May 2014
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Day May 21, 2014

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

Paolo Amore, Francisco M. Fernández, Javier Garcia

We discuss space-time symmetric Hamiltonian operators of the form \(H=H_{0}+igH^{\prime}\), 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 \(0<g<g_c\), 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.
Quantum Physics (quant-ph)

Invisibility and PT Symmetry: A Simple Geometrical Viewpoint

Luis L. Sanchez-Soto, Juan J. Monzon

We give a simplified account of the properties of the transfer matrix for a complex one-dimensional potential, paying special attention to the particular instance of unidirectional invisibility. In appropriate variables, invisible potentials appear as performing null rotations, which lead to the helicity-gauge symmetry of massless particles. In hyperbolic geometry, this can be interpreted, via Mobius transformations, as parallel displacements, a geometric action that has no Euclidean analogy.

Quantum Physics (quant-ph)