On the Role of the Normalization Factors $kappa_n$ and of the Pseudo-Metric ${cal P} neq {cal P}^dag$ in Crypto-Hermitian Quantum Models
Among ${cal P}$-pseudo-Hermitian Hamiltonians $H ={cal P}^{-1} H^dagger {cal P}$ with real spectra, the ''weaklypseudo-Hermitian" ones (i.e., those employing non-self-adjoint ${cal P} neq {cal P}^dagger$) form a remarkable subfamily. Welist some reasons why it deserves a special atte...
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| Format: | Article |
| Language: | English |
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National Academy of Science of Ukraine
2008-01-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Online Access: | http://dx.doi.org/10.3842/SIGMA.2008.001 |
| _version_ | 1818503924422803456 |
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| author | Miloslav Znojil |
| author_facet | Miloslav Znojil |
| author_sort | Miloslav Znojil |
| collection | DOAJ |
| description | Among ${cal P}$-pseudo-Hermitian Hamiltonians $H ={cal P}^{-1} H^dagger {cal P}$ with real spectra, the ''weaklypseudo-Hermitian" ones (i.e., those employing non-self-adjoint ${cal P} neq {cal P}^dagger$) form a remarkable subfamily. Welist some reasons why it deserves a special attention. Inparticular we show that whenever ${cal P} neq {cal P}^dagger$,the current involutive operator of charge ${cal C}$ getscomplemented by a nonequivalent alternative involutive quasiparityoperator ${cal Q}$. We show how, in this language, the standardquantum mechanics is restored via the two alternative innerproducts in the physical Hilbert space of states, with $langle psi_1 |{cal PQ}| psi_2 rangle = langle psi_1 | {cal CP} | psi_2 rangle $. |
| first_indexed | 2024-12-10T21:30:18Z |
| format | Article |
| id | doaj.art-dd429fb487e6464fb230716e0b26d2f8 |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-12-10T21:30:18Z |
| publishDate | 2008-01-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-dd429fb487e6464fb230716e0b26d2f82022-12-22T01:32:50ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592008-01-014001On the Role of the Normalization Factors $kappa_n$ and of the Pseudo-Metric ${cal P} neq {cal P}^dag$ in Crypto-Hermitian Quantum ModelsMiloslav ZnojilAmong ${cal P}$-pseudo-Hermitian Hamiltonians $H ={cal P}^{-1} H^dagger {cal P}$ with real spectra, the ''weaklypseudo-Hermitian" ones (i.e., those employing non-self-adjoint ${cal P} neq {cal P}^dagger$) form a remarkable subfamily. Welist some reasons why it deserves a special attention. Inparticular we show that whenever ${cal P} neq {cal P}^dagger$,the current involutive operator of charge ${cal C}$ getscomplemented by a nonequivalent alternative involutive quasiparityoperator ${cal Q}$. We show how, in this language, the standardquantum mechanics is restored via the two alternative innerproducts in the physical Hilbert space of states, with $langle psi_1 |{cal PQ}| psi_2 rangle = langle psi_1 | {cal CP} | psi_2 rangle $.http://dx.doi.org/10.3842/SIGMA.2008.001${cal PT}$-symmetrynon-self-adjoint pseudo-metrics${cal PQ}$-crypto-Hermiticity${cal CP}$-crypto-Hermiticity |
| spellingShingle | Miloslav Znojil On the Role of the Normalization Factors $kappa_n$ and of the Pseudo-Metric ${cal P} neq {cal P}^dag$ in Crypto-Hermitian Quantum Models Symmetry, Integrability and Geometry: Methods and Applications ${cal PT}$-symmetry non-self-adjoint pseudo-metrics ${cal PQ}$-crypto-Hermiticity ${cal CP}$-crypto-Hermiticity |
| title | On the Role of the Normalization Factors $kappa_n$ and of the Pseudo-Metric ${cal P} neq {cal P}^dag$ in Crypto-Hermitian Quantum Models |
| title_full | On the Role of the Normalization Factors $kappa_n$ and of the Pseudo-Metric ${cal P} neq {cal P}^dag$ in Crypto-Hermitian Quantum Models |
| title_fullStr | On the Role of the Normalization Factors $kappa_n$ and of the Pseudo-Metric ${cal P} neq {cal P}^dag$ in Crypto-Hermitian Quantum Models |
| title_full_unstemmed | On the Role of the Normalization Factors $kappa_n$ and of the Pseudo-Metric ${cal P} neq {cal P}^dag$ in Crypto-Hermitian Quantum Models |
| title_short | On the Role of the Normalization Factors $kappa_n$ and of the Pseudo-Metric ${cal P} neq {cal P}^dag$ in Crypto-Hermitian Quantum Models |
| title_sort | on the role of the normalization factors kappa n and of the pseudo metric cal p neq cal p dag in crypto hermitian quantum models |
| topic | ${cal PT}$-symmetry non-self-adjoint pseudo-metrics ${cal PQ}$-crypto-Hermiticity ${cal CP}$-crypto-Hermiticity |
| url | http://dx.doi.org/10.3842/SIGMA.2008.001 |
| work_keys_str_mv | AT miloslavznojil ontheroleofthenormalizationfactorskappanandofthepseudometriccalpneqcalpdagincryptohermitianquantummodels |