Anisotropy and asymptotic degeneracy of the physical-Hilbert-space inner-product metrics in an exactly solvable crypto-unitary quantum model

Miloslav Znojil

doi:10.3390/sym16030353

paper

Anisotropy and asymptotic degeneracy of the physical-Hilbert-space inner-product metrics in an exactly solvable crypto-unitary quantum model

arXiv:1212.0734 · doi:10.3390/sym16030353

Abstract

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 $Λ_{j}^{†} Θ = Θ Λ_{j}$ , 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.

28 pp., 1 figure, 4 tables

Anisotropy and asymptotic degeneracy of the physical-Hilbert-space inner-product metrics in an exactly solvable crypto-unitary quantum model

Abstract

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