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 $\Box ϕ$ 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.