Selection (relational algebra)

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:

${\frac {1}{AB}}=\int _{0}^{1}{\frac {du}{\left[uA+(1-u)B\right]^{2}}}$

which simplifies evaluating integrals like:

$\int {\frac {dp}{A(p)B(p)}}=\int dp\int _{0}^{1}{\frac {du}{\left[uA(p)+(1-u)B(p)\right]^{2}}}=\int _{0}^{1}du\int {\frac {dp}{\left[uA(p)+(1-u)B(p)\right]^{2}}}.$

More generally, using the Dirac delta function:

${\frac {1}{A_{1}\cdots A_{n}}}=(n-1)!\int _{0}^{1}du_{1}\cdots \int _{0}^{1}du_{n}{\frac {\delta (u_{1}+\dots +u_{n}-1)}{\left[u_{1}A_{1}+\dots +u_{n}A_{n}\right]^{n}}}.$

Even more generally, provided that ${\text{Re}}(\alpha _{j})>0$ for all $1\leq j\leq n$:

${\frac {1}{A_{1}^{\alpha _{1}}\cdots A_{n}^{\alpha _{n}}}}={\frac {\Gamma (\alpha _{1}+\dots +\alpha _{n})}{\Gamma (\alpha _{1})\cdots \Gamma (\alpha _{n})}}\int _{0}^{1}du_{1}\cdots \int _{0}^{1}du_{n}{\frac {\delta (\sum _{k=1}^{n}u_{k}-1)u_{1}^{\alpha _{1}-1}\cdots u_{n}^{\alpha _{n}-1}}{\left[u_{1}A_{1}+\cdots +u_{n}A_{n}\right]^{\sum _{k=1}^{n}\alpha _{k}}}}.$

See also Schwinger parametrization.

Derivation

${\frac {1}{AB}}={\frac {1}{A-B}}\left({\frac {1}{B}}-{\frac {1}{A}}\right)={\frac {1}{A-B}}\int _{B}^{A}{\frac {dz}{z^{2}}}.$

Now just linearly transform the integral using the substitution,

$u=(z-B)/(A-B)$ which leads to $du=dz/(A-B)$ so $z=uA+(1-u)B$

and we get the desired result:

${\frac {1}{AB}}=\int _{0}^{1}{\frac {du}{\left[uA+(1-u)B\right]^{2}}}.$

References

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