Quantum properties of the Dirac field on BTZ black hole backgrounds

TL;DR

This paper analyzes the quantum behavior of a Dirac field in a 2+1 dimensional BTZ black hole background, focusing on self-adjointness, spectral properties, and the absence of quantum angular momentum loss.

Contribution

It provides a detailed analysis of the Dirac Hamiltonian's self-adjointness, spectral properties, and quantum angular momentum transfer in BTZ black hole backgrounds, highlighting differences from higher-dimensional cases.

Findings

Self-adjointness depends on a relation between mass and cosmological radius.

No quantum bound states exist for the Dirac field in the non-extremal case.

No quantum loss of angular momentum occurs via particle pair production.

Abstract

We consider a Dirac field on a $(1 + 2)$-dimensional uncharged BTZ black hole background. We first find out the Dirac Hamiltonian, and study its self-adjointness properties. We find that, in analogy to the Kerr-Newman-AdS Dirac Hamiltonian in $(1+3)$ dimensions, essential self-adjointness on $C_0^{\infty}(r_+,\infty)^2$ of the reduced (radial) Hamiltonian is implemented only if a suitable relation between the mass $\mu$ of the Dirac field and the cosmological radius $l$ holds true. The very presence of a boundary-like behaviour of $r=\infty$ is at the root of this problem. Also, we determine in a complete way qualitative spectral properties for the non-extremal case, for which we can infer the absence of quantum bound states for the Dirac field. Next, we investigate the possibility of a quantum loss of angular momentum for the $(1 + 2)$-dimensional uncharged BTZ black hole. Unlike the corresponding stationary four-dimensional solutions, the formal treatment of the level crossing mechanism is much simpler. We find that, even in the extremal case, no level crossing takes place. Therefore, no quantum loss of angular momentum via particle pair production is allowed.