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 ight]

where a and b are positive constants and

:$$X stackrel{mathrm{def{=} frac{1}{2}eta^{mu u}partial_mu phi partial_ u 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.