By Forshaw J.
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Additional info for An introduction to QED and QCD
This is not really a serious restriction since it’s up to us how we define the coupling. We could equally well have defined the qed coupling off-shell, it’s just that the value of 1/137 is easy to extract from low energy experiments which are close to the on-shell limit. Now the renormalised coupling depends on how we define it and therefore on at least one momentum scale (in almost all practical cases, only one momentum scale). The renormalised strong coupling is thus written gˆ(q 2 ). When physical quantities are expressed in terms of gˆ(q 2 ) the coefficients of the perturbation series are finite.
43) I refer to Dave Dunbar’s course for more details on gauge symmetries. For us, the important thing is that we have got our hands on the qed Lagrangian density. From it we can figure out the Feynman rules. We have already written down the fermion and photon propagators and can more or less write down the other Feynman rules without further ado (I refer back to Dave Dunbar’s course, Section 4, for the detailed discussion of how to get Feynman rules from the Lagrangian density). 1. The rule for the vertex can be obtained directly from L int .
In general, you should also show that there is an identity element and that every U has an inverse, but these are both obvious. U(N) is the group of complex unitary N × N matrices and SU(N) is the subgroup of matrices with determinant +1. Let U be a U(N) matrix close to the identity. Write U = 1l + iG where G has infinitesimal entries. Show that G is hermitian. If, in addition, U has determinant 1, so U ∈ SU(N), show that G is traceless. Any N × N traceless hermitian matrix can be written as a linear combination of a chosen basis set.