Understanding Renormalisation Calculations in Field Theory

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In summary, the conversation is discussing a calculation involving the derivative of a function with respect to a certain parameter. The speaker is having trouble reconciling their method with the one presented in the notes, and is wondering about the use of an identity in the notes. They then clarify that the author did use their method, but applied an identity to simplify the result.
  • #1
latentcorpse
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On p 51 of these notes:
http://www.damtp.cam.ac.uk/user/ho/Notes.pdf ,

I'm trying to follow the calculation [itex]\int \frac{d^dk}{(2 \pi )^d} \frac{1}{(k^2+m^2)^2}= - \frac{\partial}{\partial m^2} \dots[/itex]

It looks to me like we can just use the calculation above and then take the derivative at the end but somehow this isn't working as whilst he has lowered the power on the m^2, he hasn't multiplied through by the old power. Additionally, the argument of the Gamma function has changed!

What's going on?
 
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  • #2
Have you tried computing the ratio of your result to his?
 
  • #3
fzero said:
Have you tried computing the ratio of your result to his?

You mean that if I expand out the gamma function in both expressions, I should find they are the same?

Even so though, how did he end up with his version rather than mine?
 
  • #4
He just used an identity to tidy up the result a bit.
 
  • #5
fzero said:
He just used an identity to tidy up the result a bit.

So he does it my way then uses [itex]\Gamma(\alpha+1)= \alpha \Gamma(\alpha)[/itex]?
 

What is renormalisation?

Renormalisation is a mathematical procedure used in theoretical physics and other fields to correct for infinite or undefined quantities in calculations, allowing for more accurate predictions and understanding of physical phenomena.

Why is renormalisation important?

Renormalisation is important because many physical theories, such as quantum field theory, involve calculations that produce infinite or undefined results. By using renormalisation, these calculations can be made finite and meaningful, allowing for more accurate predictions and a deeper understanding of the underlying physics.

How is renormalisation carried out?

Renormalisation is carried out by introducing a mathematical parameter known as a renormalisation scale, which acts as a cutoff for the infinite or undefined quantities. This scale is then adjusted to produce finite and meaningful results, and the final results are independent of the specific value chosen for the scale.

What are the limitations of renormalisation?

Renormalisation is a powerful tool in theoretical physics, but it does have its limitations. It is not always possible to renormalise a theory, and even when it is possible, the resulting calculations can be complex and difficult to interpret. Additionally, renormalisation does not always provide a unique solution, and different choices of renormalisation scale can lead to different predictions.

What are some real-world applications of renormalisation?

Renormalisation has many applications in theoretical physics, including quantum electrodynamics, quantum chromodynamics, and the standard model of particle physics. It is also used in condensed matter physics, cosmology, and other fields to study diverse physical systems and phenomena. In addition, renormalisation techniques have been applied to economics, biology, and other disciplines to model complex systems and phenomena.

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