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Proceedings of the National Academy of Sciences

Hamiltonian Systems and Transformation in Hilbert Space

Bernard Osgood Koopman

Published May 1, 1931• 2,389 citations
Publisher
DOI
10.1073/pnas.17.5.315

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The non-triviality of Hilbert space geometries in non-Hermitian quantum systems sometimes blurs the underlying physics. We present a systematic study of the vielbein formalism which transforms the Hilbert spaces 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, described by a non-trivial Hilbert space, reduces to a Hermitian system within the standard formalism of quantum mechanics for a Hilbert space.
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.
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