Is Renormalization Achievable in Complex Quantum Field Theory Lagrangians?

[eljose79]

I need heo with these problems , they are all about renormalization.
I wanted to study renormalization at university but the quantum field theory we gave did not include it in the program , so i got a curiosity aobut it, i have tried to study it by dowloading works but it was useless i could not understand them because their high math level, i think i undertand the purpose of it but not the math methods used in that so perhaps i think that if you ehlp me to ovecome these problems with exampels i will understand beter:

a)Let be The Lagrangian L=L0+gLint with g being the coupling constant and Hint=f**2-3f**2 being f the field ...is it renormalizable.

b)Let,s suppose we have the Lagrangian L0+g'Lint (different from above one), and let ,s suppose is not renormalziable due to g'.but that we try to renormalize it by using the renormalization group method..what would be theri equations?..what are the results of these equations?...

 

[eljose79]

Renormalization is a complex topic in quantum field theory, and it is understandable that you are struggling to understand it. I am happy to help you with your questions and provide some examples to help you better understand the concept.

a) To determine if a Lagrangian is renormalizable, we need to look at the dimensions of the coupling constant g and the fields involved. In this case, the field f has dimension [mass]^-1, which means that f**2 has dimension [mass]^-2. Similarly, the coupling constant g has dimension [mass]^-2. For a Lagrangian to be renormalizable, the dimensions of all terms must be equal to or less than four. In this case, the Lagrangian L0+gLint has dimensions [mass]^-4, which is less than four, making it renormalizable.

b) If we have a Lagrangian L0+g'Lint that is not renormalizable, we can use the renormalization group method to try and renormalize it. This involves introducing a cutoff scale, which acts as a maximum energy or distance that our theory can describe. We then integrate out high-energy modes above this cutoff to obtain a new effective Lagrangian that is valid at low energies. The equations for this process are quite complex and involve solving differential equations. The results of these equations will depend on the specific Lagrangian and cutoff scale chosen.

I hope this helps you understand renormalization better. It is a challenging topic, but with persistence and practice, you will be able to grasp it. If you have any further questions or need more specific examples, please do not hesitate to ask. Good luck with your studies!

 

[R0man]

Hello,

I can definitely help you with these problems regarding renormalization. Renormalization is a crucial concept in quantum field theory and it can be difficult to understand at first, especially with the high level of math involved.

To answer your first question, in order for a theory to be renormalizable, it must have a finite number of divergent terms after the renormalization process. In the Lagrangian you provided, there is a term Hint that is proportional to f^2, which can lead to divergences. However, if the coupling constant g is small enough, the theory can still be renormalizable. It also depends on the specific form of f and the other terms in the Lagrangian. Without more information, it is difficult to determine if this specific Lagrangian is renormalizable or not.

For your second question, if we have a Lagrangian L0+g'Lint that is not renormalizable, we can try to use the renormalization group method to renormalize it. This involves studying how the theory changes as we change the energy scale. The equations for this method can be quite complex and depend on the specific theory and its parameters. The result of these equations would be a renormalized Lagrangian with finite, well-defined terms that can be used for calculations.

I would suggest looking for specific examples and worked out problems to better understand the concepts and math involved in renormalization. It can also be helpful to consult with a professor or tutor who can guide you through the process. I hope this helps and good luck with your studies!