Quantum Mechanics and Black Holes in Four-Dimensional String Theory

TL;DR

This paper extends the concept of W-hair, a set of conserved quantum numbers, from two-dimensional to four-dimensional string theories, showing how they help preserve quantum coherence in black hole evaporation.

Contribution

It introduces a four-dimensional model of string black holes with W-symmetry, demonstrating how topological string states maintain quantum coherence during evaporation.

Findings

W-hair symmetry extends to four dimensions with SU(1,1)/U(1) structure.

Topological string states account for information preservation.

Black hole entropy scales quadratically with mass.

Abstract

In previous papers we have shown how strings in a two-dimensional target space reconcile quantum mechanics with general relativity, thanks to an infinite set of conserved quantum numbers, ``W-hair'', associated with topological soliton-like states. In this paper we extend these arguments to four dimensions, by considering explicitly the case of string black holes with radial symmetry. The key infinite-dimensional W-symmetry is associated with the $\frac{SU(1,1)}{U(1)}$ coset structure of the dilaton-graviton sector that is a model-independent feature of spherically symmetric four-dimensional strings. Arguments are also given that the enormous number of string {\it discrete (topological)} states account for the maintenance of quantum coherence during the (non-thermal) stringy evaporation process, as well as quenching the large Hawking-Bekenstein entropy associated with the black hole. Defining the latter as the measure of the loss of information for an observer at infinity, who - ignoring the higher string quantum numbers - keeps track only of the classical mass,angular momentum and charge of the black hole, one recovers the familiar a quadratic dependence on the black-hole mass by simple counting arguments on the asymptotic density of string states in a linear-dilaton background.