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arXiv:2605.17507

Emergent supersymmetry in a time-space inverted quantum mechanics

Marcus W Beims, Arlans JS de Lara

quant-phhep-thSubmitted May 17, 2026• Announced May 19, 2026• 0 citations
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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.

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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 P^±(H^,t^;q)=±2m[H^−V^(q)]\hat{\mathcal{P}}^{\pm}(\hat{\mathcal{H}},\hat t;q)=\pm\sqrt{2m[\hat{\mathcal{H}}-\mathcal{\hat V}(q)]}P^±(H^,t^;q)=±2m[H^−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 1/21/21/2-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.
The concept of supersymmetry developed in particle physics has been applied to various fields of modern physics. In quantum mechanics, the supersymmetric systems refer to the systems involving two supersymmetric partner Hamiltonians, whose energy levels are degeneracy except one of the systems has an extra ground state possibly, and the eigenstates of the partner systems can be mapped onto each other. Recently, an interferometric scheme has been proposed to show this relationship in ultracold atoms [Phys. Rev. A 96 043624 (2017)]. Here this approach is generalized to linear optics for observing the supersymmetric dynamics with photons. The time evolution operator is simulated approximately via Suzuki–Trotter expansion with considering the realization of the kinetic and potential terms separately. The former is realized through the diffraction nature of light and the later is implemented using a phase plate. Additionally, we propose an interferometric approach which can be implemented perfectly using an amplitude alternator to realize the non-unitary operator. The numerical results show that our scheme is universal and can be realized with current technologies.
Supersymmetry is a symmetry between a boson and a fermion. Although there is no apparent supersymmetry in nature, its mathematical consistency and appealing properties have led many people to believe that supersymmetry may exist in nature in the form of a spontaneously broken symmetry. In this paper, we explore an alternative possibility by which supersymmetry is realized in nature, that is, supersymmetry dynamically emerges in the low-energy limit of a nonsupersymmetric condensed matter system. We propose a (2+1)(2+1)(2+1)-dimensional lattice model which exhibits an emergent space-time supersymmetry at a quantum critical point. It is shown that there is only one relevant perturbation at the supersymmetric critical point in the ϵϵϵ expansion and the critical theory is the two copies of the Wess-Zumino theory with four supercharges. Exact critical exponents are predicted.
We review supersymmetry (SUSY) in nonrelativistic quantum mechanics emphasizing algebraic aspects.We discuss the Hamiltonian subgroup implementing supersymmetry as well as the corresponding algebra of the SUSY generators.In the SUSYQM framework, the two distinct partner potentials are connected by a superpotential satisfying a Riccati differential equation.A full Hamiltonian operator in an extended Hilbert space is defined in order to render the supersymmetry manifest.As a result, the eigenfunctions of the original potentials are connected by generalized ladder operators.We provide an explicit realization of the abstract supersymmetry group for SUSYQM depending on one real and two Grassmann parameters.
This undergraduate thesis contains one dimensional introductory treatment of Supersymmetric Quantum Mechanics (SUSY QM). The factorisation method is used to obtain Partner Hamiltonians, and the notions of partner potentials and superpotential are introduced. It demonstrated how energy spectra and eigenfunctions of partner potentials can be correlated for the cases of unbroken and broken supersymmetry. Generalisations of these cases attributed to Sets of Hamiltonians and shape invariant potentials are introduced. The devel oped formalisation is used to obtain energy eigenfunctions and spectra of partner potentials of some well-known quantum systems, namely the Infinite Square Well and the Free Particle. In these systems, the partner potentials are trigonometric and hyperbolic functions, respectively. A comparison of these two physically distinct but mathematically related partner potentials is provided. A new philosophical interpretation of SUSY QM is introduced, with an ontological discussion of supersymmetric notions and SUSY QM. The current observation problem of SUSY is discussed, and on that basis, two opposing arguments are presented. The method of Falsification by Karl Popper is briefly introduced, and an argument is produced regarding its application to SUSY.
Understanding whether the mathematical structure of quantum mechanics is fundamental or emergent remains a central question in the foundations of physics. In particular, the special role played by time in quantum theory, appearing as an external evolution parameter rather than a dynamical observable, suggests that the formalism itself may arise from deeper structural considerations. In this work, we investigate the emergence of quantum mechanical formalism from classical wave dynamics by adopting a dimensional framework in which time is treated as a +1 evolution parameter relative to the dimensions through which physical phenomena (fields or disturbances of a field) propagate and interact. Within this perspective, different fields may evolve with respect to different effective dimensions, while remaining embedded in a common higher-dimensional space, allowing time to acquire a relational and context-dependent role. This means that in our proposed model, time is not a fixed dimension which is experienced the same way for every field or field interaction of any dimensionality. In that sense, time for one physical phenomenon can behave as space for a higher dimensional physical phenomenon, whose time is a different +1 dimension. The central objective of this paper is to determine how a higher-dimensional deterministic field can be consistently represented by a lower-dimensional description that lacks direct access to its full set of evolution parameters and evolves through a spatial (for the higher-dimensional field) dimension. To this end, we introduce a general projection framework in which a higher-dimensional field is mapped to a reduced field through an interaction-based recording process. Crucially, we do not assume the form of this mapping a priori. Instead, we impose the requirement that it preserve the maximum amount of physically accessible information. In particular, we demand the faithful encoding of phase relations, interference structure, and spectral composition, including the relative contributions of different Fourier modes and their superposition. We first demonstrate, within a purely classical 3+1-dimensional wave framework that these constraints severely restrict the admissible form of the reduced description and naturally lead to complex amplitudes, linear superposition, Hilbert space structure, and canonical operator relations. This analysis provides an intuitive and mathematically explicit route to quantum-like descriptions without assuming quantum postulates. We then generalize the construction to a 4+1-dimensional framework, introducing an additional evolution parameter and showing that under the same information-preserving constraints, the Schrödinger equation appears as an effective low-energy description of the reduced dynamics, while a relativistic dispersion relation emerges simultaneously through the encoding of the hidden evolution parameter as an invariant frequency scale. In this way, both quantum mechanical and relativistic structures arise from the same underlying requirement: the consistent and information-preserving representation of higher-dimensional wave propagation in a lower-dimensional observational framework. The results suggest that the formal structure of quantum mechanics need not be postulated a priori, but may instead be understood as the unique mathematical language required to encode the observable remnant of a higher-dimensional deterministic dynamics under strict constraints of symmetry, invariance, and information preservation.