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ThanksThis can be done in the path-integral formalism as follows. We first assume that A is the single interval ##x\in[u, v]## at ## t_E=0 ## in the flat Euclidean coordinates ##(t_E , x) ∈ \mathbb R^2## . The ground state wave function ## \Psi ## can be found by path-integrating from ##t_E = −\infty## to ##t_E = 0## in the Euclidean formalism

## \Psi(\phi_0(x))=\int_{t_E=-\infty}^{\phi(t_E=0,x)=\phi_0(x)} D\phi e^{-S(\phi)} ##

, where ## \phi(t_E,x) ## denotes the field which defines the 2D CFT. The values of the field at the boundary ## \phi_0 ## depends on the spatial coordinate x. The total density matrix ##\rho## is given by two copies of the wave function ## [\rho]_{\phi_0 \phi_0'}=\Psi(\phi_0)\bar{\Psi}(\phi_0') ##. The complex conjugate one ## \bar{\Psi} ## can be obtained by path-integrating from ##t_E = \infty## to ##t_E = 0##. To obtain the reduced density matrix ## \rho_A ## , we need to integrate ## \phi_0 ## on B assuming ##\phi_0(x)=\phi'_0(x)## when ##x \in B##.

##\displaystyle [\rho_A]_{\phi_+\phi_-}=(Z_1)^{-1}\int_{t_E=-\infty}^{t_E=\infty}D \phi e^{-S(\phi)} \Pi_{x\in A} \delta(\phi(+0,x)-\phi_+(x)) \delta(\phi(-0,x)-\phi_-(x)) ##

where ## Z_1 ## is the vacuum partition function on ## \mathbb R^2 ## and we multiply its inverse in order to normalize ## \rho_A ## such that ##tr_A \rho_A=1##.