Feynman parametrization is a technique for evaluating loop integrals which arise from Feynman diagrams with one or more loops. However, it is sometimes useful in integration in areas of pure mathematics as well.
Richard Feynman observed that:
which simplifies evaluating integrals like:
More generally, using the Dirac delta function:
Even more generally, provided that for all :
See also Schwinger parametrization.
Now just linearly transform the integral using the substitution,
- which leads to so
and we get the desired result:
In more general cases, derivations can be done very efficiently using the Schwinger parametrization. For example, in order to derive the Feynman parametrized form of :, we first reexpress all the factors in the denominator in their Schwinger parametrized form:
Then we perform the following change of integration variables,
The next step is to perform the integration.
where we have defined
Substituting this result, we get to the penultimate form,
and, after introducing an extra integral, we arrive at the final form of the Feynman parametrization, namely,
Similarly, in order to derive the Feynman parametrization form of the most general case, : one could begin with the suitable different Schwinger parametrization form of factors in the denominator, namely,
and then proceed exactly along the lines of previous case.
A symmetric form of the parametrization is occasionally used, where the integral is instead performed on the interval , leading to:
- Kristjan Kannike. "Notes on Feynman Parametrization and the Dirac Delta Function". Archived from the original on 2007-07-29. Retrieved 2011-07-24.
|This applied mathematics-related article is a stub. You can help Wikipedia by expanding it.|
|This quantum mechanics-related article is a stub. You can help Wikipedia by expanding it.|