- Feynman parametrization
Feynman parametrization is a technique for evaluating
loop integral s which arise fromFeynman diagram s with one or more loops. However, it is sometimes useful in integration in areas ofpure mathematics too.Richard Feynman observed that::frac{1}{AB}=int^1_0 frac{du}{left [uA +(1-u)B ight] ^2}
which simplifies evaluating integrals like:
:int frac{dp}{A(p)B(p)}=int dp int^1_0 frac{du}{left [uA(p)+(1-u)B(p) ight] ^2}=int^1_0 du int frac{dp}{left [uA(p)+(1-u)B(p) ight] ^2}.
More generally, using the
Dirac delta function ::frac{1}{A_1cdots 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 ight] ^n}.
Even more generally, provided that "Re("alpha_j ")">0 for all 1 ≤ "j" ≤ "n":
: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^1_0 du_1 cdots int^1_0 du_n frac{delta(u_1+dots+u_n-1)u_1^{alpha_1-1}cdots u_n^{alpha_n-1{left [u_1 A_1+dots +u_n A_n ight] ^{alpha_1+dots+alpha_n.
See also
Schwinger parametrization .
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