Regularization of integral by substraction

1. Aug 23, 2010

zetafunction

given the divergent integral in n-variables

$$\int_{V} f(q1,q2,....,qn)dq1,dq2,....dqn$$

my question is if in general one can substract a Polynomial K in the variables $$q1,q2,...,qn$$ so the integral

$$\int_{V} (f(q1,q2,....,qn)-K(q1,q2,.........,qn))dq1,dq2,....dqn$$

is FINITE , then it would appear divergent integrals related to $$\int (q1)^{m}dq1$$

for positive 'm'

2. Aug 25, 2010

Eynstone

No, consider the integrand exp(q_1+q_2).

3. Aug 25, 2010

vanhees71

I guess what the OP is after is the BPHZ renormalization, where divergent integrals appearing in perturbative calculations of one-particle irreducible Green's functions are not regularized in any way but made finite directly by subtracting the integral with certain values of the external momenta (determining the renormalization point). This of course rests on Weinberg's theorem on the asymptotic behavior of such integrals. You find a quite detailed description of this technique in my qft writeup:

http://theorie.physik.uni-giessen.de/~hees/publ/lect.pdf

Last edited by a moderator: Apr 25, 2017