Dyson series
In scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative expansion of the time evolution operator in the interaction picture. Each term can be represented by a sum of Feynman diagrams. This series diverges asymptotically, but in quantum electrodynamics (QED) at the second order the difference from experimental data is in the order of 10−10. This close agreement holds because the coupling constant  (also known as the fine-structure constant) of QED is much less than 1. Notice that in this article Planck units are used, so that ħ = 1 (where ħ is the reduced Planck constant). The Dyson operator 
Suppose that we have a Hamiltonian , which we split into a free part   and an interacting part , i.e. . We will work in the interaction picture here, that is, 
 
where  is time-independent and  is the possibly time-dependent interacting part of the Schrödinger picture. To avoid subscripts,  stands for  in what follows. We choose units such that the reduced Planck constant  is 1. In the interaction picture, the evolution operator  defined by the equation:

is called the Dyson operator. We have a few properties:
 Identity and normalization: 
 Composition: 
 Time Reversal: 
 Unitarity: 
and from these is possible to derive the time evolution equation of the propagator:

We notice again that in the interaction picture the Hamiltonian is the same as the interaction potential . This equation is not to be confused with the Tomonaga–Schwinger equation

Consequently:

which is ultimately a type of Volterra equation. Derivation of the Dyson series 
An iterative solution of the Volterra equation above leads to the following Neumann series:

Here we have , 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  sub-regions defined by , , etc. Due to the symmetry of K, the integral in each of these sub-regions is the same and equal to  by definition. So it is true that

Returning to our previous integral, the following identity holds

Summing up all the terms, we obtain the Dyson series which is a simplified version of the Neumann series above and which includes the time ordered products:

This result is also called Dyson's formula, given he explicitly described the time ordered product and its relation to causality. Application on state vectors 
One can then express the state vector at time t in terms of the state vector at time t0, for t > t0,

Then, the inner product of an initial state (ti = t0) with a final state (tf = t) in the Schrödinger picture, for tf > ti, is as follows:

See also 
Schwinger–Dyson equation
Magnus series
Peano–Baker series
Picard iteration

References

Charles J. Joachain, Quantum collision theory, North-Holland Publishing, 1975,  (Elsevier)

Scattering theory
Quantum field theory
Freeman Dyson