The propagator has the following transitivity property
<br />
\int\limits_{-\infty}^{\infty} K(x,t,x',t') K(x',t',x'',t'')dx' = K(x,t,x'',t'')<br />
The meaning and importance of this identity become clear if you realize that the propagator (Green function) K(x,t : y,t_{0})can be interpreted as transition amplitude. That is the probability amplitude for the system prepared at t_{0} with position eigenvalue y to be found at a later time t at x ;
K(x,t : y,t_{0}) = \langle x | e^{-iH(t-t_{0})}| y \rangle = \langle x,t | y,t_{0}\rangle \ \ \ (1)
where \langle x,t| and |y,t_{0}\rangle are the eigenket and the eigenbra of the position operator in the Hiesenberg picture. Because at any given time these eigen vectors form a complete set, we can insert the identity operator;
\int dz |z,t_{1}\rangle \langle z,t_{1}| = \hat{1}
at any place we want. So, by dividing the evolution interval (t_{0},t_{2}) into two parts (t_{0},t_{1}) and (t_{1},t_{2}), we can decompose the transition amplitude as :
\langle x , t_{2} | y , t_{0} \rangle = \int dz \langle x ,t_{2} | z , t_{1} \rangle \langle z , t_{1}| y , t_{0}\rangle \ \ (2)
(t_{2} > t_{1} > t_{0})
This composition property of the transition amplitude is an important consistency requirement underlying the whole formalism of path integration. Indeed, we can use it to show that the transition amplitude \langle x , t| y , t_{0} \rangle satisfies Schrödinger equation in the variables (x,t), just as the propagater K(x,t : y,t_{0}) ;
i\partial_{t}\langle x , t | y , t_{0} \rangle = H(t_{0})\langle x ,t | y ,t_{0} \rangle \ \ \ (3)
( argue that you can write \langle x , t + \epsilon | y , t_{0} \rangle = \delta (x - y) - i \epsilon\delta(x-y)H(t_{0})
and use eq(2) for t_{1} = t and t_{2} = t + \epsilon)
When he path integration method is carried over to Brownian motion, eq(2) is known as the Chapman-Kolmogorov equation, and in diffusion theory, the Smoluchowsky equation.
regards
sam