Emergent supersymmetry in a time-space inverted quantum mechanics

TL;DR

This paper demonstrates that supersymmetry naturally arises in the time-space inverted quantum mechanics framework, revealing inherent algebraic structures and zero-mode state properties without phenomenological imposition.

Contribution

It uncovers the inherent supersymmetric structure in TSI quantum mechanics and explores properties of relativistic and non-relativistic partner Hamiltonians.

Findings

Supersymmetry is inherent to TSI quantum mechanics.

Zero-mode states in relativistic Momentunian are evanescent and independent of potential.

Partner Hamiltonians have zero-mode states with vanishing momenta.

Abstract

This Letter shows that a supersymmetric structure is inherent to the time space inverted (TSI) quantum mechanics (QM) framework, where the spatial evolution of states is generated by the operator $\hat{\mathcal{P}}^{\pm}(\hat{\mathcal{H}},\hat t;q)=\pm\sqrt{2m[\hat{\mathcal{H}}-\mathcal{\hat V}(q)]}$ [\href{https://doi.org/10.1103/PhysRevA.95.032133}{Phys. Rev. A. {\bf 95}, 032133 (2017)}], named here Momentunian, whose square-root structure that can be factorized. Such factorization leads directly to a supersymmetric algebra with supercharges and partner Hamiltonians. For the relativistic Momentunian the zero mode states are shown to be evanescent states, \textit{independent} of the physical potential. Furthermore, the existence of non-relativistic and relativistic Momentunian \textit{partners} is demonstrated, whose zero-mode states are no longer necessarily zero energies, but vanishing momenta states. The natural emergence of the $1/2$-fractional time derivatives in the TSI QM, leads to supercharges which incorporate memory effects into the supersymmetric wave functions. Results indicate that supersymmetry emerges as a structural property of the TSI QM rather than being imposed phenomenologically.