3 - Regularisation and renormalisation

Summary

So far everything has been extremely formal, a euphemism for the fact that almost any diagram that we write down will be represented by an integral that does not exist. This is because products of the distributions Δ_F with coincident space-time points are not defined. Moreover, the presence of singularities arising from such formal products is not peculiar to the scalar theories that we have considered so far, but to all local field theories.

Before considering tactics for handling this problem there is a question of philosophy. One fact that has been hammered home by models that unify the different forces of nature is that, in each step towards a common unity, a new and yet higher mass scale is introduced. With each breakthrough in our understanding the ‘best’ current model has been seen to be a ‘low’-energy effective model for a yet more complete theory that contains it as a subsector. Further, a field that is ‘elementary’ at one level may be composite (e.g. a bound state of new elementary fields) at the next.

It is true that models giving rise to unavoidable infinities may still permit useful calculations at ‘tree’ level (e.g. the four-Fermi model of weak interactions). Nonetheless, the desire for a consistent theory with no infinities at each step has been a crucial force in model-making.