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A query on the (old) motivation for renormalizable theories

by metroplex021
Tags: quantum field theory, renormalization
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metroplex021
#1
Jul13-11, 04:51 AM
P: 127
I've just realized I don't understand something pretty fundamental about the need to renormalize. Popular wisdom has it (or had it - forget the shift towards an effective framework) that theories that were not renormalizable had no predictive power, on account of the fact each n-point vertex function in such theories need to be renormalized anew, requiring new parameters to be measured at each n (see, e.g., Maggiore p139).

But can't one say the following: say I am interested in studying only 2->2 interactions. Then presumably I only need to renormalize the 2, 3 and 4-point functions in order to derive predictions for these sorts of interactions. The infinitely many parameters apparently needed for a renormalizable theory (and once again, forget about EFTs) would only arise in the case that we study n->m particle relations in the limit that n & m go to infinity, which we never do. So why *were* renormalizable theories regarded as non-predictive?

Any help much appreciated!
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tiny-tim
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Jul13-11, 06:24 AM
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hi metroplex021!
Quote Quote by metroplex021 View Post
But can't one say the following: say I am interested in studying only 2->2 interactions. Then presumably I only need to renormalize the 2, 3 and 4-point functions in order to derive predictions for these sorts of interactions.
no, the number of input and output particles is irrelevant

even with 2->2, there are infinitely many terms in the Dyson expansion

forget Feynman diagrams, it's those infinitely many terms that need to have a finite sum
metroplex021
#3
Jul13-11, 10:32 AM
P: 127
Quote Quote by tiny-tim View Post
hi metroplex021!


no, the number of input and output particles is irrelevant

even with 2->2, there are infinitely many terms in the Dyson expansion

forget Feynman diagrams, it's those infinitely many terms that need to have a finite sum
Well, I think that's how I was thinking before, but isn't the Dyson series written out for a single n-point function? (So that if you were ignoring interactions with less than n external legs, you wouldn't need to worry about that series - or indeed any series for any n'-point vertex function with n'>n?) Thanks!

tiny-tim
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Jul13-11, 05:21 PM
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A query on the (old) motivation for renormalizable theories

hi metroplex021!
Quote Quote by metroplex021 View Post
isn't the Dyson series written out for a single n-point function? (So that if you were ignoring interactions with less than n external legs, you wouldn't need to worry about that series - or indeed any series for any n'-point vertex function with n'>n?) Thanks!
sorry, i'm not understanding your terminology

by "external legs", i assume you mean eg 2->2 has 4 external legs?

but what do you mean by an "n-point function"?
Avodyne
#5
Jul13-11, 09:33 PM
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An n-point vertex function is a sum of one-particle-irreducible diagrams with external propagators removed. (There is also a nonperturbative definition in terms of Legendre transforms of the functional integral with a source, but I don't remember it precisely enough to quote it.)

For 2-2 scattering, you need the 4-point vertex function. But when you compute it, at high enough orders, you will have sub-diagrams that involve n-point vertices for arbitrarily high n. And these have to be renormalized, so you will need the corresponding parameters.
metroplex021
#6
Jul14-11, 04:58 AM
P: 127
That's awesome - thanks very much. I had a suspicion that was the case but have only ever worked at such a miniscule order I wasn't sure if it was the case. Thanks mate!

Quote Quote by Avodyne View Post
An n-point vertex function is a sum of one-particle-irreducible diagrams with external propagators removed. (There is also a nonperturbative definition in terms of Legendre transforms of the functional integral with a source, but I don't remember it precisely enough to quote it.)

For 2-2 scattering, you need the 4-point vertex function. But when you compute it, at high enough orders, you will have sub-diagrams that involve n-point vertices for arbitrarily high n. And these have to be renormalized, so you will need the corresponding parameters.


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