Path integral derivation of the thermofield double state in causal diamonds

Abstract

In this article, we adopt the framework developed by R. Laflamme in \textit{Physica A}, \textbf{158}, pp. 58-63 (1989) to analyze the path integral of a massless -- conformally invariant -- scalar field defined on a causal diamond of size $2\alpha$ in 1+1 dimensions. By examining the Euclidean geometry of the causal diamond, we establish that its structure is conformally related to the cylinder $S^{1}_{\beta} \otimes \mathbb{R}$, where the Euclidean time coordinate $\tau$ has a periodicity of $\beta$. This property, along with the conformal symmetry of the fields, allows us to identify the connection between the thermofield double (TFD) state of causal diamonds and the Euclidean path integral defined on the two disconnected manifolds of the cylinder. Furthermore, we demonstrate that the temperature of the TFD state, derived from the conditions in the Euclidean geometry and analytically calculated, coincides with the temperature of the causal diamond known in the literature. This derivation highlights the universality of the connection between the Euclidean path integral formalism and the TFD state of the causal diamond, as well as it further establishes causal diamonds as a model that exhibits all desired properties of a system exhibiting the Unruh effect.