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Ghost condensate
In particle physics, a ghost condensate is a speculative proposal in which a ghost, an excitation of a field with a wrong sign of the kinetic term, acquires a vacuum expectation value. This phenomenon breaks Lorentz invariance spontaneously. Around the new vacuum state, all excitations have a positive norm, and therefore the probabilities are positive definite.
We have a real scalar field φ with the following action
\( S=\int d^4x \left[aX^2-bX\right] \)
where a and b are positive constants and
\( X\ \stackrel{\mathrm{def}}{=}\ \frac{1}{2}\eta^{\mu\nu}\partial_\mu \phi \partial_\nu \phi \)
in the +--- sign convention.
The theories of ghost condensate predict specific non-Gaussianities of the cosmic microwave background. These theories have been proposed by Nima Arkani-Hamed, Markus Luty, and others.
Unfortunately, this theory allows for superluminal propagation of information in some cases and has no lower bound on its energy. This model doesn't admit a Hamiltonian formulation (the Legendre transform is multi-valued because the momentum function isn't convex) because it is acausal. Quantizing this theory leads to problems.
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