Living with ghosts

By S. Hawking, T. Hertog. 2001

268 citations recorded by the source corpus.

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, $\phi$ and $\Box \phi $ 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 $\phi$ and $\phi_{,\tau}$. To calculate probabilities for observations, one has to trace out over $\phi_{,\tau}$ 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.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4f2155342fe2dc7ad6d9b9ae762ec68f8a8e8906

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Citations in the corpus

The asymptotic safety scenario in quantum gravity: an introduction (2006, background): “Second, even for higher derivative theories on flat space a well-defined Euclidean functional integral can exist, free of negative norm states or negative probabilities [75].”
Dynamically induced Planck scale and inflation (2015, background): “The gravitational kinetic terms suppressed by the dimensionless constants f0 and f2 contain four derivatives: thereby the graviton contains a massive spin-2 ghost component [87], which is possibly problematic for energies above its mass M2 = |f2|M̄Pl/ √ 2: we do not address the issue of finding a se”
f(R) Theories (2010, background): “Another possible approach to dealing with the ghost degrees of freedom consists of using the Euclidean-action path formalism, for which, one can indeed introduce a notion of probability amplitude for these spurious degrees of freedom [294, 162].”
Quadratic Gravity (2018, background): “At this point it is good to mention that Hawking and Hertog [55] proposed a way to deal with four-derivative degrees of freedom, but they ended up with non-normalizable wave functions.”