By Stetz A.W.
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Extra resources for A very short introduction to quantum field theory
1: The reaction k1 + k2 → p1 + p2 • A factor 1/ 2ωi (2π)3 for each of the external momenta. By external momenta I mean the momenta of particles observed in the initial and final states. This particular diagram has no internal momenta. We’ll get to that later. This particular diagram has only one vertex and one factor of (−iλ) because we are using the lowest-order term in the perturbation theory expansion. For the same reason we did not have to deal with the complications of time ordering. First-order perturbation theory doesn’t cut much mustard, however.
20), and there are many subtle physical questions as well. ” We use this formula and it works. It is also true that equivalent formulas can be derived using the path integral formalism. This also goes way beyond the limits of conventional mathematics, but the same result is obtained in the end. 4, and calculate the four-point Green’s function G(x1 , x2 , x3 , x4 ) to first order in λ. 24) to first order is S =1− iλ 4! 85) becomes G(x1 , x2 , x3 , x4 ) = 0|T [ϕin (x1 )ϕin (x2 )ϕin (x3 )ϕin (x4 )]|0 − iλ 0|T ϕin (x1 )ϕin (x2 )ϕin (x3 )ϕin (x4 ) 4!
I also put arrows on the lines indicating the flow of momentum. Although this has not been relevant so far, lines representing antiparticles carry arrows that point in the opposite direction to the flow of momentum. • Suppose you are doing the n-th order term in the perturbation expansion. Draw all topologically distinct diagrams with n vertices connected by the prerequisite number of lines and with the external lines entering and leaving the diagram appropriately. In addition to the external lines that begin or terminate at a vertex, there will be lines that connect two vertices.