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:

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

which simplifies evaluating integrals like:

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

More generally, using the Dirac delta function:

$\frac{1}{A_1\cdots A_n}=(n-1)!\underset{0}{\overset{1}{\int }}d{u}_1\cdots \underset{0}{\overset{1}{\int }}d{u}_n\frac{\delta (u_1+\dots +u_n-1)}{{[u_1{A}_1+\dots +u_n{A}_n]}^n}.$

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

$\frac{1}{A_1^{{\alpha }_1}\cdots A_n^{{\alpha }_n}}=\frac{\mathrm{\Gamma }({\alpha }_1+\dots +{\alpha }_n)}{\mathrm{\Gamma }({\alpha }_1)\cdots \mathrm{\Gamma }({\alpha }_n)}\underset{0}{\overset{1}{\int }}d{u}_1\cdots \underset{0}{\overset{1}{\int }}d{u}_n\frac{\delta (\sum _{k=1}^n{u}_k-1)u_1^{{\alpha }_1-1}\cdots u_n^{{\alpha }_n-1}}{{[u_1{A}_1+\cdots +u_n{A}_n]}^{\sum _{k=1}^n{\alpha }_k}}.$

See also Schwinger parametrization.

Derivation

$\frac{1}{AB}=\frac{1}{A-B}(\frac{1}{B}-\frac{1}{A})=\frac{1}{A-B}\underset{B}{\overset{A}{\int }}\frac{dz}{z^2}.$

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:

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

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Template:Applied-math-stub Template:Quantum-stub