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

Abstract

We discuss space-time symmetric Hamiltonian operators of the form , where $H_0$ is Hermitian and $g$ real. $H_0$ is invariant under the unitary operations of a point group $G$ while $H^{\prime }$ is invariant under transformation by elements of a subgroup $G^{\prime }$ 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 , where $g_c$ is the exceptional point closest to the origin. Point-group symmetry and perturbation theory enable one to predict whether 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.

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• On the symmetry of three identical interacting particles in a one-dimensional box