Unitary Time Evolution and Vacuum for a Quantum Stable Ghost

TL;DR

This paper demonstrates that a quantized system with a harmonic oscillator coupled to a ghost field remains stable, unitary, and well-defined due to a positive integral of motion, confirmed by numerical solutions.

Contribution

It introduces a quantization approach for a classically stable ghost system, establishing stability and unitarity through a positive integral of motion.

Findings

Hamiltonian has a pure point spectrum unbounded in both directions

Evolution is manifestly unitary

Vacuum state is well-defined and expectation values are bounded

Abstract

We quantize a classically stable system of a harmonic oscillator polynomially coupled to a ghost with negative kinetic energy. We prove that due to an integral of motion with a positive discrete spectrum: i) the Hamiltonian has a pure point spectrum unbounded in both directions, ii) the evolution is manifestly unitary, iii) the vacuum is well-defined, iv) expectation values for squares of canonical variables are bounded. Numerical solutions of the Schr\"odinger equation confirm these results. We argue that the discrete spectrum of the integral of motion enforces stability for extended interactions.