Oscillation probabilities for a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mi>PT</mml:mi> </mml:mrow> </mml:mrow> </mml:math> -symmetric non-Hermitian two-state system

Abstract There is growing interest in viable quantum theories with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">PT</mml:mi> </mml:mrow> </mml:mrow> </mml:math> -symmetric non-Hermitian Hamiltonians, but a formulation of transition matrix elements consistent with positivity and perturbative unitarity has so far proved elusive. This Letter provides such a formulation, which relies crucially on the ability to span the state space in such a way that the interaction and energy eigenstates are orthonormal with respect to the same positive-definite inner product. We apply this non-Hermitian approach to two-neutrino flavour oscillations, and show how it can accommodate the seesaw mechanism.