In the case of the electron propagator each current has a photon which is summed with the vertex and the propagator has own sum over q, so no further sums have to be it’s done to get a scalar result for the matrix element probability. q is often written as e) Add an integral over the internal four momentum d4q/(2)4 and a four dimensional delta function for each vertex and use one to integrate over the internal four momentum and cancel the other since it corresponds to overall four momentum conservation. f) antisymmetrization. Crossed diagrams will exist for identical sets of particles in the final state. The second diagram will get a negative sign (Fermi statistics). Diagrams can have their electrons moved from the initial to the final state where they occur as a positron producing an equivalent in probability diagram ( a type of crossing symmetry). To have a consistent convention you have to be careful about the signs in those cases. 1) Electron and positron annihilation to photons. For electron positron annihilation to photons you have two identical particles in the end state and thus you get two diagrams. There is no relative minus sign between the diagrams since the photons are not fermions. Consider the non-crossed diagram Note the matrix element must give a scalar result for the probability of the electron and positron to annihilate. However how to calculate this is not obvious since the epsilon polarization vector are 3x1 and we have to take the dot product of those with the gamma Matrix. For the full matrix element. Consider this in the ultra relativistic limit where we can neglect the mass of the electron The first term squared. Note that we take the complex conjugate of the ... as one unit since that type of term must be a scalar. Group the photon terms separately. To do this we move the through the and gamma matrix terms which introduces a negative sign for each one, 7.138. The negative signs cancel out.
In the case of the electron propagator each current has a photon which is summed with the vertex and the propagator has it’s own sum over q, so no further sums have to be done to get a scalar result for the matrix element probability. q is often written as e) Add an integral over the internal four momentum d4q/(2)4 and a four dimensional delta function for each vertex and use one to integrate over the internal four momentum and cancel the other since it corresponds to overall four momentum conservation. f) antisymmetrization. Crossed diagrams will exist for identical sets of particles in the final state. The second diagram will get a negative sign (Fermi statistics). Diagrams can have their electrons moved from the initial to the final state where they occur as a positron producing an equivalent in probability diagram ( a type of crossing symmetry). To have a consistent convention you have to be careful about the signs in those cases. 1) Electron and positron annihilation to photons. For electron positron annihilation to photons you have two identical particles in the end state and thus you get two diagrams. There is no relative minus sign between the diagrams since the photons are not fermions. Consider the non-crossed diagram Note the matrix element must give a scalar result for the probability of the electron and positron to annihilate. However how to calculate this is not obvious since the epsilon polarization vector are 3x1 and we have to take the dot product of those with the gamma Matrix. For the full matrix element. Consider this in the ultra relativistic limit where we can neglect the mass of the electron The first term squared. Note that we take the complex conjugate of the ... as one unit since that type of term must be a scalar. Group the photon terms separately. To do this we move the through the and gamma matrix terms which introduces a negative sign for each one, 7.138. The negative signs cancel out.