Dyson series

Summary

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]

Notice that in this article Planck units are used, so that ħ = 1 (where ħ is the reduced Planck constant).

The Dyson operatorEdit

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

We will work in the interaction picture here, that is,

 

where   is the possibly time-dependent interacting part of the Schrödinger picture. To avoid subscripts,   stands for   in what follows. We choose 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 time evolution equation of the propagator:

 

This is not to be confused with the Tomonaga–Schwinger equation

Consequently:

 

Which is ultimately a type of Volterra equation

Derivation of the Dyson seriesEdit

This leads to the following Neumann series:

 

Here we have  , 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   sub-regions defined by  ,  , 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, the following identity holds

 

Summing up all the terms, we obtain Dyson's theorem for the Dyson series:[clarification needed]

 

WavefunctionsEdit

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

 

Returning to the Schrödinger picture, for tf > ti,

 

See alsoEdit

ReferencesEdit

  • Charles J. Joachain, Quantum collision theory, North-Holland Publishing, 1975, ISBN 0-444-86773-2 (Elsevier)