Test of Transitivity in Quantum Field theory using Rindler spacetime

Abstract

We consider a massless scalar field in Minkowski spacetime $\mathcal{M}$ 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 $R_2\subset R_1\subset \mathcal{M}$ 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 $\mathcal{M}$ which yields a thermal density matrix, b) by first evaluating the reduced state in $R_1$ from $\mathcal{M}$ 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.

6 pages, 3 figures

References in corpus (3)

• The Unruh effect and its applications
• The entropy of Hawking radiation
• Holographic Evolution of Entanglement Entropy

Cited by in corpus (1)

• Selective Thermalization, Chiral Excitations, and a Case of Quantum Hair in the Presence of Event Horizons