The Case for a Quantum Theory on a Hilbert Space with an Inner Product of Indefinite Signature

We present the theoretical considerations for the case of looking into a generalization of quantum theory corresponding to having an inner product with an indefinite signature on the Hilbert space. The latter is essentially a direct analog of having the Minkowski spacetime with an indefinite signature generalizing the metric geometry of the Newtonian space. In fact, the explicit physics setting we have in mind is exactly a Lorentz covariant formulation of quantum mechanics, which has been discussed in the literature for over half a century yet without a nice full picture. From the point of view of the Lorentz symmetry, indefiniteness of the norm for a Minkowski vector may be the exact correspondence of the indefiniteness of the norm for a quantum state vector on the relevant Hilbert space. That, of course, poses a challenge to the usual requirement of unitarity. The related issues will be addressed.