Renormalization group and cut-off

In summary: Green function depending on the value of the cutoff. These quantities become well-defined when the cutoff is finite. He suggests considering the cutoff as a physical field with a physical meaning, or using it to measure mass and charge. However, this approach can lead to issues when the energy approaches the mass of the particle, indicating the need for a more fundamental theory.
  • #1
Sangoku
20
0
Hi.. in what sense do you intrdouce the cut-off inside the action

[tex] \int_{|p| \le \Lambda} \mathcal L (\phi, \partial _{\mu} \phi ) [/tex]

then all the quantities mass [tex] m(\Lambda) [/tex] charge [tex] q(\Lambda) [/tex] and Green function (every order 'n') [tex] G(x,x',\Lambda) [/tex]

will depend on the value of cut-off, and are well defined whereas this cut-off is finite now what else can be done ??.. could we consider this cut-off [tex] \Lambda [/tex] to be some kind of 'physical' field (or have at least a physical meaning, or can we make this finite measuring 'm' 'q' or similar
 
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  • #2
Sangoku said:
Hi.. in what sense do you intrdouce the cut-off inside the action

[tex] \int_{|p| \le \Lambda} \mathcal L (\phi, \partial _{\mu} \phi ) [/tex]

then all the quantities mass [tex] m(\Lambda) [/tex] charge [tex] q(\Lambda) [/tex] and Green function (every order 'n') [tex] G(x,x',\Lambda) [/tex]

will depend on the value of cut-off, and are well defined whereas this cut-off is finite now what else can be done ??.. could we consider this cut-off [tex] \Lambda [/tex] to be some kind of 'physical' field (or have at least a physical meaning, or can we make this finite measuring 'm' 'q' or similar

I am not sure I understand your question but the cutoff represents the energy scale at which new physics becomes important.
Consider for example the Fermi model of the weak interaction. It`s an effective theory which can be used as long as the energy of the reaction is below the mass of the W boson. So you could construct an effective theory and integrate up to the mass of the W and renormalize and you would get a well defined expansion of any observable. but of the energy gets close to the mass of the W, the expansion breaks down because an infinite number of terms would have to be taken into account, signaling the need to use a mre fundamental theory.

hope this helps

Patrick
 
  • #3
)

The renormalization group and cut-off are important concepts in theoretical physics, particularly in the field of quantum field theory. The cut-off is a mathematical tool used to regulate the divergences that arise in certain calculations of quantum field theories. It essentially sets a limit on the maximum energy or momentum that can be exchanged in a given process.

By introducing the cut-off into the action, we are essentially limiting the range of integration in the Lagrangian, which in turn affects the values of all the quantities that depend on it, such as mass, charge, and Green functions. This is necessary because without a cut-off, these quantities would be infinite and therefore meaningless.

The cut-off can be thought of as a way to make the theory more well-defined and manageable. It is not a physical field in the traditional sense, but it does have a physical meaning in the context of quantum field theory. It represents the energy scale at which our current understanding of the theory breaks down and we need to consider new physics.

In terms of what else can be done, there are various techniques and methods to deal with the cut-off and its dependence on physical quantities. One approach is to use a renormalization group, which allows us to study how the theory changes as we vary the cut-off. Another approach is to use the concept of renormalization, which involves adjusting the parameters of the theory to absorb the effects of the cut-off.

In summary, the cut-off is a crucial tool in quantum field theory, allowing us to make sense of the otherwise divergent calculations. It may not have a direct physical interpretation, but it plays a crucial role in understanding the behavior of the theory at different energy scales.
 

1. What is the renormalization group?

The renormalization group is a theoretical framework used in physics to understand how physical systems behave at different length scales. It helps us study how the properties of a system change as we zoom in or out, and how these changes are related to each other.

2. What is the purpose of a cut-off in renormalization group theory?

A cut-off is used in renormalization group theory to define the length scale at which we are studying a system. By choosing an appropriate cut-off, we can focus on the relevant length scales and ignore the irrelevant ones, making our calculations more manageable.

3. How does the renormalization group help us understand critical phenomena?

The renormalization group allows us to study how a system behaves near a critical point, where its properties change drastically. By zooming in on the critical point and using the concept of universality, we can understand the universal features of different systems and predict their behavior.

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

Yes, the renormalization group has been applied to various fields such as biology, economics, and computer science. It can be used to study complex systems and understand how they behave at different levels of organization.

5. What are the limitations of renormalization group theory?

One limitation of renormalization group theory is that it is a theoretical framework and does not provide exact numerical predictions. It also relies on certain assumptions and approximations, which may not always hold for all systems. Additionally, it can be challenging to apply to systems with strong interactions or complex geometries.

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