69. Happy Halloween!
Save This Post as PDFSince it is Halloween, today’s story is about ghosts…
Let us follow the analysis of massive gravity theory in de Sitter background by Izumi and Tanaka.
The action of the theory is given by
,
and if the graviton is sufficiently light (
), the helicity 0 mode is the ghost. Naively, this fact implies that the theory is badly defined. Indeed, recall what happens with theories admitting ghosts in Minkowski spacetime. The energy of the ghost mode is negative, so that if ghosts interact with ordinary matter, pair production of matter quanta takes place (the energy taken from the ghost condensate is transferred to the excitations of the ordinary matter). One would argue that non-linearities may stop the particle production at some moment due to the backreaction, however, this argument is wrong. If the initial |in> vacuum state is Lorentz invariant, the probability of pair creation is the same for all processes boosted by Lorentz transformation w.r.t. each other. The full creation rate is given by the integral over momentum p and is divergent in UV, that makes ghosts so dangerous creatures.
Note that UV divergence of this kind is not what we expect in theories with physically relevant instabilities leading to particle production. Indeed, we expect from a healthy theory that the UV vacuum structure remains intact (in other words, there is good Minkowski limit for the pair correlation functions, when one takes the splitting between two points to zero). It is therefore necessary to have the following condition satisfied:
,
where 
is the Bogolyubov coefficient for the mode with momentum 
(its square gives the number of particles in the given mode), i.e., we expect that there is no particle production in the UV for a healthy theory.
Let us return to ghosts in de Sitter. Do these considerations automatically imply that a theory with ghost in de Sitter background is sick? My naive expection for the correct answer to this question is to be certainly positive, but Izumi and Tanaka argue that my expectation is wrong: all one can conclude is that theories with ghosts do not admit de Sitter invariant vacuum state. (Note: Hmm, so what? Light scalar field with 
also does not allow for a de Sitter invariant vacuum state.)
Trying to exploit this idea, the authors calculate the number and energy density of produced particles and find of course that both diverge in the UV. To make them convergent, they introduce the UV cutoff in three-dimensional momentum space – the step I do not feel to be very reasonable: the limit of adjacent points for the pair Green function in de Sitter should be still the usual Minkowski limit, and cutoff in 3d momentum space does not make much sense in this limit.
So, I have to conclude that no ghosts exist in Nature, even in de Sitter space. Unless, of course, I shall see one of these ghosts visiting me tonight 
Happy Halloween!
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And yet another piece of poetry from James Kotsybar:
Hodge-podge of religious superstition
And pagan ritual to raise the dead
Lasts malingeringly into our age
Like some cadaverous apparition
Or revenant, confused and full of dread,
Without a sense of purpose, drunk with rage,
Eternally lost in poor rendition,
Ever mocked by grimacing pumpkin-head,
Not laid to rest, but dressed as for the stage.