Fourfold path to Thermality: Inequivalent purifications of Rindler wedge

TL;DR

This work demonstrates that thermal spectra in accelerated frames need not arise from entanglement across horizons. By comparing four inequivalent paths—spatial and null shifts between Rindler wedges—the authors show that Unruh-type thermality (Gibbsian and entanglement-based) and non-Gibbsian, Bogoliubov-induced thermality can be inequivalently realized. In null-shift constructions, thermal occupation numbers appear in a globally pure state, and the observed thermality is a kinematic consequence of Bogoliubov mixing rather than horizon-induced mixedness. The study employs both Bogoliubov transformations and the Virasoro anomaly in 2D CFT to diagnose particle content and energy flux, revealing path-dependent stress-tensor behavior under different vacua and reinforcing the conceptual distinction between thermality and entanglement in relativistic QFT.

Abstract

We investigate thermal behaviour in quantum fields by analysing a hierarchy of null-shifted Rindler wedges in Minkowski spacetime. Starting from the Minkowski vacuum restricted to an initial Rindler wedge, we construct several inequivalent transformation paths, including direct Minkowski-Rindler mappings, spatial translations, and sequential null displacements, and analyse the resulting particle content using Bogoliubov transformations. In the standard Unruh effect, entanglement between left- and right-moving sectors across the Rindler horizon produces Gibbsian thermality, with both sectors described by mixed thermal states. In contrast, we show that null-shifted wedge constructions lead to a selective and non-Gibbsian form of thermality: only a single chiral sector develops Bose-Einstein-distributed occupation numbers, while the complementary sector remains in the vacuum. Along composite transformation paths, the global Minkowski state remains pure, and the induced states associated with null-shifted wedges are pure tensor-product states. The observed thermal behaviour arises from Bogoliubov mixing and modular time evolution rather than horizon-induced entanglement or Gibbsian mixedness. These results demonstrate the existence of inequivalent purifications of thermal spectra and clarify the distinct roles of horizon structure, observer dependence, Bogoliubov transformations, and entanglement in relativistic quantum field theory. The null-shifted construction may be viewed as a converse of the Unruh effect, in which thermal spectra arise without entanglement-induced mixedness, highlighting the operational independence of thermality and entanglement.

Figures (5)

• Figure 1: Null-shifted Rindler wedges in Minkowski spacetime. The standard right Rindler wedge is successively shifted along the future-directed null $V$-direction to obtain ($R_2$) and then along the $U$-direction to obtain ($R_3$).
• Figure 2: Alternative null-shift construction. The wedge ($R_4$) is obtained from ($R_1$) by a null shift along the $U$-direction, followed by a shift along the $V$-direction to obtain ($R_3$). As in Fig. \ref{['Fig:1']}, the construction results in nested spacetime regions.
• Figure 3: Rindler spacetime with an observer experiencing acceleration a and g, shown in green and red, respectively.
• Figure 4: Complex plane
• Figure 5: Complex plane