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 VS(t) as H = H0 + VS(t).
The potential in the interacting picture is where
is the possibly time-dependent interacting part of the Schrödinger picture.
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:[4] In the interaction picture, the Hamiltonian is the same as the interaction potential
The formal solution is which is ultimately a type of Volterra integral.
An iterative solution of the Volterra equation above leads to the following Neumann series: Here,
, called the time-ordering operator, and to define The limits of the integration can be simplified.
In general, given some symmetric function
, the integral in each of these sub-regions is the same and equal to
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:[5] This result is also called Dyson's formula.
[6] The group laws can be derived from this formula.
The state vector at time
can be expressed in terms of the state vector at time
as The inner product of an initial state at
is: The S-matrix may be obtained by writing this in the Heisenberg picture, taking the in and out states to be at infinity:[7] Note that the time ordering was reversed in the scalar product.