Anisotropy and asymptotic degeneracy of the physical-Hilbert-space inner-product metrics in an exactly solvable crypto-unitary quantum model
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arXiv
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| Format: | Preprint |
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2012
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| _version_ | 1866911796333379584 |
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| author | Znojil, Miloslav |
| author_facet | Znojil, Miloslav |
| contents | In quantum mechanics (formulated, say, in Schrödinger picture) only the knowledge of a complete set of observables $Λ_j$ enables us to declare the related physical inner product (i.e., the Hilbert-space metric $Θ$ such that $Λ_j^\dagger Θ=Θ\,Λ_j$, i.e., such that $Θ=Θ(Λ_j)$) unique. In many applications people simplify the model and consider just a single input observable (mostly an energy-representing Hamiltonian $Λ_1=H$) and pick up, out of all of the eligible metrics $Θ=Θ(H)$, just the simplest candidate (typically, in the case of the special self-adjoint input $H$ we virtually always work with trivial $Θ=I$). As long as this forces us to admit only the self-adjoint forms of any other input observable $Λ_j$, the scope of the theory is, without any truly meaningful phenomenological reason, restricted. In our present paper we describe a strictly non-numerical $N$ by $N$ matrix model in which such a restriction is replaced by another, phenomenologically non-equivalent restriction in which $Θ\neq I$ and in which the system reaches a collapse (i.e., a loss-of-bservability catastrophe) via unitary evolution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1212_0734 |
| institution | arXiv |
| publishDate | 2012 |
| record_format | arxiv |
| spellingShingle | Anisotropy and asymptotic degeneracy of the physical-Hilbert-space inner-product metrics in an exactly solvable crypto-unitary quantum model Znojil, Miloslav Quantum Physics Mathematical Physics 81Q12 81S99 81R40 81Q80 81P10 81Q10 In quantum mechanics (formulated, say, in Schrödinger picture) only the knowledge of a complete set of observables $Λ_j$ enables us to declare the related physical inner product (i.e., the Hilbert-space metric $Θ$ such that $Λ_j^\dagger Θ=Θ\,Λ_j$, i.e., such that $Θ=Θ(Λ_j)$) unique. In many applications people simplify the model and consider just a single input observable (mostly an energy-representing Hamiltonian $Λ_1=H$) and pick up, out of all of the eligible metrics $Θ=Θ(H)$, just the simplest candidate (typically, in the case of the special self-adjoint input $H$ we virtually always work with trivial $Θ=I$). As long as this forces us to admit only the self-adjoint forms of any other input observable $Λ_j$, the scope of the theory is, without any truly meaningful phenomenological reason, restricted. In our present paper we describe a strictly non-numerical $N$ by $N$ matrix model in which such a restriction is replaced by another, phenomenologically non-equivalent restriction in which $Θ\neq I$ and in which the system reaches a collapse (i.e., a loss-of-bservability catastrophe) via unitary evolution. |
| title | Anisotropy and asymptotic degeneracy of the physical-Hilbert-space inner-product metrics in an exactly solvable crypto-unitary quantum model |
| topic | Quantum Physics Mathematical Physics 81Q12 81S99 81R40 81Q80 81P10 81Q10 |
| url | https://arxiv.org/abs/1212.0734 |