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

Abstract

In quantum mechanics (formulated, say, in Schr\"{o}dinger picture) only the knowledge of a complete set of observables $\Lambda_j$ enables us to declare the related physical inner product (i.e., the Hilbert-space metric $\Theta$ such that $\Lambda_j^\dagger \Theta=\Theta\,\Lambda_j$, i.e., such that $\Theta=\Theta(\Lambda_j)$) unique. In many applications people simplify the model and consider just a single input observable (mostly an energy-representing Hamiltonian $\Lambda_1=H$) and pick up, out of all of the eligible metrics $\Theta=\Theta(H)$, just the simplest candidate (typically, in the case of the special self-adjoint input $H$ we virtually always work with trivial $\Theta=I$). As long as this forces us to admit only the self-adjoint forms of any other input observable $\Lambda_j$, 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 $\Theta \neq I$ and in which the system reaches a collapse (i.e., a loss-of-bservability catastrophe) via unitary evolution.