Re: Renormalization

Igor Khavkine wrote:
>
> All functions involved here are analytic in d. But they have poles at
> some integer values of d, like d=4. If two analytic functions agree on a
> finite volume (open) subset of the complex plain, then they agree
> everywhere. Even if you only know that they agree on some finite (open)
> interval of the real line, they still agree everywhere. Hence if you've
> convinced yourself that the equality holds for d<2, then it must hold
> everywhere, including as d->4.
>
> If nothing else convinces you, consider simply plotting the LHS and the
> RHS of the equality for a range of d. Simply fix all quantities other
> than d to some reasonable values and evaluate the integral as well as
> the Gamma function numerically. Numerical integrals are easily
> performed, especially with the help of a computer algebra system.
>
> Igor

Please carefully look at Eq. 11.2.12 of Weinberg's book. Carefully
examine the integral on the left hand side. Let d=3.9 for example. It
should be evident by inspection that the integrand is positive over the
range of integration (from 0 to infinity) and the the integrand
increases in magnitude as the variable of integration approaches
infinity. Note that the numerator goes as k^4.9 and the denominator
goes as k^4 (for large k). Therefore integrand goes as k^(.9).
Obviously the integral is divergent. One does not need to do numerical
integration to reach this conclusion. Now substitute d=3.9 on the right
hand side. All of the quantities on this side of the equation are well
defined and finite. Therefore the quantity on the right hand side is
finite and the quantity on the left hand side is divergent. Therefore
they are not equal for d=3.9. They are equal for d<2 for which case
the integral is finite.

The equality given by Eq. 11.2.12 does not hold for d>2 where d is a
real number. For all these values of d the integral is divergent which
is obvious from inspection. The quantities on the right are all finite
except for possible poles at integer values of d. This is the case for
d being a real number. I am not sure what happens when d is a complex
number.

Dan Solomon

.

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