Test of Transitivity in Quantum Field theory using Rindler spacetime

TL;DR

This paper investigates whether different methods of deriving reduced states in Rindler spacetime yield consistent results, revealing a discrepancy that has implications for understanding quantum states near black holes.

Contribution

It introduces a novel approach to test transitivity of quantum states in Rindler spacetime by comparing two independent methods of state reduction.

Findings

Different methods produce inconsistent reduced states in Rindler wedges

Discrepancy suggests non-transitivity of quantum states in this setup

Implications for quantum black hole physics and information paradox

Abstract

We consider a massless scalar field in Minkowski spacetime $\cal{M} $ in its vacuum state, and consider two Rindler wedges $R_1$ and $R_2$ in this space. $R_2$ is shifted to the right of $R_1$ by a distance $\Delta$. We therefore have $R_2\subset R_1 \subset \cal{M}$ with the symbol $\subset$ implying a quantum subsystem. We find the reduced state in $R_2$ using two independent ways: a) by evaluation of the reduced state from vacuum state in $\cal{M}$ which yields a thermal density matrix, b) by first evaluating the reduced state in $R_1$ from $\cal{M} $ yielding a thermal state in $R_1$, and subsequently evaluate the reduced state in $R_2$ in that order of sequence. In this article we attempt to address the question whether both these independent ways yield the same reduced state in $R_2$. To that end, we devise a method which 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, and discuss the possible implications of our result in the context of quantum states outside a non-extremal black hole formed by collapsing matter.