Superselected ghost theory: real spectrum

Abstract

Quantum field theories with ghosts can be unitary and perturbatively stable, yet the negative-norm states of their conserved indefinite inner product obstruct a probabilistic interpretation. This problem is especially relevant to renormalizable quantum gravity. The focus of this paper is the real-spectrum regime, in which the interacting Hamiltonian has an exact $\mathbb{Z}_2$ symmetry $Q$, called exact ghost parity. Its eigenvalue on each energy eigenstate equals the sign of the norm, and it reduces to free ghost parity at zero coupling. Imposing $Q$ as a superselection charge defines a new theory in which the physical states have definite ghost parity and the observables commute with $Q$. The native Born rule then yields non-negative probabilities, while the optical theorem acquires a direct probabilistic interpretation. The propagator decomposes into $Q$-sector spectral representations with fixed-sign spectral functions and no complex poles on the physical sheet. Finally, a similarity transformation yields a Hermitian perturbation theory that preserves free ghost parity order by order, making the exact superselection structure perturbatively manifest.