Diamonds's Temperature: Unruh effect for bounded trajectories and thermal time hypothesis

TL;DR

The paper addresses whether the Unruh effect extends to observers with finite lifetimes and how the thermal time hypothesis interprets time and temperature in this setting. It employs algebraic quantum field theory tools—local observable algebras, modular automorphisms, and the KMS condition—together with conformal geometry to map diamonds to wedges and derive a geometric modular flow. The main contributions are the explicit construction of the diamond's modular group, the local temperature $\beta(s)$ along the finite-lifetime worldline, and the closed-form inertial diamond temperature $T_D=\frac{2\hbar}{\pi k_B \mathcal{T}}$, thereby generalizing Unruh thermality to finite lifetimes. This work links finite-lifetime thermodynamics to the access limitations of quantum fields, offering a concrete framework for analyzing detector responses and thermodynamic time in curved or constrained spacetimes.

Abstract

We study the Unruh effect for an observer with a finite lifetime, using the thermal time hypothesis. The thermal time hypothesis maintains that: (i) time is the physical quantity determined by the flow defined by a state over an observable algebra, and (ii) when this flow is proportional to a geometric flow in spacetime, temperature is the ratio between flow parameter and proper time. An eternal accelerated Unruh observer has access to the local algebra associated to a Rindler wedge. The flow defined by the Minkowski vacuum of a field theory over this algebra is proportional to a flow in spacetime and the associated temperature is the Unruh temperature. An observer with a finite lifetime has access to the local observable algebra associated to a finite spacetime region called a "diamond". The flow defined by the Minkowski vacuum of a (four dimensional, conformally invariant) quantum field theory over this algebra is also proportional to a flow in spacetime. The associated temperature generalizes the Unruh temperature to finite lifetime observers. Furthermore, this temperature does not vanish even in the limit in which the acceleration is zero. The temperature associated to an inertial observer with lifetime T, which we denote as "diamond's temperature", is 2hbar/(pi k_b T).This temperature is related to the fact that a finite lifetime observer does not have access to all the degrees of freedom of the quantum field theory.

Figures (1)

• Figure 1: The inverse temperature ${\beta}$ as a function of the proper time $\tau$ seen by the accelerated observer with finite, but long, lifetime $L$ (acceleration is $a = 1$).