Thermal States and Black holes III

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 -- Schrodinger functional from Schrodinger representation of Quantum field theory

 Following Liu's notes  and arxiv paper combined

$<\phi_2(\vec{x}, t_2)| \phi_1(\vec{x}, t_1)> = \int_{\phi(t_1,\vec{x})} ^{\phi(t_1,\vec{x})}  D \phi(\vec{x},t)e^{i S[\phi]}$

Vacuum functional

$\Psi_0[\phi(\vec{x})] = <\phi(\vec{x})| 0> = \int_{\phi(t_E < 0)} ^{\phi(t_E =0, \vec{x})}  D \phi(\vec{x},t)e^{- S_E [\phi]}$


$H = \int dx [\frac{1}{2}(\frac{\partial \phi}{\partial t_E})^2 + \frac{1}{2}(\nabla \phi)^2 + \frac{1}{2} m^2 \phi^2]$

$\Psi_0 [\phi(X)] = \langle \phi_R|e^{-i(-i \pi) H}| \phi_L \rangle$


$| 0\rangle  \propto \sum_n  e^{- \pi  E_n }| n \rangle \otimes | n \rangle$

$\rho_{Rindler}$ is then obvious

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