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)

    “The professional celebrity, male and female, is the crowning result of the star system of a society that makes a fetish of competition. In America, this system is carried to the point where a man who can knock a small white ball into a series of holes in the ground with more efficiency than anyone else thereby gains social access to the President of the United States.”
    —C. Wright Mills (1916–1962)