|
|
| Line 1: |
Line 1: |
| '''Feynman parametrization''' is a technique for evaluating [[loop integral]]s which arise from [[Feynman diagram]]s with one or more loops. However, it is sometimes useful in integration in areas of [[pure mathematics]] as well. | | I would like to introduce myself to you, I am Jayson Simcox but I don't like when individuals use my complete title. He is an info officer. Alaska is the only location I've been residing in but now I'm considering other choices. The preferred hobby for him and his kids is to perform lacross and he would never give it up.<br><br>Review my web site accurate psychic readings - [http://cartoonkorea.com/ce002/1093612 cartoonkorea.com] - |
| | |
| [[Richard Feynman]] observed that:
| |
| | |
| :<math>\frac{1}{AB}=\int^1_0 \frac{du}{\left[uA +(1-u)B\right]^2}</math>
| |
| | |
| which simplifies evaluating integrals like:
| |
| | |
| :<math>\int \frac{dp}{A(p)B(p)}=\int dp \int^1_0 \frac{du}{\left[uA(p)+(1-u)B(p)\right]^2}=\int^1_0 du \int \frac{dp}{\left[uA(p)+(1-u)B(p)\right]^2}.</math>
| |
| | |
| More generally, using the [[Dirac delta function]]:
| |
| | |
| :<math>\frac{1}{A_1\cdots A_n}=(n-1)!\int^1_0 du_1 \cdots \int^1_0 du_n \frac{\delta(u_1+\dots+u_n-1)}{\left[u_1 A_1+\dots +u_n A_n\right]^n}.</math>
| |
| | |
| Even more generally, provided that <math> \text{Re} ( \alpha_{j} ) > 0 </math> for all <math> 1 \leq j \leq n </math>:
| |
| | |
| :<math>\frac{1}{A_{1}^{\alpha_{1}}\cdots A_{n}^{\alpha_{n}}}=\frac{\Gamma(\alpha_{1}+\dots+\alpha_{n})}{\Gamma(\alpha_{1})\cdots\Gamma(\alpha_{n})}\int_{0}^{1}du_{1}\cdots\int_{0}^{1}du_{n}\frac{\delta(\sum_{k=1}^{n}u_{k}-1)u_{1}^{\alpha_{1}-1}\cdots u_{n}^{\alpha_{n}-1}}{\left[u_{1}A_{1}+\cdots+u_{n}A_{n}\right]^{\sum_{k=1}^{n}\alpha_{k}}}
| |
| .</math> <ref>
| |
| {{cite web
| |
| | author= Kristjan Kannike
| |
| | title= Notes on Feynman Parametrization and the Dirac Delta Function
| |
| | url= http://www.physic.ut.ee/~kkannike/english/science/physics/notes/feynman_param.pdf
| |
| | archiveurl= http://web.archive.org/web/20070729015208/http://www.physic.ut.ee/~kkannike/english/science/physics/notes/feynman_param.pdf
| |
| |archivedate=2007-07-29
| |
| | work=
| |
| | publisher=
| |
| | date=
| |
| | accessdate=2011-07-24
| |
| }} </ref>
| |
| | |
| See also [[Schwinger parametrization]].
| |
| | |
| ==Derivation==
| |
| :<math>\frac{1}{AB} = \frac{1}{A-B}\left(\frac{1}{B}-\frac{1}{A}\right)=\frac{1}{A-B}\int_B^A \frac{dz}{z^2}.</math>
| |
| Now just linearly transform the integral using the substitution,
| |
| :<math>u=(z-B)/(A-B)</math> which leads to <math>du = dz/(A-B)</math> so <math>z = uA + (1-u)B</math>
| |
| and we get the desired result:
| |
| :<math>\frac{1}{AB} = \int_0^1 \frac{du}{\left[uA + (1-u)B\right]^2}.</math>
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| [[Category:Quantum field theory]]
| |
| | |
| | |
| {{applied-math-stub}}
| |
| {{quantum-stub}}
| |
I would like to introduce myself to you, I am Jayson Simcox but I don't like when individuals use my complete title. He is an info officer. Alaska is the only location I've been residing in but now I'm considering other choices. The preferred hobby for him and his kids is to perform lacross and he would never give it up.
Review my web site accurate psychic readings - cartoonkorea.com -