# Feynman rule for closed fermion loop in QED

One of the Feynman rules of QED is the following:

For a closed fermionic loop, the Feynman rule is to start at an arbitrary vertex or propagator, follow the line until we get back to the starting point, multiply all the vertices and the propagators in the order of the line, then take the trace of the matrix product. In addition, we include a negative sign for every closed fermionic loop.

For an open fermionic line, we must trace from the head of the line to the tail. In other words, we must start by writing down the polarization spinor for the line with outgoing charge, ..., and finally the polarization spinor for the line with the incoming charge.

Is there also a rule for closed fermionic loop that we must trace the loop in the direction opposite to the direction of the charge flow?

PeterDonis
Mentor
2020 Award
One of the Feynman rules of QED is the following

Can you give the source you are getting this from?

PeterDonis
Mentor
2020 Award
For an open fermionic line, we must trace from the head of the line to the tail. In other words, we must start by writing down the polarization spinor for the line with outgoing charge, ..., and finally the polarization spinor for the line with the incoming charge.

You write down the spinors because the head and tail are external legs in the diagram.

Is there also a rule for closed fermionic loop that we must trace the loop in the direction opposite to the direction of the charge flow?

You don't write down spinors corresponding to external legs for a closed fermion loop, because it has no external legs.

vanhees71
Gold Member
One of the Feynman rules of QED is the following:

For a closed fermionic loop, the Feynman rule is to start at an arbitrary vertex or propagator, follow the line until we get back to the starting point, multiply all the vertices and the propagators in the order of the line, then take the trace of the matrix product. In addition, we include a negative sign for every closed fermionic loop.

For an open fermionic line, we must trace from the head of the line to the tail. In other words, we must start by writing down the polarization spinor for the line with outgoing charge, ..., and finally the polarization spinor for the line with the incoming charge.

Is there also a rule for closed fermionic loop that we must trace the loop in the direction opposite to the direction of the charge flow?
Yes, you always have to read the Feynman diagrams against the direction of the arrows. That's why it is most convenient to let run time from bottom to top of your page and then read the diagram from top to bottom. The additional sign in the closed loop comes from the fact that you have to reorder the fields to be contracted in Wick's theorem being on the ends of the expression to get a propagtor. Due to the fermion nature of the fields this reordering under the time-ordering symbol just gives an additional factor (-1) as mentioned in the fermion-loop Feynman rule.

Yes, you always have to read the Feynman diagrams against the direction of the arrows. That's why it is most convenient to let run time from bottom to top of your page and then read the diagram from top to bottom. The additional sign in the closed loop comes from the fact that you have to reorder the fields to be contracted in Wick's theorem being on the ends of the expression to get a propagtor. Due to the fermion nature of the fields this reordering under the time-ordering symbol just gives an additional factor (-1) as mentioned in the fermion-loop Feynman rule.

Consider the following:

Please ignore the fact that there are no polarization spinors and other junk. It's best if you concentrate on the trace.

I've written the gamma matrices in the direction of the arrows. Is this wrong?