Isomorphic Hilbert spaces associated with different Complex Contours of the $\mathcal{PT}$-Symmetric $(-x^{4}) $ Theory

TL;DR

This paper demonstrates that different complex contours in the $ ext{PT}$-symmetric $-x^4$ theory are connected through isomorphisms, with the metric operator depending on the contour but the Hilbert spaces remaining isomorphic, ensuring consistent physical predictions.

Contribution

It introduces a parametrization of complex contours and shows the existence of isomorphic Hilbert spaces with contour-dependent metrics in $ ext{PT}$-symmetric quantum mechanics.

Findings

Different contours yield the same transition amplitudes.

The metric operator depends on contour parameters.

Hilbert spaces are isomorphic despite different contours.

Abstract

In this work, we stress the existence of isomorphisms which map complex contours from the upper half to contours in the lower half of the complex plane. The metric operator is found to depend on the chosen contour but the maps connecting different contours are norm-preserving. To elucidate these features, we parametrized the contour $z=-2i\sqrt{1+ix}$ considered in Phys.Rev.D73:085002 (2006) for the study of wrong sign $x^{4}$ theory. For the parametrized contour of the form $z=a\sqrt{b+i c x}$, we found that there exists an equivalent Hermitian Hamiltonian provided that $a^{2} c$ is taken to be real. The equivalent Hamiltonian is $b$-independent but the metric operator is found to depend on all the parameters $a$, $b$ and $c$. Different values of these parameters generate different metric operators which define different Hilbert spaces . All these Hilbert spaces are isomorphic to each other even for parameters values that define contours with ends in two adjacent wedges. As an example, we showed that the transition amplitudes associated with the contour $z=-2i\sqrt{1+ix}$ are exactly the same as those calculated using the contour $z=\sqrt{1+ix}$, which is not $\mathcal{PT}$-Symmetric and has ends in two adjacent wedges in the complex plane.