Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 001, 19 pages      arXiv:0901.0700      https://doi.org/10.3842/SIGMA.2009.001
Contribution to the Proceedings of the VIIth Workshop ''Quantum Physics with Non-Hermitian Operators''

Three-Hilbert-Space Formulation of Quantum Mechanics

Miloslav Znojil
Nuclear Physics Institute ASCR, 250 68 Rez, Czech Republic

Received October 29, 2008, in final form December 31, 2008; Published online January 06, 2009

Abstract
In paper [Znojil M., Phys. Rev. D 78 (2008), 085003, 5 pages, arXiv:0809.2874] the two-Hilbert-space (2HS, a.k.a. cryptohermitian) formulation of Quantum Mechanics has been revisited. In the present continuation of this study (with the spaces in question denoted as H(auxiliary) and H(standard)) we spot a weak point of the 2HS formalism which lies in the double role played by H(auxiliary). As long as this confluence of roles may (and did!) lead to confusion in the literature, we propose an amended, three-Hilbert-space (3HS) reformulation of the same theory. As a byproduct of our analysis of the formalism we offer an amendment of the Dirac's bra-ket notation and we also show how its use clarifies the concept of covariance in time-dependent cases. Via an elementary example we finally explain why in certain quantum systems the generator H(gen) of the time-evolution of the wave functions may differ from their Hamiltonian H.

Key words: formulation of Quantum Mechanics; cryptohermitian operators of observables; triplet of the representations of the Hilbert space of states; the covariant picture of time evolution.

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