Path integral derivation of the thermofield double state in causal diamonds

In this article, we follow the framework given in the article Physica A, 158, pg 58-63 (1989) by R. Laflamme to derive the thermofield double state for a causal diamond using the Euclidean path integral formalism, and subsequently derive the causal diamond temperature. The interpretation of the physical and fictitious system in the thermofield double state arises naturally from the boundary conditions of the fields defined on the Euclidean sections of the cylindrical background geometry $S^{1}_{\beta}\times \mathbb{R}$, where $\beta$ defines the periodicity of the Euclidean time coordinate and $S^{1}_{\beta}$ is the one-dimensional sphere (circle). The temperature detected by a static diamond observer at $x=0$ matches with the thermofield double temperature derived via this path integral procedure.