Three-Hilbert-Space Formulation of Quantum Mechanics

• Published in: Symmetry, integrability and geometry, methods and applications integrability and geometry, methods and applications, 2009-01, Vol.5, p.001
• Main Author: Znojil, Miloslav
• Format: Article
• Language: English
• Subjects: cryptohermitian operators of observables; formulation of Quantum Mechanics; the covariant picture of time evolution; triplet of the representations of the Hilbert space of states
• ISSN: 1815-0659; 1815-0659
• DOI: 10.3842/SIGMA.2009.001

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.