On the Unruh effect and the thermofield double state
Gustavo Valdivia-Mera
TL;DR
The work provides a pedagogical synthesis of the Unruh effect and the thermofield double state by building from Rindler spacetime to a Euclidean derivation of thermal entanglement. It shows how Minkowski and Rindler modes mix through Bogoliubov coefficients, making the Minkowski vacuum appear thermal to an accelerated observer with temperature $T=\frac{g}{2\pi}$. The analysis connects to Schwarzschild thermality via near-horizon conformal structures and uses Unruh–DeWitt detectors and Euclidean path integrals to derive a thermofield double state $|0_M(\beta)\rangle$, with reduced density matrix $\rho_R=e^{-\beta H_R}/Z(\beta)$. Overall, the paper illuminates the observer-dependent nature of the vacuum, the role of horizons, and how thermal behavior emerges from quantum fields in curved or accelerated backgrounds, with implications for black hole thermodynamics and quantum information.”
Abstract
The purpose of this review is to provide a pedagogical development of the Unruh effect and the thermofield double state. In Section 2, we construct Rindler spacetime and analyze the perspective of an observer undergoing constant acceleration in Minkowski spacetime, which motivates the establishment of the relationship between the Fourier modes in both geometries using the Bogoliubov-Valatin transformation. In Section 3, we explore the underlying physics leading to the Unruh effect, its analogy with the thermal radiation observed around a Schwarzschild black hole, and its manifestation through the coupling of a particle detector to the scalar field. Finally, in Section 4, we derive the thermofield double state by conducting a Euclidean analysis of the field and geometry.