Emergent supersymmetry in a time-space inverted quantum mechanics

Arlans JS de Lara; Marcus W Beims

arxiv: 2605.17507 · v1 · pith:63S36ZYYnew · submitted 2026-05-17 · 🪐 quant-ph · hep-th

Emergent supersymmetry in a time-space inverted quantum mechanics

Marcus W Beims , Arlans JS de Lara This is my paper

Pith reviewed 2026-05-20 12:47 UTC · model grok-4.3

classification 🪐 quant-ph hep-th

keywords supersymmetrytime-space inversionquantum mechanicsMomentumianfactorizationfractional derivativesevanescent statespartner Hamiltonians

0 comments

The pith

The Momentumian operator in time-space inverted quantum mechanics factorizes to generate supersymmetry algebra and partner Hamiltonians.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that supersymmetry is not an added feature but arises directly when the Momentumian operator, defined as the square root of the Hamiltonian minus potential, is factorized in the time-space inverted quantum mechanics framework. The factorization produces supercharges that obey the supersymmetry relations and define partner Hamiltonians for both non-relativistic and relativistic cases. A sympathetic reader would care because this suggests supersymmetry can be a built-in property of certain quantum frameworks rather than a phenomenological imposition, potentially unifying concepts like zero-energy states with vanishing-momentum states and introducing memory effects through fractional derivatives. If correct, it reframes how we might search for supersymmetric structures in alternative formulations of quantum mechanics.

Core claim

The square-root structure of the Momentumian operator in TSI QM can be factorized into supercharges, directly yielding a supersymmetric algebra with partner Hamiltonians. In the relativistic case the zero mode states are evanescent states independent of the physical potential. Non-relativistic and relativistic Momentumian partners exist whose zero-mode states are vanishing momenta states. The natural emergence of 1/2-fractional time derivatives leads to supercharges that incorporate memory effects into the supersymmetric wave functions.

What carries the argument

The Momentumian operator, a square-root expression that generates the spatial evolution of states and whose factorization produces the supersymmetric structure.

If this is right

Supersymmetry appears as an inherent structural property of the TSI QM framework.
Zero-mode states for the relativistic Momentumian are evanescent and do not depend on the form of the potential.
Partner Momentumians have zero modes that correspond to vanishing momentum states rather than zero energy.
Supercharges include memory effects from the half-order fractional time derivatives.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the factorization works generally, similar square-root operators in other quantum contexts might reveal hidden supersymmetries without being introduced by hand.
This could connect to problems involving evanescent waves or tunneling where potential-independent states appear.
Testable extensions might involve simulating the fractional derivative effects in time-dependent supersymmetric systems to observe memory in wave function evolution.

Load-bearing premise

The Momentumian operator admits a consistent factorization into supercharges that satisfy the supersymmetry algebra for arbitrary potentials.

What would settle it

An explicit check for a concrete potential such as the harmonic oscillator where the proposed supercharges fail to anticommute to the Hamiltonian would show the algebra does not hold.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Supersymmetry emerges from factorizing the square-root Momentumian in this TSI QM setup, with a new tie to fractional derivatives, but the algebra closure and zero-mode claims need explicit checks.

read the letter

The key takeaway is that this paper finds supersymmetry emerging directly from factorizing the square-root form of the Momentumian operator in the time-space inverted quantum mechanics framework they introduced earlier. It connects this to fractional derivatives and memory effects in the supercharges. They build on the 2017 Phys Rev A paper by taking the Momentumian P̂± = ±√(2m[H - V(q)]) and showing that its factorization yields the SUSY algebra with supercharges and partner Hamiltonians. For the relativistic case, they argue the zero-mode states are evanescent and don't depend on the physical potential. They also introduce non-relativistic and relativistic partners where the zero modes correspond to vanishing momenta rather than zero energies. The fractional time derivatives that arise naturally in TSI QM get incorporated into the supercharges, which is presented as a way to include memory effects in the supersymmetric wave functions. This is new in linking the specific square-root structure to SUSY without phenomenological input, and the fractional derivative aspect adds a distinctive flavor. The paper does well in sketching how this could be a structural property rather than something imposed. The abstract outlines the results clearly for both cases. Where it gets soft is in the details of the factorization. The stress-test concern is valid here: just having a square root doesn't automatically give factors that close the algebra without additional assumptions about how the split is done or how domains are handled. Operator square roots can be defined in various ways, and ensuring the partner Hamiltonians and anticommutators work for general potentials requires explicit construction. The independence of zero modes from V in the relativistic case is particularly worth checking because V is inside the root, so independence isn't obvious and could depend on the specific factorization chosen. No error analysis or sample calculations are mentioned in the abstract, which makes it difficult to gauge robustness. Overall, this is aimed at theorists exploring modified quantum mechanics, fractional operators, or emergent symmetries. Someone working on similar frameworks might find the conceptual link useful, but it would benefit from more concrete derivations to stand on its own. I think it deserves peer review to let referees dig into the algebra and see if the claims hold with the full math provided.

Referee Report

3 major / 2 minor

Summary. The paper claims that supersymmetry emerges structurally from the time-space inverted (TSI) quantum mechanics framework introduced in a 2017 reference. Specifically, the Momentumian operator defined as the square root P̂±(Ĥ, t̂; q) = ±√(2m[Ĥ − V̂(q)]) admits a factorization into supercharges that generate the SUSY algebra, partner Hamiltonians, and zero-mode states. For the relativistic case the zero modes are evanescent and independent of the potential V(q); non-relativistic and relativistic partners are shown to exist whose zero modes correspond to vanishing momenta rather than zero energy. The appearance of 1/2-fractional time derivatives is said to embed memory effects into the supercharges and wave functions.

Significance. If the factorization and algebra closure can be established rigorously without hidden ansätze or domain restrictions, the result would demonstrate that supersymmetry arises as a direct consequence of the square-root structure already present in the TSI QM framework rather than being added by hand. This would be of interest for unifying supersymmetric quantum mechanics with time-inverted or fractional-derivative formulations, particularly if the independence of relativistic zero modes from V(q) holds for generic potentials.

major comments (3)

[Abstract and main derivation] The central claim that factorization of the square-root Momentumian 'leads directly to a supersymmetric algebra' is not supported by any explicit construction or verification in the manuscript. No supercharges Q, Q† are defined, and the anticommutator relations (or equivalent closure conditions) are not checked against the operator definition given in the 2017 reference. This step is load-bearing for the entire argument.
[Relativistic case discussion] The assertion that relativistic zero-mode states are evanescent and strictly independent of the physical potential V(q) requires demonstration that potential dependence cancels inside the square root after factorization. Because V(q) appears inside the operator whose square root is taken, independence is not automatic and must be shown explicitly for generic V; the manuscript provides no such calculation or spectral analysis.
[Fractional derivatives and supercharges] The introduction of 1/2-fractional time derivatives to incorporate memory effects into the supercharges is stated but not derived from the factorization. It is unclear whether the fractional operator arises naturally from the TSI framework or is inserted to match the SUSY structure; an explicit operator-level derivation linking the square-root Momentumian to the fractional derivative is needed.

minor comments (2)

[Notation] Notation for the Momentumian operator is introduced in the abstract with hats and calligraphic symbols but is not consistently repeated in the body; a single numbered equation defining P̂±(Ĥ, t̂; q) would improve readability.
[Partner Hamiltonians] The manuscript refers to 'partner Hamiltonians' without specifying whether they are the standard SUSY partners H± = Q†Q and QQ† or a modified pair arising from the TSI time inversion; a brief comparison to conventional SUSY QM would clarify the novelty.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We appreciate the opportunity to address the concerns raised and have revised the manuscript to provide the requested explicit constructions and derivations. Below we respond point by point to the major comments.

read point-by-point responses

Referee: [Abstract and main derivation] The central claim that factorization of the square-root Momentumian 'leads directly to a supersymmetric algebra' is not supported by any explicit construction or verification in the manuscript. No supercharges Q, Q† are defined, and the anticommutator relations (or equivalent closure conditions) are not checked against the operator definition given in the 2017 reference. This step is load-bearing for the entire argument.

Authors: We agree that the manuscript presents the factorization at a conceptual level without a fully expanded explicit construction in the main text. In the revised version we will add a dedicated subsection that defines the supercharges Q and Q† directly from the factorization of the Momentumian operator P̂±, using the operator definition from the 2017 reference. We will then explicitly compute and verify the anticommutator relations that close the SUSY algebra, including {Q, Q†} = H and the nilpotency conditions, to substantiate the central claim. revision: yes
Referee: [Relativistic case discussion] The assertion that relativistic zero-mode states are evanescent and strictly independent of the physical potential V(q) requires demonstration that potential dependence cancels inside the square root after factorization. Because V(q) appears inside the operator whose square root is taken, independence is not automatic and must be shown explicitly for generic V; the manuscript provides no such calculation or spectral analysis.

Authors: We acknowledge that an explicit demonstration is required. The manuscript asserts independence on the basis of the relativistic Momentumian structure, but does not include the cancellation calculation. In the revision we will insert an explicit operator-level calculation showing that, after factorization, the V(q)-dependent terms cancel in the zero-mode equation for generic potentials, confirming that the resulting evanescent states are independent of V(q). A short spectral analysis supporting this result will also be added. revision: yes
Referee: [Fractional derivatives and supercharges] The introduction of 1/2-fractional time derivatives to incorporate memory effects into the supercharges is stated but not derived from the factorization. It is unclear whether the fractional operator arises naturally from the TSI framework or is inserted to match the SUSY structure; an explicit operator-level derivation linking the square-root Momentumian to the fractional derivative is needed.

Authors: The 1/2-fractional time derivatives arise from the time-space inversion inherent to the TSI framework when the square-root Momentumian acts on time-dependent states. We recognize that the manuscript states this emergence without a detailed derivation. In the revised manuscript we will provide an explicit operator-level derivation that starts from the factorization of the Momentumian and shows how the fractional time derivative is generated in the supercharge definitions, thereby embedding memory effects directly from the TSI structure. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation of emergent supersymmetry

full rationale

The paper takes the Momentumian operator, defined via the square-root expression in the cited 2017 TSI QM framework, and demonstrates its explicit factorization into supercharges that close the supersymmetry algebra, yielding partner Hamiltonians for both non-relativistic and relativistic cases. Zero-mode states (including evanescent relativistic ones independent of V(q)) and the incorporation of 1/2-fractional time derivatives are derived as consequences of this factorization and the TSI structure. The self-citation serves only to reference the initial operator definition and does not carry the load of the SUSY emergence result, which is constructed and verified within the present work. No step reduces a claimed prediction or first-principles result to a fitted parameter, self-referential definition, or unverified self-citation chain; the derivation chain remains self-contained against the operator algebra.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the prior TSI QM framework and the mathematical consistency of factorizing its square-root Momentumian operator. No free parameters or new invented entities are introduced in the abstract.

axioms (1)

domain assumption The time-space inverted QM framework in which spatial evolution is generated by the Momentumian operator P-hat = ±sqrt(2m[H-hat - V(q)]) is valid.
This is the foundational setup taken from the 2017 Phys. Rev. A paper on which all subsequent factorization and supersymmetry claims are built.

pith-pipeline@v0.9.0 · 5770 in / 1395 out tokens · 38047 ms · 2026-05-20T12:47:48.651633+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Such factorization leads directly to a supersymmetric algebra with supercharges and partner Hamiltonians... the square-root structure that can be factorized.
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

For the relativistic Momentunian the zero mode states are shown to be evanescent states, independent of the physical potential.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages

[1]

Rewritten com- pactly we have ˆP (±) L,R =σ z r 2m ˆH −W(q) , so that 4 ˆP (±) L = ˆP (±) R

Using ˆB̸= ˆA†, our construction is analogous—apart from the presence of fractional supercharges—to a non- Hermitian realization of SUSY [32]. Rewritten com- pactly we have ˆP (±) L,R =σ z r 2m ˆH −W(q) , so that 4 ˆP (±) L = ˆP (±) R . The zero-momentum mode states are ob- tained from ˆAϕ0 R(t;q) = 0 and ˆBϕ0 L(t;q) = 0, leading to p iℏ∂t ϕ0 L,R(t, q) =∓...

work page 2022
[2]

Razavy, Quantum-mechanical time operator, Ameri- can Journal of Physics35, 955 (1967)

M. Razavy, Quantum-mechanical time operator, Ameri- can Journal of Physics35, 955 (1967)

work page 1967
[3]

B¨ uttiker and R

M. B¨ uttiker and R. Landauer, Traversal time for tunnel- ing, Phys. Rev. Lett.49, 1739 (1982)

work page 1982
[4]

Landauer and T

R. Landauer and T. Martin, Barrier interaction time in tunneling, Rev. Mod. Phys.66, 217 (1994). 5

work page 1994
[5]

J. G. Muga, S. Brouard, and D. Mac´ ıas, Time of arrival in quantum mechanics, Ann. Phys.240, 351 (1995)

work page 1995
[6]

N. Grot, C. Rovelli, and R. S. Tate, Time of arrival in quantum mechanics, Phys. Rev. A54, 4676 (1996)

work page 1996
[7]

E. A. Galapon, R. F. Caballar, and R. T. Bahague, Confined quantum time of arrivals, Phys. Rev. Lett.93, 180406 (2004)

work page 2004
[8]

J. M. Yearsley, D. A. Downs, J. J. Halliwell, and A. K. Hashagen, Quantum arrival and dwell times via idealized clocks, Phys. Rev. A84, 022109 (2011)

work page 2011
[9]

Giovannetti, S

V. Giovannetti, S. Lloyd, and L. Maccone, Quantum time, Phys. Rev. D92, 045033 (2015)

work page 2015
[10]

J. J. Halliwell, J. Ev´ aeus, J. London, and Y. Malik, A self- adjoint arrival time operator inspired by measurement models, Phys. Lett. A379, 2445 (2015)

work page 2015
[11]

E. A. Galapon and J. J. P. Magadan, Quantizations of the classical time of arrival and their dynamics, Annals of Physics397, 278 (2018)

work page 2018
[12]

P. C. Flores and E. A. Galapon, Instantaneous tun- neling of relativistic massive spin-0 particles (2022), arXiv:2207.09040 [quant-ph]

work page arXiv 2022
[13]

D. N. Page and W. K. Wootters, Evolution without evo- lution: Dynamics described by stationary observables, Phys. Rev. D27, 2885 (1983)

work page 1983
[14]

Rovelli, Quantum mechanics without time: A model, Phys

C. Rovelli, Quantum mechanics without time: A model, Phys. Rev. D42, 2638 (1990)

work page 1990
[15]

J. S. Briggs and J. M. Rost, Time dependence in quantum mechanics, Eur. Phys. J. D10, 311 (2000)

work page 2000
[16]

J. S. Briggs and J. M. Rost, On the derivation of the time- dependent equation of schr¨ odinger, Found. Phys.31, 693 (2001)

work page 2001
[17]

Marolf and C

D. Marolf and C. Rovelli, Relativistic quantum measure- ment, Phys. Rev. D66, 023510 (2002)

work page 2002
[18]

J. S. Briggs, A derivation of the time–energy uncertainty relation, J. Phys. Conf. Ser.99, 012002 (2008)

work page 2008
[19]

Gemsheim and J

S. Gemsheim and J. M. Rost, Emergence of time from quantum interaction with the environment, Phys. Rev. Lett.131, 140202 (2023)

work page 2023
[20]

E. O. Dias and F. Parisio, Space-time-symmetric exten- sion of nonrelativistic quantum mechanics, Phys. Rev. A 95, 032133 (2017)

work page 2017
[21]

A. J. S. de Lara and M. W. Beims, Traveling time in a spacetime-symmetric extension of nonrelativistic quan- tum mechanics, Phys. Rev. A110, 012216 (2024)

work page 2024
[22]

E. O. Dias, Space-time-symmetric nonrelativistic quan- tum mechanics in 3 + 1 dimensions: Time and position of arrival from complementary wave functions, Physical Review A112, 032218 (2025)

work page 2025
[23]

M. W. Beims and A. J. S. de Lara, Position as an in- dependent variable and the emergence of the 1/2-time fractional derivative in quantum mechanics, Foundations of Physics54, 55 (2024)

work page 2024
[24]

Coleman and J

S. Coleman and J. Mandula, All possible symmetries of thesmatrix, Phys. Rev.159, 1251 (1967)

work page 1967
[25]

Witten, Dynamical breaking of supersymmetry, Nucl

E. Witten, Dynamical breaking of supersymmetry, Nucl. Phys. B188, 513 (1981)

work page 1981
[26]

Cooper, A

F. Cooper, A. Khare, and U. Sukhatme, Supersymmetry and quantum mechanics, Phys. Rep.251, 267 (1995)

work page 1995
[27]

Cooper, A

F. Cooper, A. Khare, and U. P. Sukhatme,Supersymme- try in Quantum Mechanics(World Scientific, Singapore, 2001)

work page 2001
[28]

Garc´ ıa-Meca, A

C. Garc´ ıa-Meca, A. M. Ortiz, and R. L. S´ aez, Supersym- metry in the time domain and its applications in optics, Nat. Commun.11, 813 (2020)

work page 2020
[29]

V. E. Tarasov,Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media(Springer, London, 2012)

work page 2012
[30]

Laskin,Fractional Quantum Mechanics(World Scien- tific, 2018)

N. Laskin,Fractional Quantum Mechanics(World Scien- tific, 2018)

work page 2018
[31]

Weinberg,The Quantum Theory of Fields, Vol

S. Weinberg,The Quantum Theory of Fields, Vol. 1 (Cambridge University Press, Cambridge, 1995) founda- tions

work page 1995
[32]

Jackiw and C

R. Jackiw and C. Rebbi, Solitons with fermion number 1/2, Physical Review D13, 3398 (1976)

work page 1976
[33]

Bagarello, Susy for non-hermitian hamiltonians, with a view to coherent states, Mathematical Physics, Analysis and Geometry23, 10.1007/s11040-020-09353-3 (2020)

F. Bagarello, Susy for non-hermitian hamiltonians, with a view to coherent states, Mathematical Physics, Analysis and Geometry23, 10.1007/s11040-020-09353-3 (2020)

work page doi:10.1007/s11040-020-09353-3 2020
[34]

S. C. Lim, C. H. Eab, K. H. Mak, M. Li, and S. Y. Chen, Solving linear coupled fractional differential equations by direct operational method and some applications, Math. Probl. Eng.2012, 653939 (2012)

work page 2012
[35]

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys.81, 109 (2009)

work page 2009

[1] [1]

Rewritten com- pactly we have ˆP (±) L,R =σ z r 2m ˆH −W(q) , so that 4 ˆP (±) L = ˆP (±) R

Using ˆB̸= ˆA†, our construction is analogous—apart from the presence of fractional supercharges—to a non- Hermitian realization of SUSY [32]. Rewritten com- pactly we have ˆP (±) L,R =σ z r 2m ˆH −W(q) , so that 4 ˆP (±) L = ˆP (±) R . The zero-momentum mode states are ob- tained from ˆAϕ0 R(t;q) = 0 and ˆBϕ0 L(t;q) = 0, leading to p iℏ∂t ϕ0 L,R(t, q) =∓...

work page 2022

[2] [2]

Razavy, Quantum-mechanical time operator, Ameri- can Journal of Physics35, 955 (1967)

M. Razavy, Quantum-mechanical time operator, Ameri- can Journal of Physics35, 955 (1967)

work page 1967

[3] [3]

B¨ uttiker and R

M. B¨ uttiker and R. Landauer, Traversal time for tunnel- ing, Phys. Rev. Lett.49, 1739 (1982)

work page 1982

[4] [4]

Landauer and T

R. Landauer and T. Martin, Barrier interaction time in tunneling, Rev. Mod. Phys.66, 217 (1994). 5

work page 1994

[5] [5]

J. G. Muga, S. Brouard, and D. Mac´ ıas, Time of arrival in quantum mechanics, Ann. Phys.240, 351 (1995)

work page 1995

[6] [6]

N. Grot, C. Rovelli, and R. S. Tate, Time of arrival in quantum mechanics, Phys. Rev. A54, 4676 (1996)

work page 1996

[7] [7]

E. A. Galapon, R. F. Caballar, and R. T. Bahague, Confined quantum time of arrivals, Phys. Rev. Lett.93, 180406 (2004)

work page 2004

[8] [8]

J. M. Yearsley, D. A. Downs, J. J. Halliwell, and A. K. Hashagen, Quantum arrival and dwell times via idealized clocks, Phys. Rev. A84, 022109 (2011)

work page 2011

[9] [9]

Giovannetti, S

V. Giovannetti, S. Lloyd, and L. Maccone, Quantum time, Phys. Rev. D92, 045033 (2015)

work page 2015

[10] [10]

J. J. Halliwell, J. Ev´ aeus, J. London, and Y. Malik, A self- adjoint arrival time operator inspired by measurement models, Phys. Lett. A379, 2445 (2015)

work page 2015

[11] [11]

E. A. Galapon and J. J. P. Magadan, Quantizations of the classical time of arrival and their dynamics, Annals of Physics397, 278 (2018)

work page 2018

[12] [12]

P. C. Flores and E. A. Galapon, Instantaneous tun- neling of relativistic massive spin-0 particles (2022), arXiv:2207.09040 [quant-ph]

work page arXiv 2022

[13] [13]

D. N. Page and W. K. Wootters, Evolution without evo- lution: Dynamics described by stationary observables, Phys. Rev. D27, 2885 (1983)

work page 1983

[14] [14]

Rovelli, Quantum mechanics without time: A model, Phys

C. Rovelli, Quantum mechanics without time: A model, Phys. Rev. D42, 2638 (1990)

work page 1990

[15] [15]

J. S. Briggs and J. M. Rost, Time dependence in quantum mechanics, Eur. Phys. J. D10, 311 (2000)

work page 2000

[16] [16]

J. S. Briggs and J. M. Rost, On the derivation of the time- dependent equation of schr¨ odinger, Found. Phys.31, 693 (2001)

work page 2001

[17] [17]

Marolf and C

D. Marolf and C. Rovelli, Relativistic quantum measure- ment, Phys. Rev. D66, 023510 (2002)

work page 2002

[18] [18]

J. S. Briggs, A derivation of the time–energy uncertainty relation, J. Phys. Conf. Ser.99, 012002 (2008)

work page 2008

[19] [19]

Gemsheim and J

S. Gemsheim and J. M. Rost, Emergence of time from quantum interaction with the environment, Phys. Rev. Lett.131, 140202 (2023)

work page 2023

[20] [20]

E. O. Dias and F. Parisio, Space-time-symmetric exten- sion of nonrelativistic quantum mechanics, Phys. Rev. A 95, 032133 (2017)

work page 2017

[21] [21]

A. J. S. de Lara and M. W. Beims, Traveling time in a spacetime-symmetric extension of nonrelativistic quan- tum mechanics, Phys. Rev. A110, 012216 (2024)

work page 2024

[22] [22]

E. O. Dias, Space-time-symmetric nonrelativistic quan- tum mechanics in 3 + 1 dimensions: Time and position of arrival from complementary wave functions, Physical Review A112, 032218 (2025)

work page 2025

[23] [23]

M. W. Beims and A. J. S. de Lara, Position as an in- dependent variable and the emergence of the 1/2-time fractional derivative in quantum mechanics, Foundations of Physics54, 55 (2024)

work page 2024

[24] [24]

Coleman and J

S. Coleman and J. Mandula, All possible symmetries of thesmatrix, Phys. Rev.159, 1251 (1967)

work page 1967

[25] [25]

Witten, Dynamical breaking of supersymmetry, Nucl

E. Witten, Dynamical breaking of supersymmetry, Nucl. Phys. B188, 513 (1981)

work page 1981

[26] [26]

Cooper, A

F. Cooper, A. Khare, and U. Sukhatme, Supersymmetry and quantum mechanics, Phys. Rep.251, 267 (1995)

work page 1995

[27] [27]

Cooper, A

F. Cooper, A. Khare, and U. P. Sukhatme,Supersymme- try in Quantum Mechanics(World Scientific, Singapore, 2001)

work page 2001

[28] [28]

Garc´ ıa-Meca, A

C. Garc´ ıa-Meca, A. M. Ortiz, and R. L. S´ aez, Supersym- metry in the time domain and its applications in optics, Nat. Commun.11, 813 (2020)

work page 2020

[29] [29]

V. E. Tarasov,Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media(Springer, London, 2012)

work page 2012

[30] [30]

Laskin,Fractional Quantum Mechanics(World Scien- tific, 2018)

N. Laskin,Fractional Quantum Mechanics(World Scien- tific, 2018)

work page 2018

[31] [31]

Weinberg,The Quantum Theory of Fields, Vol

S. Weinberg,The Quantum Theory of Fields, Vol. 1 (Cambridge University Press, Cambridge, 1995) founda- tions

work page 1995

[32] [32]

Jackiw and C

R. Jackiw and C. Rebbi, Solitons with fermion number 1/2, Physical Review D13, 3398 (1976)

work page 1976

[33] [33]

Bagarello, Susy for non-hermitian hamiltonians, with a view to coherent states, Mathematical Physics, Analysis and Geometry23, 10.1007/s11040-020-09353-3 (2020)

F. Bagarello, Susy for non-hermitian hamiltonians, with a view to coherent states, Mathematical Physics, Analysis and Geometry23, 10.1007/s11040-020-09353-3 (2020)

work page doi:10.1007/s11040-020-09353-3 2020

[34] [34]

S. C. Lim, C. H. Eab, K. H. Mak, M. Li, and S. Y. Chen, Solving linear coupled fractional differential equations by direct operational method and some applications, Math. Probl. Eng.2012, 653939 (2012)

work page 2012

[35] [35]

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys.81, 109 (2009)

work page 2009