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:

Read more about this topic:  Dyson Series

Famous quotes containing the words dyson and/or series:

    “The question that will decide our destiny is not whether we shall expand into space. It is: shall we be one species or a million? A million species will not exhaust the ecological niches that are awaiting the arrival of intelligence.”
    —Freeman Dyson (b. 1923)

    “Every Age has its own peculiar faith.... Any attempt to translate into facts the mission of one Age with the machinery of another, can only end in an indefinite series of abortive efforts. Defeated by the utter want of proportion between the means and the end, such attempts might produce martyrs, but never lead to victory.”
    —Giuseppe Mazzini (1805–1872)