Appropriate Inner Product for PT-Symmetric Hamiltonians

A Hamiltonian H that is not Hermitian can still have a real and complete energy eigenspectrum if it instead is PT symmetric. For such Hamiltonians three possible inner products have been considered in the literature, the V norm, the PT norm, and the C norm. Here V is the operator that implements VHV^-1=H^dagger, the PT norm is the overlap of a state with its PT conjugate, and C is a discrete linear operator that always exists for any Hamiltonian that can be diagonalized. Here we show that it is the V norm that is the most fundamental as it is always chosen by the theory itself. In addition we show that the V norm is always equal to the PT norm if one defines the PT conjugate of a state to contain its intrinsic PT phase. We discuss the conditions under which the V norm coincides with the C operator norm, and show that in general one should not use the linear C operator but for the purposes that it is used one can instead use the antilinear PT operator itself.