Emergent supersymmetry in a time-space inverted quantum mechanics

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.

• The paper demonstrates that the factorized Momentunian operator naturally yields SUSY structures, establishing a momentum-based spectral organization.
• The study reveals the emergence of fractional time derivatives that introduce non-local memory effects in both non-relativistic and relativistic frameworks.
• The work extends to relativistic quantum mechanics, showing evanescent zero-modes and potential-independent supersymmetric pairing in momentum space.

Emergent Supersymmetry in Time-Space Inverted Quantum Mechanics

Background and Motivation

The persistent asymmetry between space and time in Quantum Mechanics (QM) has posed significant conceptual challenges, particularly regarding the operator status of observables: while position is represented as an operator on a Hilbert space, time remains an external parameter. This disparity complicates physically meaningful definitions concerning temporal observables and uncertainty relations, and precludes the construction of a canonical time operator conjugate to energy. Space-time-symmetric (STS) formalisms, which elevate time to operator status and invert the standard roles of space and time, offer a promising route to address these foundational issues.

Prior work by Dias and Parisio formalized a TSI (time-space inverted) quantum theory, introducing the concept of a "Momentunian" operator generating spatial evolution in a new Hilbert subspace where position acts parametrically. However, structural questions remained regarding the underlying symmetries and organizational principles of this inverted framework.

Main Results and Theoretical Structure

This work demonstrates that supersymmetry (SUSY) is an intrinsic feature of TSI quantum mechanics, specifically that the square-rooted structure of the Momentunian operator allows an automatic factorization into SUSY algebraic elements. The key technical finding is the realization that the Momentunian, defined as

$$\hat{P}_\pm(\hat{H}, \hat{t}; q) = \pm \sqrt{2m[\hat{H} - \hat{V}(q)]},$$

serves as a generator for spatial evolution, with space as a dynamical variable and time promoted to operator status. In this context, the factorization of the Momentunian yields SUSY supercharges and corresponding partner Hamiltonians. Unlike conventional SUSY, where the superpotential is a mathematical auxiliary, here the superpotential is tied directly to the physical potential, reflecting the foundational reversal of roles in the theory.

Non-relativistic Sector

For non-relativistic systems, the TSI formalism leads to a pair of non-Hermitian but structurally supersymmetric partner "Momentunian" operators. A salient consequence is that the spectra are organized in momentum doublets, not the traditional energy doublets, with zero-mode (vacuum) states corresponding to zero-momentum, not merely zero energy. This aligns with QFT-like vacuum definitions and admits well-defined physical interpretations (in particular, configurations that remain static unless perturbed).

Notably, zero-momentum states decouple the time and space evolution equations and, when normalizable, manifest unbroken SUSY—non-normalizable cases correspond to broken SUSY, as in the harmonic oscillator example discussed in the paper. These results carry over to more general potentials due to the direct identification of the superpotential $W(q)$ with $V(q)$.

A marked technical distinction of the TSI formalism is the appearance of 1/2-order (fractional) time derivatives, in the Caputo sense, in the dynamical equations. This naturally introduces non-local temporal memory effects at the level of the supersymmetric wavefunctions, giving an explicit algebraic basis to temporal non-Markovian behavior within this class of quantum theories.

Relativistic Extension

The study extends the TSI supersymmetric structure to relativistic quantum mechanics. Here, the Momentunian is generalized to

$$\hat{P}_\pm(\hat{H}, \hat{t}; q) = \pm \sqrt{[\hat{H} - V(t,q)]^2 - (m_0c^2)^2},$$

mirroring the quadratic mass dependence of relativistic kinematics. The construction leads to a Dirac-type equation with factorized forms corresponding to relativistic SUSY partners.

A crucial finding is that the zero-mode solutions for the relativistic Momentunian correspond to evanescent states—wavefunctions decaying exponentially in space—regardless of the specific physical potential. This is a result of the constraint that, for zero available kinetic energy, the momentum becomes imaginary. Such solutions are structurally analogous to the Jackiw-Rebbi zero-modes for position-dependent mass Dirac systems.

Generalized SUSY Algebra and Partner Construction

The algebraic structure of VT-SUSY (the authors' designation for their supersymmetry in TSI quantum mechanics) is formalized through generalizations of conventional supercharges, which now include fractional time derivatives and are realized via non-Hermitian operators. The resulting SUSY algebra closes as expected, with the spectrum symmetry enforced by the action of explicit parity-like operators.

The construction also highlights that, due to the symmetry $\hat{S}\hat{P}_\pm \hat{S}^{-1} = -\hat{P}_\pm$, the momentum spectrum is symmetric around zero, and isolated zero-momentum states are protected unless paired states with $\pm p$ emerge—an analogy to Dirac node protection under chiral symmetry.

Numerical and Analytical Results

Explicit spectra and wavefunctions are provided for archetypal systems, such as the harmonic oscillator. Ground state and excited solutions reflect the shifted focus from energy quantization to quantized momenta due to the redefinition of the evolution generator. The analysis demonstrates unbroken SUSY for the normalizable zero-momentum ground state, with the Witten index corresponding to the physical sector count.

For the fractional time evolution equation, solutions are given in terms of Mittag-Leffler functions, emblematic of fractional dynamics and distinguishing VT-SUSY from purely local-time quantum evolutions.

Implications and Prospects

The core implication is that supersymmetry, often introduced by hand for analytical convenience or model-building, here emerges as a natural and unavoidable feature due to the algebraic structure of TSI quantum mechanics itself. This insight has several notable ramifications:

• Structural Origin of SUSY: Instead of being a modeled symmetry, SUSY is embedded in the foundational geometry of time-space-inverted quantum theory.
• Momentum-based Spectral Organization: The reorganization of vacuum and excitation structure in terms of momentum, rather than energy, recontextualizes known results from both SUSY QM and QFT, with possible implications for the interpretation of vacuum and excited states in open and dynamically fluctuating systems.
• Fractional Dynamics Integration: The unification of supersymmetry with temporal fractional derivatives provides a direct algebraic source for memory effects, inviting further study of their impact on coherence, decoherence, and non-Markovian processes in extended quantum systems.
• Relativistic Evanescence: The universal emergence of potential-independent evanescent zero-modes in the relativistic sector signals possible connections to protected edge modes and defects in topological condensed matter systems.

Future Directions

The natural extension of this framework entails probing interacting many-body systems, the interplay of TSI SUSY with environmental decoherence, and exploring the physical realization of momentum doublets in engineered quantum platforms. The generalized non-Hermitian and fractional supercharges invite speculation on the classification of new dynamical phases in open quantum systems, and motivate application to temporally disordered and nonlocal-in-time quantum field theories.

One particularly intriguing avenue is the realization of observable signatures of TSI SUSY, for example via the manipulation of time-space symmetry in quantum optical, cold atom, or solid-state systems where effective Hamiltonians and time-operator protocols can be engineered.

Conclusion

This paper provides a rigorous theoretical advancement by establishing that supersymmetry is a structural, rather than phenomenological, feature of time-space-inverted quantum mechanics. Through careful algebraic factorization of the Momentunian operator, the work reveals VT-SUSY with both non-relativistic and relativistic sectors, underpinned by fractional temporal evolution and non-Hermitian supercharges. The theoretical results have meaningful implications for foundational quantum theory, algebraic classifications of quantum states, and the analysis of memory effects and topological protections, suggesting several productive directions for continuing investigation and application.