On the state space of the dipole ghost (Q1057997)

The irreducible nonunitary representation D(1,1/2,1/2) of SO(4,2) is identified with the state space of the free dipole ghost. As Minkowski space will not suffer the global action of the group SO(4,2), a spacetime locally isomorphic to Minkowski space is taken instead. An explicit realization as the solution space of a 4th order wave equation is given to the above defined representation. The solution space of the wave equation combines the representations D(1,1/2,1/2) and D(0,0,0) to form an indecomposable SO(4,2) module. An invariant inner product is suggested, indefinite on the modes which carry D(1,1/2,1/2). The modes of the dipole field and those of the massless, scalar field which carries the D(1,0,0) irreducible unitary representation of SO(4,2) are suitable connected. The wave equation for the dipole ghost in Minkowski space is obtained by a particular coordinate transformation. It is finally concluded that the dipole ghost is not unitarizable in a Poincaré invariant manner.