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.^{[clarification needed]}

In the interaction picture, a Hamiltonian H, can be split into a free part H₀ and an interacting part V_S(t) as H = H₀ + V_S(t).

The potential in the interacting picture is

where ${\displaystyle H_{0}}$ is time-independent and ${\displaystyle V_{\mathrm {S} }(t)}$ is the possibly time-dependent interacting part of the Schrödinger picture. To avoid subscripts, ${\displaystyle V(t)}$ stands for ${\displaystyle V_{\mathrm {I} }(t)}$ 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: