The idea behind renormalization group.

In summary, the conversation discusses the idea behind renormalization group and its use in finding finite values for quantities in quantum field theory. The renormalization group is based on implementing the concept of scale invariance and can be used to determine leading log corrections in calculations. However, understanding renormalization is necessary before discussing the renormalization group.
  • #1
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What is the idea behind renormalization group ??

i believe 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 ??
 
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  • #2
mhill said:
What is the idea behind renormalization group ??

i believe 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.
 
  • #3
yes, thank you very mach KDV for your response. The main problem is that i only know a bit about Lie Group theory , i know about the Beta function (how is defined) and the Callan-Szymanzik equation, but i do not know how could be RG used to obtain finite results in the renormalization process.
 
  • #4
mhill said:
yes, thank you very mach KDV for your response. The main problem is that i only know a bit about Lie Group theory , i know about the Beta function (how is defined) and the Callan-Szymanzik equation, but i do not know how could be RG used to obtain finite results in the renormalization process.

group theory is really a completely separate concept. We can discuss renormalization without ever talking about group theory. Of course, if you have a gauge theory, some group theory stuff will creep in the discussion but it has nothing to do with renormalization per se.

Second, the renormalization group equations don't have anything to do with obtaining finite results in the renormalization process! You can renormalize a theory without ever talking about the renormalization group. If you want to discuss renormalization, that's a separate issue. It would be important for you to first understand renormalization before tackling the renormalization group.

Regards
 

1. What is the purpose of the renormalization group?

The renormalization group is a theoretical framework used in physics to study systems at different length scales. Its main purpose is to understand how the physical properties of a system change as we zoom in or out, by taking into account the interactions between its different components.

2. How does the renormalization group work?

The renormalization group works by breaking down a system into smaller components and analyzing how their interactions change as we change the scale at which we are looking at the system. This is done through a mathematical process called "renormalization", which involves rescaling the system's parameters and equations.

3. What is the significance of the renormalization group in physics?

The renormalization group has been a crucial tool in the development of theoretical physics, especially in fields such as quantum field theory and statistical mechanics. It has allowed scientists to better understand the behavior of complex systems and make predictions about their properties at different scales.

4. Can the renormalization group be applied to other fields besides physics?

Yes, the concept of renormalization group has been applied to other areas such as economics and biology. It can be used to study complex systems in any field where interactions between different components play a significant role.

5. Are there any limitations to the renormalization group?

While the renormalization group has been a powerful tool in understanding many physical systems, it does have certain limitations. It relies on certain assumptions and simplifications, and may not be applicable to all systems. Additionally, it can be a complex and time-consuming process to apply it to certain problems.

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