Finite-Dimensional PT-Symmetric Hamiltonians

This paper investigates finite-dimensional representations of PT-symmetric Hamiltonians. In doing so, it clarifies some of the claims made in earlier papers on PT-symmetric quantum mechanics. In particular, it is shown here that there are two ways to extend real symmetric Hamiltonians into the complex domain: (i) The usual approach is to generalize such Hamiltonians to include complex Hermitian Hamiltonians. (ii) Alternatively, one can generalize real symmetric Hamiltonians to include complex PT-symmetric Hamiltonians. In the first approach the spectrum remains real, while in the second approach the spectrum remains real if the PT symmetry is not broken. Both generalizations give a consistent theory of quantum mechanics, but if D>2, a D-dimensional Hermitian matrix Hamiltonian has more arbitrary parameters than a D-dimensional PT-symmetric matrix Hamiltonian.