Relation between the Hadamard Expansion and the vacuum state in question? I am trying to understand the paper "Off-diagonal coefficients of the DeWitt-Schwinger and Hadamard representations of the Feynman propagator" by Decanini and Folacci.
Consider a ($m$)assive scalar field $\phi$ living in a spacetime with metric $g$ (coupled to the Ricci scalar $R$ with coupling $\xi$). I specify to $D=4$ dimensions in what follows here. The paper concerns itself with the short-distance behaviour of the Feynman propagator $G^{\mathrm{F}}$ obeying
$$
( \Box_x - m^2 - \xi R ) G^{\mathrm{F}}(x,x') \ = \ - \frac{\delta^{(4)}(x-x')}{\sqrt{-g(x)}} \ ,
$$
which I think here is the time-ordered correlation function where $G^{\mathrm{F}}(x,x') \sim \langle \Omega | \mathscr{T}\big( \phi(x) \phi(x') \big) | \Omega \rangle$ for some vacuum state $| \Omega \rangle$.
The authors say that the Hadamard expansion for $G^{\mathrm{F}}(x,x')$ is
$$
G^{\mathrm{F}}(x,x') = \frac{i}{8\pi^2} \bigg[ \frac{\sqrt{ \Delta(x,x') }}{\sigma(x,x') + i \epsilon}  + V(x,x') \log\big( \sigma(x,x') + i \epsilon \big)+ W(x,x') \bigg]
$$
where $\sigma(x,x')$ is the Synge world function (aka. half the square of the geodesic distance between the points $x$ and $x'$), and $\Delta(x,x')$ is the Van Vleck-Morette determinant. The parameter $\epsilon \to 0^{+}$ is a regulator with mass dimension $-2$, and is there so that the singularity structure of $G_{\mathrm{F}}$ is consistent with the definition of the Feynman propagator.
The functions $V(x,x')$ and $W(x,x')$ are symmetric biscalars, which are regular for $x \to x'$, and possess expansions of the form
$$
V(x,x') = \sum_{n=0}^{\infty} V_{n}(x,x') \ \sigma(x,x')^{n} \\
W(x,x') = \sum_{n=0}^{\infty} W_{n}(x,x') \ \sigma(x,x')^{n} 
$$
The so-called Hadamard coefficients $V_{n}$ and $W_{n}$ are also symmetric and regular as $x \to x'$. The functions $V_{n}$ are determined via the recursion relation
$$
(n+1)(2n+4) V_{n+1} + 2 (n+1) V_{n+1;\mu}\sigma^{;\mu} - 2 (n+1) V_{n+1} \Delta^{-1/2}\Delta^{1/2}_{\ ;\mu} \sigma^{;\mu} + ( \Box_{x} - m^2 - \xi R ) V_{n}=0
$$
for all $n \in \mathbb{N}$, subject to the boundary condition at $n=0$ given by
$$
2 V_0 + 2 V_{0\;\mu} \sigma^{;\mu}  - 2 V_0 \Delta^{-1/2} \Delta^{1/2}_{ ;\mu} \sigma^{;\mu} + ( \Box_{x} - m^2 - \xi R )\Delta^{1/2} = 0\ .
$$
Similarly the coefficients $W_{n}$ satisfy the recursion relations
$$
(n+1)(2n+4) W_{n+1} + 2 (n+1) W_{n+1;\mu}\sigma^{;\mu} - 2 (n+1) W_{n+1} \Delta^{-1/2}\Delta^{1/2}_{\ ;\mu} \sigma^{;\mu} + (4n+6) V_{n+1} + 2 V_{n+1;\mu} \sigma^{;\mu} - 2 V_{n+1} \Delta^{-1/2}\Delta^{1/2}_{\ ;\mu} \sigma^{;\mu} + ( \Box_{x} - m^2 - \xi R ) W_{n}=0
$$
for $n \in \mathbb{N}$, however the biscalar $W_0(x,x')$ (boundary condition) is unrestrained by the recursion relation.
Usually, I have seen $W_0(x,x')=0$ being set, and it's somewhat implied that it is a choice to do so.
Question 1: Is the above function $G^{\mathrm{F}}(x,x')$ truely the time-ordered correlator $\sim \langle \Omega | \mathscr{T}\big( \phi(x) \phi(x') \big) | \Omega \rangle$? Sometimes in the literature I see a Hadamard expansion referred to for retarded correlation functions (see for example equation (2.4) of Ottewill and Wardell's  Quasilocal contribution to the scalar self-force: Nongeodesic motion), and the use of the same phrase ``Hadamard Expansion'' for these two different functions confuses me.
Question 2: Does the choice of $W_0(x,x')$ correspond to a choice of vacuum state $|\Omega \rangle$? I am very confused about this --- for example, if I chose $W_0=0$ in Schwarszchild space, would this correspond to the Boulware vacuum, or the Hartle-Hawking vacuum, or the Unruh vacuum? It seems it would have to pick out one of those three vacuums (surely not all of them?). Is there a statement regarding this? Does some other choice of $W_0(x,x') \neq 0$ correspond to different vacuum states then? (From page 162 of "Semiclassical and Stochastic Gravity" by Hu and Verdaguer, it seems that there are restrictions on what $W_0(x,x')$ can be ie. you need to pick it such that all $W_{n}(x,x')$ are still symmetric in $x$ and $x'$)
 A: In both cases the answer is NO.
Regarding the first issue: You are dealing with a parametrix and not with a true state. Usually the series does not even converge. 
Regarding the second issue: in general it is impossible to obtain a final sum $W(x,y)=W(y,x)$ by summing the series with a suitable choice of $w_0$ as a true two-point function would deserve. This is the reason for the appearance of the trace anomaly when renormalizing the stress energy tensor with the point-splitting procedure. 
Decanini and Folacci cite a paper of mine in their paper
V. Moretti:  Comments on the stress-energy tensor operator in curved spacetime
Commun. Math. Phys. 232, 189 (2003)
about the stress energy tensor operator, there is a discussion on all these issues therein.
Again regarding your second question. The ``almost modern'' prescription is to remove completely the part of the asympthotic series containing all the terms $W_k$ obtaining a parametrix $H$ since it is useless to fix a state. The remaining part $H$ is therefore universal and a Gaussian state is said to be  of Hadamard type is its two-point function $G$ (or its Feynman propagator depending on which version of the expansion you consider) subtracted to the parametrix
$$G(x,y) - H(x,y) = w(x,y)$$
 is a smooth function $w$. This function, if the state is Hadamard, completely determines it.  
Excluding very particular cases it is impossible to obtain $G$ as a true complete convergent expansion suitably fixing $W_0$. 
The case of the vacua in Schwarzschild-Kruskal manifold is very delicate and the approach in terms of a parametrix is even more difficult. It can be tackled (to prove that the state is Hadamard) with the `'modern'' approach of mircolocal analysis of Hoermander introduced by Radzikowski (wavefront sets technology).
Here is a complete discussion and rigorous construction on the Unruh state as a Hadamard state,
C. Dappiaggi, V. Moretti and N. Pinamonti: Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime.
Adv. Theor. Math. Phys. 15, vol 2, 355-448 (2011)  (93 pages!)
It could be also useful the chapter of a book I wrote with I. Khavkine,
I.Khavkine and V. Moretti: Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction  (freely downloadable at arXiv:1412.5945)
And also this paper with T. Hack written into a more physically minded fashion 
T.-P. Hack and V. Moretti: On the stress-energy tensor of QFT in curved spacetime - Comparison of different regualrization schemes and symmetry of the Hadamard/Seeley-DeWitt coefficients J. Phys. A: Math.Theor. 45 374019 (2012) 
