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

Abstract

We discuss space-time symmetric Hamiltonian operators of the form H = H 0 +igH ′ , where H 0 is Hermitian and g real. H 0 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 < g c , where g c 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.

Non-Hermitian Hamiltonians and Similarity Transformations

“…the so-called space-time symmetry did not prove to be so robust in producing nonHermitian operators with real spectra [9][10][11][12].…”

Three PT-symmetric Hamiltonians with completely different spectra

“…In the multidimensional case H 0 and H may exhibit more complex symmetry that is conveniently described by means of group theory. In this way Fernández and Garcia [17,18] and Amore et al [19,20] found that some ST-symmetric Hamiltonians exhibit broken ST symmetry for all values of g. The main conjecture was that ST symmetry may be unbroken for some values of g provided that S is the only member of a class in the point group for H 0 [20]. This appeared to be the case when S = P .…”

“…The latter case may only take place when the spectrum of H is degenerate. In many cases it suffices to calculate the simplest, straightforward perturbation correction of first order E (1) [17][18][19][20].…”

“…The resulting Hamiltonian H exhibits cylindrical symmetry and may be viewed as a kind of Stark effect with imaginary electric field. In Section 2 we outline the main ideas of PT symmetry as well as a simple argument based on perturbation theory [17][18][19][20]. In Section 3 we briefly discuss the general case.…”

Perturbation theory by the moment method and point-group symmetry

“…However, such treatments of symmetry look rather rudimentary when compared with the more rigorous approach carried out, for example, by Pullen and Edmonds [25,26]. Those enlightening papers motivated the application of point-group symmetry (PGS) to several multidimensional non-Hermitean anharmonic oscillators that led to most interesting conclusions [27][28][29][30].…”