Indefinite metric vectors are absolutely required as the physical states in Minkowski space because that is indefinite metric space and the physical space-time. For example, Maxwell equations are wave equations in Minkowski space. However, traditional Quantum theory ordinarily has been studied only in definite metric space, i.e., Hilbert space. There are no clear expression for indefinite metric vectors. Here we show a wave function example using Dirac's delta function for indefinite metric vectors in Minkowski space. In addition, we show the vectors can interfere with itself. This example also suggests indefinite metric will be absolutely required.
A new proof is given for why the non-Hermitian, PT-Invariant cubic oscillator with imaginary coupling has real eigenvalues. The proof consists of two steps. In the first step, it is shown that for many PT-Invariant Hamiltonians, one can define corresponding Hermitian Hamiltonians that have the same eigenvalues when quantized using indefinite metric. The second step is to show that the indefinite-metric eigenvalues of the Hermitian cubic oscillator are real since the norms of its eigenstates do not vanish. The correspondence between PT-Invariant Hamiltonians and indefinite-metric Hermitian Hamiltonians is further discussed as a way to determine vacuum stability for certain supersymmetric gauge theories.