Renormalization and divergences

[eljose]

Check the webpage..

http://arxiv.org/ftp/math/papers/0402/0402259.pdf

specially the part of Abel-Plana formula as a renormalization tool...

\zeta(-m,\beta)-\beta ^{m}/2- i\int_{0}^{\infty}dt[ (it+\beta )^{m}-(-it+\beta )^{m}](e^{2 \pi t}-1)^{-1}=\int_{0}^{\infty}dpp^{m}

valid for every m>0 so renormalization for gravity can be possible.

\zeta(-m,\beta)-\beta ^{m}/2- i\int_{0}^{\infty}dt[ (it+\beta )^{m}-(-it+\beta )^{m}](e^{2 \pi t}-1)^{-1}=\int_{0}^{\infty}dpp^{m}

 

[CarlB]

After reading Ramanujan's letter to Hardy, I picked up a copy of Hardy's "Divergent Series" that you reference in this paper. I've always found it an interesting book.

But these methods of obtaining finite values for horribly divergent series has always struck me as rather arbitrary and unphysical. In fact, Hardy's book gives examples of sums that have more than one choice of finite sum, depending on how you group the terms and the like. This raises two questions.

First, do you have any physical explanation for why these forms should be used?

Second, do the sums you obtain this way match the usual methods of QFT? And how do they extend these methods?

Carl

 

[eljose]

-Zeta regularization...has been used before in calculations for "Casimir effect" \zeta(-3,0) and in String theory for giving a finite meaning to the series... 1+2+3+4+5+6+7+8+9+...\rightarrow \zeta(-1,0) [/tex] also an explanation of why it should work is included in Hardy,s book, the Abel-Plana formula is an exact result of complex analysis.

Note that here the "Zeta" function used is Hurwitz's zeta

 

[shiekh]

Operator-regularization is a generalization of the zeta-function that works to all loop orders.