Invisibility and PT Symmetry: A Simple Geometrical Viewpoint

TL;DR

This paper provides a geometric perspective on the transfer matrix in complex 1D potentials, linking invisibility and PT symmetry to null rotations and hyperbolic geometry, revealing new insights into unidirectional invisibility.

Contribution

It introduces a simplified geometric framework for understanding transfer matrices, unidirectional invisibility, and PT symmetry using hyperbolic geometry and Möbius transformations.

Findings

Invisibility corresponds to null rotations in the transfer matrix.

Unidirectional invisibility relates to helicity-gauge symmetry of massless particles.

Hyperbolic geometry offers a novel interpretation of optical phenomena.

Abstract

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 M\"{o}bius transformations, as parallel displacements, a geometric action that has no Euclidean analogy.