Living with ghosts

Abstract

Perturbation theory for gravity in dimensions greater than two requires higher derivatives in the free action. Higher derivatives seem to lead to ghosts, states with negative norm. We consider a fourth order scalar field theory and show that the problem with ghosts arises because in the canonical treatment, φ and 2φ are regarded as two independent variables. Instead, we base quantum theory on a path integral, evaluated in Euclidean space and then Wick rotated to Lorentzian space. The path integral requires that quantum states be specified by the values of φ and φ ,τ . To calculate probabilities for observations, one has to trace out over φ ,τ on the final surface. Hence one loses unitarity, but one can never produce a negative norm state or get a negative probability. It is shown that transition probabilities tend toward those of the second order theory, as the coefficient of the fourth order term in the action tends to zero. Hence unitarity is restored at the low energies that now occur in the universe.

Dynamically induced Planck scale and inflation

“…The gravitational kinetic terms suppressed by the dimensionless constants f 0 and f 2 contain four derivatives: thereby the graviton contains a massive spin-2 ghost component [87], which is possibly problematic for energies above its mass M 2 = |f 2 |M Pl / √ 2: we do not address the issue of finding a sensible interpretation for it (see [88][89][90][91][92][93][94][95] for some attempts). In this section, as mentioned in the introduction, we do not adopt the effective field theory approach of section 2 and eq.…”

Dynamical sectors of a relativistic two particle model

“…The Hilbert space would be the direct sum of two Hilbert spaces: one with positive norm and the second one with negative norm. As soon as one adds interaction terms in the field theory Lagrangian, instabilities can occur and special care has to be taken [25].…”

Quantum aspects of massive gravity II: non-Pauli-Fierz theory

“…1 Thus, if we manage to recover unitarity that is apparently violated by the ghost, nPF might become a viable candidate for a large distance modification of gravity: GR is restored at short scales, while the gravitational force gets weakened by Yukawa-suppression at distances ≫ m −1 , so that long wavelength sources such as the cosmological constant gravitate less, providing a solution to the cosmological constant problem in the sense of [7]. Then, the spectrum of nPF suggests to pick up, among various proposals [8] to deal with ghosts, the Lee-Wick (LW) mechanism [9] as a resolution to its ghost problem. The LW mechanism unitarizes a theory with a ghost by the decay of the ghost into ordinary fields.…”