Test of transitivity in quantum field theory using Rindler spacetime

Abstract

Consider a massless scalar field in Minkowski spacetime $\mathcal{M}$ in its vacuum state and two Rindler wedges ${R}_{1}$ and ${R}_{2}$. Here ${R}_{2}$ is shifted to the right of ${R}_{1}$ by a distance $\mathrm{\ensuremath{\Delta}}$. We, therefore, have ${R}_{2}\ensuremath{\subset}{R}_{1}\ensuremath{\subset}\mathcal{M}$ with the symbol $\ensuremath{\subset}$ implying a quantum subsystem. We find the reduced state in ${R}_{2}$ using two independent ways: (a) Evaluation of the reduced state of ${R}_{2}$ from vacuum state in $\mathcal{M}$, which yields a thermal density matrix. (b) First, evaluating the reduced state in ${R}_{1}$ from $\mathcal{M}$ yielding a thermal state in ${R}_{1}$, and subsequently evaluating the reduced state in ${R}_{2}$ from ${R}_{1}$. In this article, we attempt to address the question of whether both these independent paths yield the same reduced state in ${R}_{2}$. To that end, we devise a method that involves cleaving the Rindler wedge ${R}_{1}$ into two domains such that they form a thermofield double. One of the domains aligns itself along the wedge ${R}_{2}$ while the other is a diamond-shaped construction between the boundaries of ${R}_{1}$ and ${R}_{2}$. We conclude that both these independent methods yield two different answers, indicating that there are systems in which the reduced quantum state of a subsystem can be path dependent and hence not unique in a quantum field theoretic setting. We discuss the possible implications of our result in the context of quantum states outside a nonextremal black hole formed by collapsing matter.