Dyson operator

In scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative series, and each term is represented by 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).

Suppose that we have a Hamiltonian H, which we split into a free part H₀ and an interacting part V, i.e. H = H₀ + V.

We will work in the interaction picture here and assume units such that the reduced Planck constant ħ is 1.

In the interaction picture, the evolution operator U defined by the equation

is called the Dyson operator.

We have

and hence the Tomonaga–Schwinger equation,

Consequently,

This leads to the following Neumann series:

Here we have t₁ > t₂ > ..., > t_n, so we can say that the fields are time-ordered, and it is useful to introduce an operator ${\displaystyle {\mathcal {T}}}$ called time-ordering operator, defining

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