do not lie in the unperturbed state space. This has many manifestations such as Haag’s theorem, orthogonality catastrophe, inequivalent representations of canonical commutation relations, etc (see [@H_theorem] for a nice exposition and discussion). Typically[^1] however the problem is with the norm of the perturbed states in interaction representation which is formally infinite, while the component ratios such as (\[gen\_rat\]) are well defined.

To show that the pairing (\[TCSA\_pairing\]) and ratios (\[gen\_rat\]) associated with an RG flow arise from a particular local conformal interface we discuss states in Lagrangian formalism using wave functionals. Suppose ${\rm CFT}_{\rm UV}$ is described via a fundamental field $\varphi(x)$ and an action functional $S_{0}[\varphi]$. States in this theory can be represented by a wave functional $\Psi[\varphi_{0}]$ depending on functions $\varphi_{0}(x)$ defined on the circle $y=0$. The vacuum state is then given by a renormalised functional integral over the left half cylinder $y\le 0$ \_[vac]{}\[\_[0]{}\] = \_[(x,0)=\_[0]{}(x)]{} e\^[-S\_[0]{}\[\]]{} . The functionals describing excited states can be obtained by inserting local operators at positions with $y<0$ inside the above functional integral.

Suppose now that the perturbed theory is described by an action functional S\[\] = S\_[0]{}\[\] + dxdy V((x,y)) where $V(\varphi)$ is some potential. The vacuum of the 