# Renormalization and Infinite Series

In the renormalization of QED, one subtracts infinities to get finite results. This is not possible in gravity because of infinitely many divergences, and infinitely many counter terms.

The general view is that the problem is with physics. That the current theory is in error because it produced divergences in the expansion, therefore there must be a better approach without divergences.

1) Would a demonstration of a class of sums of infinitely many divergences in a convergent expression alter the attitude to the problem? In other words, is the strong confidence in the failure of GR mainly the consequence of the difficulty of summing infinities, and the resulting consistency issues from a field theoretical point of view, ignoring issues like firewalls.

2) one is familiar with approaches like Borel or Abel summation, Zeta function regularization through which divergent expressions are given essentially arbitrary meanings. Is there a known approach with respect to naked divergences whereby one sums infinitely many divergent terms by taking a limit in the usual sense, i.e. by summing term by term while increasing the value of each term as the sum is made to produce infinitely many divergences, say,

$$\sum^{\infty}_{n=0} f\left(n \right) \infty_{n}$$

Where the functions $f(n)$ ensure the convergence of the expression in the manner of a conditionally convergent sum as the limit is taken. Is there any example of such an expression being utilized in physics in the literature? Are physicists aware of any comparable expression including individual divergent terms.

In the manner that one thought before calculus that all sums of infinitely many terms are divergent, for instance, in the spirit of Zeno's paradox, is it conceivable that the problem is not with physics, but with an inadequate understanding of the meaning of infinity in mathematics?

• About your 1st paragraph: In QED one does not subtract infinities, one always deals with regularised objects which are always finite. Also, the problem with the counter-terms in QG is not about the number of them (being infinite), but about their nature (they include higher derivatives, which leads to instabilities). Non-renormalisable theories -- those where there are infinitely many counter-terms -- are not a problem per se, we can deal with those. The rest of post contains several misconceptions as well. Sorry, the question is rather unclear to me. – AccidentalFourierTransform Apr 5 '17 at 14:49
• -1. Renormalisation is possible in canonically quantised gravity, it is simply you need more and more counter-terms added to the Lagrangian as it is an effective field theory. – JamalS Apr 5 '17 at 14:55
• JamalS I don't think you are saying anything there. The question is simply whether a sum of divergences where pairwise cancellations do not occur has ever been considered in the solution of this problem. I think the attitude shows that the answer is no, but I would like to get a clearer answer. – Ban Garbat Apr 5 '17 at 15:32