^{2(j+1)}}\ln\bigg(\frac{\sigma_\pm (x,x^\prime)}{\ell^2}\bigg)
\nonumber \\
&\,& \qquad 
+\sum^k_{j=0}w_j(x,x^\prime)\frac{\sigma^j(x,x^\prime)}{\ell^{2(j+1)}}\bigg\}
\, \end{aligned}$$ where we have introduced a length scale $\ell$ to make $\sigma/\ell^2$ dimensionless. By $F(\sigma_\pm)$, $F$ some function, we mean $$F(\sigma_\pm) = \lim_{\epsilon\rightarrow 0^+}F(\sigma_{\pm\epsilon})$$ in the sense of distributions, where $\sigma_{\pm\epsilon}(x,x^\prime) =
\sigma(x,x^\prime)\pm 2i\epsilon(t(x)-t(x^\prime))+\epsilon^2$ and $t$ is a time function on $X$. The functions $\Delta$, known as the van Vleck-Morette determinant, $v_j$ and $w_j$ are found by fixing $x^\prime$ and applying $P\otimes\mathbbm{1}$ to $H_k^{(+)}$ and equating all the coefficients of $1/\sigma_+$, $1/\sigma^2_+$, $\ln \sigma_+$ etc to zero; moreover, they are all spinors (i.e., they carry internal indices). This determines a system of differential equations known as the *Hadamard recursion relations*[^3]. In $X$ the system of differential equations uniquely determines the $\{ v_j \}_{j=0,\dots,k}$ series. The $\{ w_j\}_{j=0,\dots,k}$ series is specified once the value of $w_0$ is fixed; we adopt Wald’s prescription that $w_0=0$ [@wald].

Let $u$ be of Hadamard form for the operator $P$, i.e. within a regular domain $X$ one has $u = H_k^{(+)}$ modulo $C^k(X)$ for each $k\in{\mathbb{N}}$; the distribution $u$ is sometimes referred to as the *auxiliary two-point function*. A state $\omega$ on the algebra of smeared (Dirac) fields is said to be *Hadamard* if its associated two-point function 