living with ghosts

" We conclude that quantum gravity with fourth order corrections can make sense,
despite apparently having negative energy solutions and ghosts. In doing this,
we seem to go against the convictions of the last 25 years ..."

Hawking and Hertog, 2001

It is well known that the perturbation theory of quantum gravity is not renormalizable, but one can 'fix' this problem by introducing higher order terms ( R² ) in the action.
Unfortunately, it is also well known that higher order derivative terms (appear to) come with
dangerous ghosts, threatening the S-matrix with states of negative probabilities.
However, in their (very clear and easy to read) paper Hawking and Hertog provide for a convincing argument that one should not be afraid of such ghosts.

In a related paper Bender and Mannheim, 2007 showed that "contrary to common belief .. theories whose field equations are higher than second order in derivatives need not be stricken with ghosts. In particular, the prototypical fourth-order derivative Pais-Uhlenbeck oscillator model is shown to be free of states of negative energy or negative norm."

Last but not least, Benedetti, Machado and Saueressig, 2009 "study the non-perturbative renormalization group flow of higher-derivative gravity employing functional renormalization group techniques" and argue that "asymptotic safety also resolves the unitarity problem typically haunting higher-derivative gravity theories."

In other words, if (for whatever reason) you don't like string theory, you could try to get used to living with ghosts...

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