Self-Adjointness of the Dirac Hamiltonian for a Class of Non-Uniformly Elliptic Boundary Value Problems

TL;DR

This paper establishes conditions under which the Dirac Hamiltonian is self-adjoint for certain boundary value problems in curved spacetime, including cases with horizons, ensuring well-defined quantum evolution.

Contribution

It constructs a self-adjoint extension of the Dirac Hamiltonian for non-uniformly elliptic boundary value problems in specific Lorentzian manifolds.

Findings

Self-adjoint extension of the Dirac Hamiltonian is achieved.

Results apply to spacetimes with horizons.

Addresses non-elliptic Hamiltonian cases.

Abstract

We consider a boundary value problem for the Dirac equation in a smooth, asymptotically flat Lorentzian manifold admitting a Killing field which is timelike near and tangential to the boundary. A self-adjoint extension of the Dirac Hamiltonian is constructed. Our results also apply to the situation that the space-time includes horizons, where the Hamiltonian fails to be elliptic.