Dyson Series - Derivation of The Dyson Series

Derivation of The Dyson Series

This leads to the following Neumann series:

\begin{array}{lcl} U(t,t_0) & = & 1 - i \int_{t_0}^{t}{dt_1V(t_1)}+(-i)^2\int_{t_0}^t{dt_1\int_{t_0}^{t_1}{dt_2V(t_1)V(t_2)}}+\cdots \\ & &{} + (-i)^n\int_{t_0}^t{dt_1\int_{t_0}^{t_1}{dt_2 \cdots \int_{t_0}^{t_{n-1}}{dt_nV(t_1)V(t_2) \cdots V(t_n)}}} +\cdots. \end{array}

Here we have t1 > t2 > ..., > tn so we can say that the fields are time ordered, and it is useful to introduce an operator called time-ordering operator, defining:

We can now try to make this integration simpler. in fact, by the following example:

Assume that K is symmetric in its arguments and define (look at integration limits):

The region of integration can be broken in n! sub-regions defined by t1 > t2 > ... > tn, t2 > t1 > ... > tn, etc. Due to the symmetry of K, the integral in each of these sub-regions is the same, and equal to n by definition. So it is true that:

Returning to our previous integral, it holds the identity:

Summing up all the terms we obtain the Dyson series:

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