perturbed theory can be represented by the functional integration over the fields defined \[perturbed\_psi\] \_[vac]{}\^[pert]{}\[\_[0]{}\] = \_[(x,0)=\_[0]{}(x)]{} e\^[-S\[\]]{} . Expanding inside the functional integral the exponent $$\exp\Bigl[{\iint\limits_{y\le 0} \!  dxdy\,  V(\varphi) }\Bigr]$$ in power series we obtain a formal expansion of the perturbed vacuum functional $\Psi_{\rm vac}^{\rm pert}[\varphi_{0}] $ in terms of unperturbed states (interaction representation). We can extend this pairing to excited states of the deformed theory by inserting into the functional integral (\[perturbed\_psi\]) additional local operators. This gives a pairing between the states in the two theories. In particular if we follow the perturbed theory to the IR fixed point we obtain a pairing of the type (\[TCSA\_pairing\]). We assume in this discussion that all divergences are renormalised including the ones that need additional boundary counterterms at $y=0$. Renormalisation of wave functionals has been discussed in [@Symanzik], [@Luscher], [@Minic_Nair] and more recently in [@FLP]. (More on renormalisation shortly.)

What becomes clear in this picture is that the pairing arises by perturbing the UV fixed point on a half space (a half cylinder) and letting the perturbed theory flow to the IR fixed point. This gives us a local conformal interface between ${\rm CFT}_{\rm UV}$ and ${\rm CFT}_{\rm IR}$ 