Quantization of unstable linear scalar fields in static spacetimes

TL;DR

This paper develops a method to quantize unstable scalar fields in static spacetimes, constructing a one-particle Hilbert space and analyzing the Hamiltonian for unstable modes, revealing their non-relativistic particle behavior.

Contribution

It introduces a novel approach to quantize unstable scalar fields in static spacetimes using a complex structure derived from classical solutions, extending previous methods to include instability.

Findings

Constructed a one-particle Hilbert space for unstable fields.

Derived the Hamiltonian for unstable modes showing non-relativistic behavior.

Confirmed that unstable degrees of freedom behave as particles in a potential barrier.

Abstract

We discuss the quantization of an unstable field through the construction of a "one-particle Hilbert space." The system considered here is a neutral scalar field evolving over a globally hyperbolic static spacetime and subject to a stationary external scalar potential. In order to prove our results we assume spacetimes without horizons and that the theory possess a "mass gap." Our strategy consists in building a complex structure, which arises from a suitable positive bilinear form defined over the space of classical solutions of the field equation. Once the space of states of the quantum field has been set, it is possible to study the effect of the time translation symmetry on it. From the time translation operator we obtain an expression for the Hamiltonian operator associated with the unstable sector of the field. This last result coincides with findings from long ago showing that the unstable degrees of freedom of the field behave as non-relativistic particles in a parabolic potential barrier.