Ghost Degrees of Freedom Without Quantum Runaway: Exact Moment Bounds from an Operator Conservation Law

TL;DR

This paper establishes an exact quantum conservation law for a harmonic oscillator coupled to a ghost degree of freedom, providing rigorous bounds on quantum state evolution without confining potentials or spectral assumptions.

Contribution

It introduces a novel operator conservation law that guarantees quantum stability and bounded second moments in a non-confining, effective field theory setting.

Findings

Quantum conservation law yields state-independent bounds.

Wavepacket confinement confirmed by three numerical methods.

System exhibits real energy spectrum and Poisson level statistics.

Abstract

We prove an exact quantum conservation law for a harmonic oscillator coupled to a ghost degree of freedom: a second classical conserved quantity lifts to a quantum operator that commutes with the Hamiltonian with no hbar corrections, yielding a rigorous, state-independent upper bound on the mean squared phase-space radius for all time and every quantum state with finite initial second moments. The proof uses only canonical commutation relations and the Leibniz rule; it requires no confining potential, no spectral assumptions, and no perturbative expansion. The interaction studied here is bounded and vanishes at large separations, the generic situation in effective field theory, yet this suffices to guarantee quantum stability in the sense of bounded second moments. Three independent numerical frameworks (Heisenberg picture, Schrodinger picture, and Fock-space diagonalization) confirm wavepacket confinement below the analytic bound, a real energy spectrum, and Poisson level statistics numerically consistent with an integrable structure. The absence of a confining potential means the proof is silent on spectral discreteness and the existence of a ground state; those questions, addressed for polynomial confining interactions in concurrent work, remain open for the interaction class studied here and represent the sharpest targets for future work. Ghost quantum instability is therefore not an inevitable consequence of a wrong-sign kinetic term but depends critically on the interaction structure.