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 and hence the time evolution equation of the propagator: This is not to be confused with the Tomonaga–Schwinger equation Consequently: Which is ultimately a type of Volterra equation Derivation of the Dyson series This 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 Dyson's theorem for the Dyson series: 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