Density matrix of black hole radiation

TL;DR

This paper compares different models of black hole evaporation, analyzing the density matrix of emitted radiation, and highlights how quantum corrections and entropy evolution differ among Hawking, Page, and semiclassical models.

Contribution

It provides a detailed comparison of the density matrices in Hawking, Page, and semiclassical models, revealing how off-diagonal corrections and entropy growth behave in each framework.

Findings

Hawking's model has power-law suppressed off-diagonal corrections.

Semiclassical corrections grow monotonically and are less suppressed.

The Rényi entropy initially grows linearly and then decreases after the Page time.

Abstract

Hawking's model of black hole evaporation is not unitary and leads to a mixed density matrix for the emitted radiation, while the Page model describes a unitary evaporation process in which the density matrix evolves from an almost thermal state to a pure state. We compare a recently proposed model of semiclassical black hole evaporation to the two established models. In particular, we study the density matrix of the outgoing radiation and determine how the magnitude of the off-diagonal corrections differs for the three frameworks. For Hawking's model, we find power-law corrections to the two-point functions that induce exponentially suppressed corrections to the off-diagonal elements of the full density matrix. This verifies that the Hawking result is correct to all orders in perturbation theory and also allows one to express the full density matrix in terms of the single-particle density matrix. We then consider the semiclassical theory for which the corrections, being non-perturbative from an effective field-theory perspective, are much less suppressed and grow monotonically in time. In this case, the R\'enyi entropy for the outgoing radiation is shown to grow linearly at early times; but this growth slows down and the entropy eventually starts to decrease at the Page time. In addition to comparing models, we emphasize the distinction between the state of the radiation emitted from a black hole, which is highly quantum, and that of the radiation emitted from a typical classical black body at the same temperature.