View Single Post
waterfall
waterfall is offline
#10
Feb21-12, 08:05 AM
P: 381
Quote Quote by bhobba View Post
Sorry if what I posted wasn't clear - its a good idea to read the link I gave. Without a cutoff the value of the power series parameter you expand about turns out to be infinite so it obviously will not work. For the variable in the power series you expand about (that's epsilon in the equation you posted) substitute infinity and the result is infinity. However if you impose a cutoff and choose a low enough value then it is small and epsilon to some power gets smaller and smaller as the power you raise it to gets bigger so the method works - each term gets smaller and smaller. That's because it secretly depends on the cutoff. As the cutoff is made larger and larger the coupling constant gets larger and larger until in the limit it is infinite. That's why you need to expand about something better - that something is called the renormalised value. When this is done it does not blow up as you take the cutoff to infinity so the method now works. All this is made clear in the paper I linked to - its a bit heavy going - but persevere.

Thanks
Bill
I tried to read the paper for more than 30 minutes and see some web references and calculus book and thinking all this for more than an hour already. But I still can't understand the very basic question whether it applies to all power series. To know what I'm asking. Let us forget about Renormalization first. In a power series like

The p-series rule:

(infinity)
sum sign 1/n^p
n = 1

for p-series p=2

1 + (1/2^2) + (1/3^2) + (1/4^2) + (1/5^2)....


So is the coupling constant equivalent to the p or n in the above equation, or the terms 2^2, 3^2, etc.?

Also about the coupling constant getting larger for longer series without cutoff and it getting normal in value or smaller when there is cutoff. Do you also apply this to nonQED thing like trajectory of a ball thrown or is it only in QED?

Just answer whether it is only in QED or present in all power series. This is all I need to know now. If it is only in QED.. then it has to do with the quantum nature or probability amplitude and all those path-integrals, etc. thing which I already understood and can relate and I will continue with the paper you gave. But if it is present in all power series.. i can't find it in a basic calculus book about power series where the equivalent of coupling constant gets infinite depending on whether you make a cut-off and will need to find it in other calculus book about power series. Again don't mention about renormalization first. Thanks.