Dyson series

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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).

The Dyson operator[edit]

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,


Derivation of the Dyson series[edit]

This leads to the following Neumann series:

Here we have t1 > t2 > ..., > tn, 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 n! sub-regions defined by t1 > t2 > ... > tn, t2 > t1 > ... > tn, 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, it holds the identity

Summing up all the terms, we obtain the Dyson series:

The Dyson series for wavefunctions[edit]

Then, going back to the wavefunction for t > t0,

Returning to the Schrödinger picture, for tf > ti,

See also[edit]