To use all functions of this page, please activate cookies in your browser.
my.chemeurope.com
With an accout for my.chemeurope.com you can always see everything at a glance – and you can configure your own website and individual newsletter.
 My watch list
 My saved searches
 My saved topics
 My newsletter
Dyson seriesIn scattering theory, the Dyson series is a perturbative series, and each term is represented by Feynman diagrams. This series diverges asymptotically, but in quantum electrodynamics at the second order the difference from experimental data is in the order of 10^{ − 10}. We get this result because the coupling constant of QED (also known as the fine structure constant) is much less than 1. Notice that in this article ħ = 1. Additional recommended knowledge
The Dyson operatorWe suppose we have a Hamiltonian H which we split into a "free" part H_{0} and an "interacting" part V i.e. H=H_{0}+V. We will work in the interaction picture here. In the interaction picture, the evolution operator U defined by the equation:
is called Dyson operator. We have and then (TomonagaSchwinger equation) Thus: Derivation of the Dyson seriesThis leads to the following Neumann series: If we assume that t > t_{1} > t_{2} > ... > t_{n} we can say that the fields are time ordered, and so it is useful to introduce an operator called timeordering operator. Defining: We can now try to make this integration simpler. in fact, in the following example: If K is symmetric in its arguments, we can define (look at integration limits): And 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 wavefunctionsThen, going back to the wavefunction for t>t_{0},
Returning to the Schrödinger picture, for t_{f} > t_{i},
References


This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Dyson_series". A list of authors is available in Wikipedia. 