Quantum Field Theory Exam Q3: Draw Feynman Diagram | Advice Needed

[latentcorpse]

Consider q3 in this exam:
http://www.maths.cam.ac.uk/postgrad/mathiii/pastpapers/2006/Paper48.pdf

I reckon I can manage a good part of the rest of the question. Unfortunately, I cannot manage the very first bit (drawing the Feynman diagram) and this is preventing me from continuing!

I have attached my best attempt so far...

The problem is in completing the loop. We are only given those to types of vertices and I just cannot get the lines to join up - I end having to add more and more lines and it just gets more and more complicated!

Any advice?

 

[grey_earl]

Don't worry too much about the distinction between $$\psi$$ and $$\bar\psi$$, since one is the other when changing the direction of time. You can just close the loop without inserting anything more. The Fermion running in the loop is a $$\psi$$, but from every interaction vertex there is one coming and one going, so you have a $$\psi \bar\psi$$ at every vertex.

 

[latentcorpse]

Don't worry too much about the distinction between $$\psi$$ and $$\bar\psi$$, since one is the other when changing the direction of time. You can just close the loop without inserting anything more. The Fermion running in the loop is a $$\psi$$, but from every interaction vertex there is one coming and one going, so you have a $$\psi \bar\psi$$ at every vertex.

I don't get it. If I close the loop as it is just now using a $\psi$ then that bottom right hand vertex has a $\psi,\psi,\Phi$ at it. Which we clearly can't have because of the interaction terms in the lagrangian!

Can you explain how this works in a bit more detail please?

Thanks!

 

[grey_earl]

In reality you don't have $$\psi$$ and $$\bar\psi$$ at any vertex, you have a $$\psi$$ entering and a $$\psi$$ leaving. But since the $$\psi$$ entering "comes from the future", from the point of view of this vertex it is a $$\bar\psi$$. Take your first vertex, the $$\Phi\psi\bar\psi$$ one. Assuming time advances from left to right, the upper $$\psi$$ leaves, advances in time, and so your naming and your arrow are correct. The lower, the one you baptised $$\bar\psi$$, needs the arrow in the other direction, since it is a $$\bar\psi$$ leaving from the vertex. If you insist on your arrow direction, however, you must baptise it $$\psi$$, since a $$\bar\psi$$ coming from the future (as the arrow shows) is a $$\psi$$ going into the future.

For the other vertices, the reasoning is the same, if you want one arrow coming in and one leaving (for the $$\psi$$'s), they are both $$\psi$$'s, but if you want to write $$\psi$$ and $$\bar\psi$$, both arrows are leaving.

Note that for the $$\Phi$$ and $$\phi$$ this doesn't make any difference, because they are their own antiparticles.

 

[paranormal]

I can understand how frustrating it can be when faced with a difficult problem, especially in a time-pressured exam setting. However, in the field of quantum field theory, Feynman diagrams are an essential tool for understanding and visualizing particle interactions. Therefore, it is important to practice and become proficient in drawing them.

My advice would be to start by understanding the rules for drawing Feynman diagrams. Each line represents a particle, with arrows indicating the direction of its momentum. Vertices represent interactions between particles, with lines coming in and out of them. The number of lines connected to a vertex indicates the number of particles involved in the interaction.

In this specific problem, q3, you are given two types of vertices - a fermion-fermion-photon vertex and a fermion-antifermion-photon vertex. Remember that fermions are represented by solid lines and photons by wavy lines. Start by drawing the fermion and photon lines, connecting them at the vertices as indicated in the question.

Next, try to identify the particles involved in the interaction and connect them accordingly. Remember that momentum must be conserved at each vertex, so make sure the arrows on the lines are pointing in the correct direction.

If you are still having trouble completing the loop, try breaking it down into smaller parts and drawing them separately before connecting them. You can also refer to textbooks or online resources for examples of similar Feynman diagrams to get a better understanding.

Practice makes perfect, so don't get discouraged if you are not able to draw the diagram perfectly on your first try. Keep practicing and you will become more comfortable with drawing Feynman diagrams. Good luck with your exam!