By Forshaw J.
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Additional resources for An introduction to QED and QCD
Since χ satisfies the wave equation, we can perform a mode expansion just like we did for the Aµ field. Thus we are free to perform the replacement aµ → aµ + λkµ . For kµ = (k, 0, 0, k) and aµ = (a0 , a1 , a2 , a0 ) the choice λ = −a0 /k transforms aµ → (0, a1 , a2 , 0) and we have finished. Having convinced ourselves that we can go ahead and quantise the Maxwell field, we can now proceed to look for the photon propagator. Again I’m going to skip the details and refer back to Dave Dunbar’s course: iDFµν (x − y) = 0|T(Aµ (x)Aν (y))|0 d4 k e−ik·(x−y) = −ig µν .
The finite value of q 2 provides the infrared regulator and the diagram has a term proportional to ln(M 2/q 2 ). Thus in qcd we can’t define a physical coupling constant from an on-shell vertex. 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).
For a sensible theory, in any relation between physical quantities the ultraviolet divergences must cancel leaving a relation which is independent of the method used to regulate divergences. This seems a very sensible demand of our theory. g. e. before M → ∞. g. Quantum Gravity does not destroy Coulombs Law! Demanding this of our theory is equivalent to saying that we want our theory to be renormalisable. 5. 2). 2) m2 where c0 , c1 and d1 are constants, determined by the calculation. In order to evaluate Mf i numerically, however, we must express it in terms of the known parameter eˆ.
An introduction to QED and QCD by Forshaw J.