of operator-valued distributions). Now, at each point $x$ in Minkowski spacetime, one would like to define a set of “thermal observables” $\boldsymbol{\Theta}_x$ formed by observables which are sensitive to intensive thermal quantities at $x$. If that quantity is, e.g., temperature, then the corresponding quantity in $\boldsymbol{\Theta}_x$ would be a thermometer “located” at spacetime point $x$. One may wonder if it makes physical sense to idealize a thermometer as being of pointlike “extension” in space and time, but as discussed in [@BOR], this isn’t really a problem.

What are typical elements of $\boldsymbol{\Theta}_x$? Let us look at a very particular example. Assume that $m= 0$, so we have the massless field. Moreover, fix some Lorentzian frame consisting of a tetrad $(e^a_0,e^a_1,e^a_2,e^a_3)$ of Minkowski vectors such that $\eta_{ab}e^a_\mu e^b_\nu  = \eta_{\mu\nu}$ and with $e_0^a$ future-directed. Relative to these data, let $\omega_{\beta,e_0}$ denote the global thermal equilibrium state — i.e. KMS-state — with respect to the time-direction $e_0 \equiv e_0^a$ at inverse temperature $\beta > 0$ [@Haag]. Evaluating $:\Phi_0^2:(x)$, the Wick-square of $\Phi_0$ at the spacetime point $x$, in the KMS state gives $$\langle : \Phi_0^2:(x) \rangle_{\omega_{\beta,e_0}} = \frac{1}{12 \beta^2}\,.$$ Recall that the Wick square is defined as $$\label{Wick1}
   :\Phi_0^2:(x) = \lim_{\zeta \to 