}^{t_1} dt_2 H(t_1)H(t_2) + \int_{-\infty}^{t} dt_1 \int_{t_1}^{t} dt_2 H(t_2)H(t_1)$
but now? I tried to change variables ... can someone help me?
I tried also to visualized it as the integral over a square $(-\infty, t_1=t]\times (-\infty, t_2=t]$ subdivided into two triangles by the diagonal $t_2=t_1$ of an operator $K(t_1,t_2) = K(t_2,t_1)$ because $T[H(t_1)H(t_2)] = T[H(t_2)H(t_1)]$