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.