PT-symmetric quantum electrodynamics and unitarity

More than 15 years ago, a new approach to quantum mechanics was suggested, in which Hermiticity of the Hamiltonian was to be replaced by invariance under a discrete symmetry, the product of parity and time-reversal symmetry, . It was shown that, if is unbroken, energies were, in fact, positive, and unitarity was satisfied. Since quantum mechanics is quantum field theory in one dimension—time—it was natural to extend this idea to higher-dimensional field theory, and in fact an apparently viable version of -invariant quantum electrodynamics (QED) was proposed. However, it has proved difficult to establish that the unitarity of the scattering matrix, for example, the Källén spectral representation for the photon propagator, can be maintained in this theory. This has led to questions of whether, in fact, even quantum mechanical systems are consistent with probability conservation when Green’s functions are examined, since the latter have to possess physical requirements of analyticity. The status of QED will be reviewed in this paper, as well as the general issue of unitarity.