By H V Klapdor-Kleingrothaus; K Zuber
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We thus follow the method used to develop timedependent perturbation theory in nonrelativistic quantum mechanics. 9 39 Time History of States Consider the time evolution equation a at i - I I,t) s =HII , t)s, where H is the full Hamiltoni an. The subscript S indicates that this state is in what is called the Schrodinger representation. We will omit the subscript for the state in the interaction representation below. We now separate H into H F + HI, where HF describes the free field evolution and HI the rest of H , which corresponds to the interactions.
You can check by explicit substitution that, for example, the quantities iiy",u = 2p", and iiu = 2m transfo rm correctly. A question we can ask is whether we can find other operators 0 that are 4 x 4 matrices, for which {f01jf is both Hermitian and has a well-defined transformation property under Lorentz transformations. Such operators, called bilinear covariants, are legitimate candidates to appear in L for terms involving only the fields and no derivatives. So far, the choices for 0 are I (scalar) and Y/l-(vector).
Thus Mj l/ = 'H Ii (0) . Note that Mj~) is Hermitian since 'HIi is; this will not be true for higher orders. The vertices we've looked at are simple, involving a small number of particles like e- ---+ e- + y. This will remain true as we look at other interactions. The S matrix element for e - ---+ e - + y, calculated to lowest order, vanishes, since this process would violate the momentum conservation enforced by the 84 (PI - Pi) factor. The first order process, however, is responsible for excited atomic states making transitions to lower energy states with the emission of a photon.