A few questions about perturbation and string theory

[QuantumKitty]

Am I correct in thinking string theory has an infinite number of terms so to prove finiteness to the first order means proving one (or the first) term to be finite?

If so, then how can we ever prove an infinite number of terms? And what exactly does it mean to say, or prove, something is finite to the first (or second, or third, etc..) approximation? What about it is finite? And what makes up one term?


Please try not to be too technical with your answers or I will be lost and have even more questions...I'm just a curious layman.

 

[javierR]

I lost the previous attempted post, so I'll be even more brief. I don't know if this is answering your questions, but I'm trying. In perturbative expansions of a quantum amplitude, you can't check term by term since it never ends. "Finite to n-th order" means that none of the 1st, 2nd,..,n-th order contributions are individually infinite (and therefore that the sum of those contributions is a finite result). In perturbative string theory, one can show that the stringy nature of the objects implies that the contributions to any order should be finite, unlike in point particle perturbation theory in which there are point-like interactions. Hence the claim that the amplitudes in string theory are finite at all orders. HOWEVER, what about the sum of all those contributions at each order? That's the business of the analysis of infinite series, and in perturbative string theory they aren't "Borel summable", which means the series diverge in a certain way that one is used to requiring in point particle perturbation theory to get physical probability amplitudes. That's a big criticism and inspired the searches for a non-perturbative formulation of string theory, in which infinite expansions are fundamentall not involved.

 

[tom.stoer]

... In perturbative string theory, one can show that the stringy nature of the objects implies that the contributions to any order should be finite, unlike in point particle perturbation theory ... Hence the claim that the amplitudes in string theory are finite at all orders.

As far as I know no such proof (beyond first order) has been published up to now!

In QFT infinities arise in amplitudes containing closed loops. They have to be renormalized which means that a subtraction scheme has to be applied which removes infinities and "hides" them in unphysical parameters (e.g. charge, coupling constant etc.). For renormalizable theories one can derive identities relating amplitudes of different orders from which one can deduce that no new types of divergencies appear in higher orders (see e.g. power-counting, BPHZ scheme). From this one can further deduced that new divergencies in higher orders can be renormalized via the same subtractions, which means that it is sufficient to "hide" the divergent contributions in the same unphysical parameters as above.

In string theory the subtraction of infinities should not be required, but nevertheless there must be a relation between different amplitudes which allows to prove finiteness to all orders.

I do not know if such relations between amplitudes of different orders exist.