= & \lim_{\zeta \to 0} 
        \partial_{\zeta^{\mu_1}} \cdots \partial_{\zeta^{\mu_n}} \Big{(}
                         \Phi_0(q_x(\zeta))\Phi_0(q_x(-\zeta))  
                                         - 
                                     \langle \Phi_0(q_x(\zeta))\Phi_0(q_x(-\zeta)) \rangle_{\rm vac} 
 \Big{)} \nonumber\end{aligned}$$ Prominent among these is the second balanced derivative because $$\label{theen}
\boldsymbol{\varepsilon}_{\mu\nu}(x) = - \frac{1}{4} \eth_{\mu\nu} :\Phi_0^2:(x)$$ is the [*thermal stress energy tensor*]{} which in the KMS state $\omega_{\beta,e_0}$ takes on the values[^1] $$\begin{aligned}
\varepsilon_{\mu \nu}(x) & =  
\langle \boldsymbol{\varepsilon}_{\mu\nu}(x)\rangle_{\omega_{\beta e_0}}  = \frac{1}{(2\pi )^3} \int_{\mathbb{R}^3} 
 \frac{p_\mu p_\nu}{({\rm e}^{\beta p_0} -1)p_0}\, d^3p\end{aligned}$$ where $p_\mu = p_a e^a_\mu$ are the covariant coordinates of $p$ with respect to the chosen Lorentz frame, and $p_0 = |\underline{p}|$ with $(p_\mu) = (p_0,\underline{p})$. Again, this quantity is independent of $x$ in the unique KMS state of our quantum field model.

Notice that the values of $\varepsilon_{\mu\nu}$ depend not only on the inverse temperature, but also on the time-direction $e_0$ of the Lorentz-frame with respect to which $\omega_{\beta,e_0}$ is a KMS-state. Since the dependence of the thermal quantities on $\beta$ and $e_0$ in expectation values of $\omega_{\beta,e_0}$ is always a function of the timelike, future-directed vector $\boldsymbol{\beta} = \beta \cdot e_0$, it is hence useful to label the KMS states correspondingly as $\omega_{\boldsymbol{\beta}}$, and to define LTE states with reference to $\boldsymbol{\beta}$.

Let $\omega$ be a state of the linear scalar 