Einstein's quantum elevator: Hermitization of non-Hermitian Hamiltonians via a generalized vielbein formalism

The formalism for non-Hermitian quantum systems sometimes blurs the underlying physics. We present a systematic study of the vielbeinlike formalism which transforms the Hilbert space bundles of non-Hermitian systems into the conventional ones, rendering the induced Hamiltonian to be Hermitian. In other words, any non-Hermitian Hamiltonian can be “transformed” into a Hermitian one without altering the physics. Thus we show how to find a reference frame (corresponding to Einstein's quantum elevator) in which a non-Hermitian system, equipped with a nontrivial Hilbert space metric, reduces to a Hermitian system within the standard formalism of quantum mechanics.

Figure 1

An illustration of “flattening” the coordinates via the vielbein formalism. This procedure flattens the space (or curve in one dimension). (Note that Regge calculus also flattens the curved manifold into flat space with deficit angles, which measure the local curvature [19].)

Figure 2

Fictitious force induced from the coordinate change. This is a classical analog of different gauge choices inducing different Hamiltonians. As usual, there is no best gauge choice for all physical systems.