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I have a request. Please clarify these doubts for me:

In the loops in quantum field theory there is a momentum $k$ which is integrated over. In a lecture, Professor Hong Liu says that this free $k$ running in these loops is integrated over all real values and thus this means that the momentum fluctuates over all these values. "My question is why don't fluctuations arise even in tree level diagrams because here too, particles are point like." :: this is clearly a wrong question because clearly undetermined momentum values occur in loops only.

However the math just says that loops arise due to contractions of fields sharing common vertices. I want to know physically why this is the root cause of fluctuations. In ordinary quantum mechanics, a point like particle will have huge fluctuations in momentum. Here, what is the physical cause for these fluctuations in the theory of quantum fields? Is it just observation that tree level diagrams don't show fluctuations but loops do ?

SX849
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    Can you draw a tree-level diagram with a loop? – Wihtedeka Mar 01 '24 at 08:19
  • Hi SX849. Do you know the $\hbar$/loop expansion of Feynman diagrams? – Qmechanic Mar 01 '24 at 10:02
  • @Qmechanic and Wihtedeka I don't know these. Actually I am just beginning to learn qft. Could you please give me references by which I can understand and learn them? – SX849 Mar 01 '24 at 10:14
  • If you work out momentum conservation on each vertex you will see that in loop diagrams, some momenta are not determined and need to be integrated over. In tree diagrams, there are no such undetermined momenta. Draw some tree and loop diagrams and work it out for yourself.
    • – Oбжорoв Mar 01 '24 at 13:04
  • Well this gist of it is that momentum is conserved at each vertex and so loop momenta can only arise if you go beyond tree-level and include more vertices, i.e. powers of the coupling constant. The number of loops is related to an expansion in powers of $\hbar$, see e.g. here https://physics.stackexchange.com/a/488523 or https://physics.stackexchange.com/a/333767 – Wihtedeka Mar 01 '24 at 13:04
  • $k$ has nothing to do with the field momentum $\pi$. It ($k$) is the momentum from the Fourier transform. In view of your 2 questions, you seem to have missed a few basic things of QFT. Maybe you need to start again from the beginning?
    • – Oбжорoв Mar 01 '24 at 13:06
  • Thanks a lot for the comments... I've almost understood and I'll do as suggested and revise my basics ... – SX849 Mar 01 '24 at 18:23