# Can zeta regularization provide FINITENESS to quantum field theory ?

Yes. It's called dimensional regularization in this context. Nobel prize of '99 is based on this.

tom.stoer

can zeta regularization provide FINITENESS to quantum field theory ??
No, strictly speaking not.

All regularization schemes simply re-write the infinite result for an amplitude or a Green's function as an infinite term + a finite term. The regularization expresses the infinite term as a regulated expression which, when the regulator is removed, becomes infinite again. The regularization scheme allowes you to drop the infinities in a consistent manner for all amplitudes you want to calculate. The remaining finite terms contain the physical information.

So no regularization scheme (zeta-function, dim. reg., Pauli-Villars, ...) provides a proof of finiteness; they simply manage to "hide" the infinities in a consistent manner. The expressions are still divergent, or - mathematically speaking - do not exist.

however for Casimir effect the zeta regularization gives the CORRECT answer $$\sum_{n=1}^{\infty}n^{3} = \zeta (-3)$$
http://en.wikipedia.org/wiki/Casimir_effect#Casimir.27s_calculation

shouldn't we expect the same for similar procedures involving zeta regularization ??? , the idea si that perhaps the REGULARIZATION procedures 'hide' the infinities away to get FINITE answers that can be contrasted by experiments is somehow as it nature 'sees analytic continuation' and in the end you get finite answers.

Also in string theory the zeta regularized value $$1+2+3+4+5+......= -1/12$$ appears to be consistent with the fact that bosoinc string is only valid for 23 dimensions.

tom.stoer

Yes, it is something like this.

Compare it to the geometric series. You have f(q) = 1+q+q²+q³+ ... and you know that you can rewrite the series as f(q) = (1-q)-1. You know that the series converges for |q| < 1. Once you have the expression f(q) = (1-q)-1 for some q with |q| > 1 it would be silly to write it as a series as that would mean to throw away a finite expression and replace it by something that diverges; nobody would do that ...

But in QFT, especially in a perturbative approach, all you have is the diverging expression. You can't re-sum it explicitly to get the finite expression, you have to live with the infinities that are created by such a stupid approach. Of course you would be happy to be able to use finite expressions only, but unfortunately nobody is able to derive them. Therefore you have to live with the infinities.

[Attention: don't get me wrong, my example with the geometric series is not in one-to-one correspondence with QFT; it is not the re-summation of the power series in the coupling constant, but applies to every individual term in the perturbation expansion; it is only an example where you replace a finite expression by something infinite]

So my guess is that the reason why regularization and hiding of infinities works is a hint that there is an underlying finite expression which is still to be discovered. The infinities arise simply because we have not yet managed to express QFT correctly, that means based on finite quantities only. The point where everything gets "wrong" is when you start to use perturbation theory. Perhaps the problem arises even earlier when you take a classical action integral and use it as a starting point for the construction of a quantum field theory.Perhaps already the replacement of fields with field operators is the wrong turn, I am not so sure about that. But it definitly becomes "wrong" when you use perturbation theory.

So we believe that there is something like f(q) = (1-q)-1 but unfortunately we only know f(q) = 1+q+q²+q³+ ... .

[Remark: there are hints that non-perturbative calculations or new theories like SUGRA or strings may cure this mess. As far as I understood the latest ideas rearding finiteness in SUGRA there could be a different turn where it usually gets wrong in QFT. It could very well be that it is not allowed to use Green's functions G(Q²) off-shell. As far as I understood the approaches to prove finiteness of SUGRA, they rely on on-shell symmetries of G(Q²). Nevertheless it means that we use the wrong expressions and that the correct ones are still to be discovered]

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thanks Tom a last question regarding this paper, can ALL logarithmic divergences be re-written as $$\int_{0}^{\infty}dx \frac{log^{k}(x)}{x+a}$$ for some constants (positive) a and k ???

and from the renormalization point of view .. would it be licit to make a change of variable ? for example

$$\int_{0}^{\infty}dx \frac{1}{x}$$ becomes under the change of variable $$e^{u}=x$$ the logarithmic divergent integral becomes the UV divergent integral $$\int_{-\infty}^{\infty}du$$

tom.stoer

... can ALL logarithmic divergences be re-written as $$\int_{0}^{\infty}dx \frac{log^{k}(x)}{x+a}$$ for some constants (positive) a and k ???
I am sorry, I don't know.

The problem is with the mass, it is still not well understood. The particle being a point and having properties is very vague. We all know the work around, anybody who works with software knows how that works.

The problem is with the mass, it is still not well understood. The particle being a point and having properties is very vague. We all know the work around, anybody who works with software knows how that works.

What do you think, Tom?

Zeta look up the paper in the link you will see a new kind of regularizaion/renormalization.

tom.stoer

Interesting, but it has not really much to do with zeta-function regularization.

Interesting, but it has not really much to do with zeta-function regularization.
You are right Tom, I just ment this section on page 5, as a general concept.

V RENORMALIZATION BY GRAVITY AND
REGULARIZATION OF SELF-ENERGY

Of course After reading more carefully, I found out that the theory is a branch of Twister theory.