NOTES ON FEYNMAN DIAGRAMS

Abstract. We introduce in a pedagogical manner how to compute probability amplitudes from Feynman diagrams, starting with φ4 model. We introduce the notion of renormalization in this model at the one-loop level. Then we review the Dirac equation, and introduce QED. We then perform several example calculations in QED. The appendices gives a survey of Gamma matrices and use of Feynman diagrams in computing decay rates. 1. Feynman Rules in a Nutshell with a Toy Model We will work in a toy model1 with massive spinless particles (so we won’t have to worry about spin). This is the easiest nontrivial example of the use of Feynman diagrams. The basic ritual of Feynman diagrams is outlined thus: (1) (Notation) Label the incoming and outgoing four-momenta p1, p2,..., pn. Label the internal momenta q1, q2,.... Put an arrow on each line, keeping track of the “positive ” direction (antiparticles move “backward ” in time). (2) (Coupling Constant) At each vertex, write a factor of −ig where g is called the “coupling constant”; it specifies the strength of the interaction. In our toy model, g will have dimensions of momentum, but in the real world it is dimensionless. (3) (Propagator) For each internal line, write a factor i q2j −m2jc2 where qj is the four-momentum of the line (q2j = q µ j qjµ; i.e. j is just a label keeping track of which internal line we are dealing with) and mj is the mass of the particle the line describes. (Note that for virtual particles, we don’t have the E2 − ~p · ~p = m2c2 relation that’s for external legs only!) (4) (Conservation of Momentum) For each vertex, write a delta function of the form (2pi)4δ(4)(k1 + k2 + k3) where the k’s are the three four-momenta coming into the vertex (if the arrow leads outward, then k is minus the four-momentum of that line). This factor imposes conservation of energy and momentum at each vertex, since the delta function is zero unless the sum of the incoming momenta equals the sum of the outgoing momenta