Derivation of The Dyson Series
This leads to the following Neumann series:
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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