Is space-time symmetry a suitable generalization of parity-time symmetry?

Abstract

We discuss space-time symmetric Hamiltonian operators of the form H = H0 + igH ′ , where H0 is Hermitian and g real. H0 is invariant under the unitary operations of a point group G while H ′ is invariant under transformation by elements of a subgroup G′ of G. If G exhibits irreducible representations of dimension greater than unity, then it is possible that H has complex eigenvalues for sufficiently small nonzero values of g. In the particular case that H is parity-time symmetric then it appears to exhibit real eigenvalues for all 0 < g < gc , where gc is the exceptional point closest to the origin. Point-group symmetry and perturbation theory enable one to predict whether H may exhibit real or complex eigenvalues for g > 0. We illustrate the main theoretical results and conclusions of this paper by means of two- and three-dimensional Hamiltonians exhibiting a variety of different point-group symmetries.