By Stetz A.W.
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Extra resources for A very short introduction to quantum field theory
THE INTERACTION PICTURE AND THE S-MATRIX because the self-interactions are virtual processes, and as such they are contained within a region roughly the size of the Compton wavelength. So long as one does not probe inside this region, the electron has it’s usual physical properties, and probing inside the region means doing a high-energy scattering experiment. The same sort of ambiguities also complicate our understanding of the vacuum state. According to perturbation theory, virtual particle-antiparticle pairs are spontaneously emitted from and reabsorbed into empty space.
In addition to the external lines that begin or terminate at a vertex, there will be lines that connect two vertices. These are called internal lines. • Label all lines with their momentum. External lines carry the momentum that is observed in the lab; in the previous example, k1 , k2 , p1 , and p2 . Label the internal lines as you wish, perhaps q1 , q2 , q3 · · · would be a good choice. Put an arrow on each of the internal lines indicating the flow of momentum. The direction you choose will be arbitrary, but you must make a choice.
18) is more difficult than it looks. 2. INTERACTIONS AND THE S MATRIX 39 the left of earlier times. This is called time ordering. We need some machinery to do this for us. 19) The generalization to three or more Hamiltonians is obvious. You should convince yourself that the following things are true: t0 dt1 t0 t2 t t1 t dt2 H(t1 )H(t2 ) = = 1 2 t0 dt2 t t t0 t0 t0 dt1 H(t2 )H(t1 ) dt1 dt2 T [H(t1 )H(t2 )], and in general t t0 tn−1 dt1 · · · t0 dtn H(t1 ) · · · H(tn ) = 1 n! t t0 t dt1 · · · t0 dtn T [H(t1 ) · · · H(tn )] So our final result for the U operator is ∞ U (t, t0 ) = n=0 (−i)n n!
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