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kdv
kdv is offline
#2
Feb24-08, 09:32 AM
P: 329
Quote Quote by mhill View Post
What is the idea behind renormalization group ??

i belive you begin with an action [tex] S[\phi] =\int d^{4}x L(\phi , \partial _{\mu} \phi ) [/tex]

then you expand the fields into its fourier components upto a propagator..

[tex] \phi (x) =C \int_{ \Lambda}d^{4}x e^{i \vec p \vec x} + c.c [/tex]

but then i do not more, i know that every quantity measured by the QFT theory will have the form:

[tex] m(\Lambda) = alog(\Lambda) + \sum_{n=1}^{\infty} b_{n} \Lambda ^{n} [/tex]

and that from the definition of propagator we should ask for 'conformal invariance' but i do not know how this theory helps to find finite (regularized) value of the quantities.

the idea i am looking for, it is to know if using renormalization group we could find a relation between the bare quantities [tex] m^{(0)} [/tex] and the renormalized ones [tex] m^{(R)} [/tex] via some differential or integral equation.

another mor informal question, if we found a method to obtain finite values for integrals.

[tex] \int_{0}^{T}dx x^{n} [/tex] as T-->oo and for every positive 'n'

would the problem of renormalization be solved ??
I will meep itshort just in case you are not around anymore or not interested in this issue anymore.

The renormalization group is simply the implementation of the fact that changing the scale at which the renormalization procedure is applied should not change the physics. It's neat because even if you do the calculation at a specific order, imposing the renormalizatipn scale invariance allows you to determine the leading log correction of the next orders and solving the renormalization group differential equation essentially sums up those leading logs.
I could say much more about your questions but will wait to see if you are still interested.