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.

Dyson operator

In the interaction picture, a Hamiltonian H, can be split into a free part H0 and an interacting part VS(t) as . The potential in the interacting picture is where H_0 is time-independent and is the possibly time-dependent interacting part of the Schrödinger picture. To avoid subscripts, V(t) stands for in what follows. In the interaction picture, the evolution operator U is defined by the equation: This is sometimes called the Dyson operator. The evolution operator forms a unitary group with respect to the time parameter. It has the group properties: and from these is possible to derive the time evolution equation of the propagator: In the interaction picture, the Hamiltonian is the same as the interaction potential and thus the equation can also be written in the interaction picture as Caution: this time evolution equation is not to be confused with the Tomonaga–Schwinger equation. The formal solution is which is ultimately a type of Volterra integral.

Derivation of the Dyson series

An iterative solution of the Volterra equation above leads to the following Neumann series: Here,, and so the fields are time-ordered. It is useful to introduce an operator \mathcal T, called the time-ordering operator, and to define The limits of the integration can be simplified. In general, given some symmetric function one may define the integrals and The region of integration of the second integral can be broken in n! sub-regions, defined by. Due to the symmetry of K, the integral in each of these sub-regions is the same and equal to S_n by definition. It follows that Applied to the previous identity, this gives Summing up all the terms, the Dyson series is obtained. It is a simplified version of the Neumann series above and which includes the time ordered products; it is the path-ordered exponential: This result is also called Dyson's formula. The group laws can be derived from this formula.

Application on state vectors

The state vector at time t can be expressed in terms of the state vector at time t_0, for t>t_0, as The inner product of an initial state at t_i=t_0 with a final state at t_f=t in the Schrödinger picture, for t_f>t_i is: The S-matrix may be obtained by writing this in the Heisenberg picture, taking the in and out states to be at infinity: Note that the time ordering was reversed in the scalar product.