Renormalization of QFT deformations

Consider a quantum field theory with an action invariant under some Lie algebra of symmetries .

Let us study the infinitesimal deformations of the action:


  • is an infinitesimal parameter

  • are some space-time-dependent coupling constants

  • is some set of local operators closed under

Then the expressions on the RHS of Eq. (14) form a linear representations of ; call it :

is the linear space of infinitesimal deformations (14)

We can study the deformations (14) at finite . This requires taking care of the divergencies. Suppose that the space of operators is big enough to include all the needed counterterms. After regularization and adding counterterms, Eq. (14) is well-defined in the quantum theory. Therefore we have a map:


there is a natural action of on the (non-linear) space of deformations

This is a subtle point. We need to regularize multiple integrals (to avoid collisions) and then subtract counterterms (which are also of the form , but with less integrals). Then, just acts on each ( is a local operator in the undeformed theory, acts on them).

Because we had some freedom in the choice of counterterms, the map does not necessarily commute with the action of . Maybe we can choose counterterms with some care, so the resulting does commute with ? Generally speaking we can not, there are obstacles.