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In chapter 4 of the book Quantum fields in curved space by Birrell and Davies, the authors provide a derivation of the Casimir energy for a massless scalar field (in 4-D) using the method of images and Green's function. The new improved stress-energy tensor which is traceless has been used to calculate the vacuum expectation value. The conformally invariant traceless tensor is given by the following expression (Eq. 4.35 of the book)

$$ T_{\mu\nu} = \frac{2}{3}\, \partial_\mu\phi \partial_\nu\phi\,- \frac{1}{6}\,\eta_{\mu\nu}\eta^{\rho\sigma}\partial_\rho\phi\,\partial_\sigma\phi\,-\frac{1}{3}\phi\partial_\mu\partial_\nu\phi\,+ \eta_{\mu\nu}\phi\Box\phi.$$

This for the energy density component reduces to

$$ T_{00} = \frac{1}{2} (\partial_t\phi)^2 + \frac{1}{6} (\partial_i\phi)^2 - \frac{1}{4}\phi \partial_t^2\phi - \frac{1}{12}\phi\partial_i^2\phi.$$ where the sum over the spatial index $i=1,2,3$ is implied.

In all the previous calculations the book has used the Hadamard's Green's function $D^{(1)}$ to calculate the canonical stress-energy tensor which is defined as

$$ D^{(1)}(x,x') = \langle 0| \{\phi(x),\phi(x')\}|0\rangle.$$

However, the book does not show how to find the improved stress-energy tensor from the Green's function but uses $D^{(1)}$ to calculate the Casimir energy. I am stuck with the relation between the improved stress-energy tensor and the Green's function. The first two terms of $T_{00}$ can be found easily by doing the following

$$ \lim_{x',x'' \rightarrow x} \left(\frac{1}{4}\partial_{t^{''}}\partial_{t^{'}} + \frac{1}{12}\partial_{x_i^{''}}\partial_{x_i^{'}}\right) D^{(1)}(x',x'')$$

where $x=(t,x_i)$. But I am stuck with how to get the next two terms in the expression of $T_{00}$. Any guidance is highly appreciated. Thanks.

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